Geometric classification of 4d $\mathcal{N}=2$ SCFTs

The classification of 4d $\mathcal{N}=2$ SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected $\mathbb{Q}$-factorial log-Fano variety with Hodge numbers $h^{p,q}=\delta_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring $\mathscr{R}$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension $\Delta_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$ which are realized as Coulomb branch dimensions of some rank-$k$ CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible $\{\Delta_1,\cdots,\Delta_k\}$'s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number $\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by a certain sum of the Erd\"os-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large $k$ $$ \boldsymbol{N}(k)=\frac{2\,\zeta(2)\,\zeta(3)}{\zeta(6)}\,k^2+o(k^2). $$ In the special case $k=2$ our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension $k$-tuples $\{\Delta_1,\cdots,\Delta_k\}$ are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few $k$'s.


Introduction and Overview
Following the seminal papers by Seiberg and Witten [1,2], in the last years a rich landscape of four-dimensional N = 2 superconformal field theories (SCFT) had emerged, mostly without a weakly-coupled Lagrangian formulation [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. It is natural to ask for a map of this vast territory, that is, for a classification of unitary 4d N = 2 SCFTs. The work in this direction follows roughly two approaches: the first one aims to partial classifications of N = 2 SCFTs having some specific construction [5-7, 13, 17] or particular property [3,19]. The second approach, advocated in particular by the authors of refs. [20][21][22][23][24][25], relates the classification of N = 2 SCFTs to the geometric problem of classifying the conic special geometries (CSG) which describe their IR physics along the Coulomb branch Má la Seiberg and Witten [1,2]. The present paper belongs to this second line of thought: it is meant to be a contribution to the geometric classification of CSG with applications to N = 2 SCFT.
Comparison with other classification problems in complex geometry suggests that, while describing all CSG up to isomorphism may be doable when M has very small dimension, 1 it becomes rapidly intractable as we increase the rank k. A more plausible program would be a coarse-grained classification of the CSG not up to isomorphism but rather up to some kind of "birational" equivalence, that is, neglecting the details of the geometry along "exceptional" loci of positive codimension. This is the point of view we adopt in our analysis.
The classification problem is in a much better shape than one may expect. Indeed, while a priori the Coulomb branch M is just a complex analytic space, it follows from the local properties of special Kähler geometry that a CSG must be a complex cone over a base K which is a normal projective variety. K is (birational to) an algebraic variety of a very special kind: a simply-connected log-Fano [26] with Picard number one and trivial Hodge diamond (a special instance of a Mori dream space [27]).
In practice, in the coarse-grained classification we limit our ambition to the description of the allowed rings R of holomorphic functions on M (the Coulomb branch chiral rings). Several distinct (deformation-types of) SCFTs have the same R but differ in other respects as their flavor symmetry. The simplest example of a pair of distinct SCFT with the same R is given by SU(N) SQCD with 2N fundamentals and N = 2 * with the same gauge group; see refs. [22][23][24][25] for additional examples in rank 1. The chiral ring R of a SCFT is graded by the value of the U(1) R charge (equal to the dimension ∆ for a chiral operator). The general expectation 2 is that the Coulomb branch chiral ring is a graded free polynomial ring, R = C[u 1 , u 2 , · · · , u k ].
(1.1) This is equivalent 3 to saying that the log-Fano K is a weighted projective space (WPS). The last statement is only slightly stronger than the one in the previous paragraph: all WPS are simply-connected log-Fano with Picard number one and trivial Hodge diamond [29]. Conversely, a toric log-Fano with these properties is necessarily a (fake 4 ) weighted projective space [30]. Then it appears that the log-Fano varieties which carry all the structures implied by special geometry form to a class of manifolds only slightly more general than the WPS. This explains why "most" N = 2 models have free chiral rings. Assuming (1.1), the information encoded in the ring R is just the k-tuple {∆ 1 , ∆ 2 , · · · , ∆ k } of the U(1) R charges of its free generators u i . Even if the ring is non-free the spectrum of dimensions of R is a basic invariant of the SCFT. The coarse-grained classification of CSG then aims to list the allowed Coulomb branch dimension k-tuples {∆ 1 , · · · , ∆ k } for each rank k ∈ N. An even less ambitious program is to list the finitely-many real numbers ∆ which may be the dimension of a Coulomb branch generator in a N = 2 SCFT of rank at most k. In a unitary theory, all ∆'s are rational numbers ≥ 1. If the chiral ring is non-free, but with a finite free covering, we easily reduce to the above case.
Note added (non-free chiral rings). After the submission of the present paper, the article [31] appeared in the arXiv where examples of N = 2 theories with non-free chiral rings are constructed. Those examples are in line with our geometric discussion being related to the free ring case by a finite quotient (gauging). Most of the discussion of the present paper applies to these more general situation as well (many arguments are formulated modulo finite quotients). The only point where we assume that the ring R is free 5 , in order to simplify the analysis, is in showing that the Coulomb branch contains "many" normal rays. Our argument in the present form requires just one normal ray, so the request of "many" of them is rather an overkill. It is relatively straightforward to extend the details of our analysis to finite quotients.
In the rest of this Introduction we present a non-technical survey of our main results for both the list of allowed dimensions and dimension k-tuples. In particular, §. 1.3 contains a heuristic derivation of the Universal Dimension Formula.

Coulomb branch dimensions ∆
In section 4 we present a very simple recursive algorithm to produce the list of (putative) dimensions ∆ allowed in a rank-k CSG for all k ∈ N. The dimension lists for ranks up to 13 are presented in the tables of section 6. After the completion of this paper, ref. [32] appeared on the arXiv where the list of dimensions for k = 2 is also computed. Our results are in perfect agreement with theirs. 3 Since the dimensions ∆ are positive rationals. 4 All fake WPS are quotients of WPS by finite Abelian groups. 5 (24)).
Let us describe some general property of the set of dimensions in given rank k. The number of allowed ∆'s is not greater than a certain Number-Theoretical function N (k) of the rank k ∆ ≡ dimension of a Coulomb branch free generator in a CSG of rank ≤ k ≤ N (k). (1.2) There is evidence that ≤ may be actually replaced by an equality sign. N (k) is a rather peculiar function: it is stationary for "most" k ∈ N, N (k) = N (k − 1) and, while the ratio is "typically" a small integer, it takes all integral value ≥ 2 infinitely many times. As mentioned above, we expect this bound to be optimal, that is, the actual number of Coulomb dimensions to have the above behavior for large k.
The number N (k) is vastly smaller than the number of isoclasses of CSG of rank ≤ k, showing that the coarse-grained classification is dramatically simpler than the fine one. The counting (1.2) should be compared with the corresponding one for Lagrangian N = 2 SCFTs # ∆ ∈ Q ≥1 ∆ ≡ dimension of a Coulomb branch free generator in a Lagrangian model of rank ≤ k = 3 k 2 , k ≥ 15, (1.6) which confirms the idea that the Lagrangian dimensions have "density zero" in the set of all N = 2 Coulomb dimensions. The Lagrangian dimensions are necessarily integers; the number of allowed integral dimension at rank k is (not greater than) so, for large k, roughly 38.5% of all allowed integral dimensions may be realized by a Lagrangian SCFT. Remarkably, for k ≥ 15 the ratio ̺(k) = # Lagrangian dimensions in rank k # integral dimensions in rank k (1.8) is roughly independent of k up to a few percent modulation, see e.g. table 2.

Dimension k-tuples and Dirac quantization of charge
The classification of the dimension k-tuples {∆ 1 , · · · , ∆ k } allowed in a rank-k CSG contains much more information than the list of the individual dimensions ∆. Indeed, the values of the dimensions of the various operators in a given SCFT are strongly correlated. The list of dimension k-tuples may also be explicitly determined recursively in k using our Universal Formula. The problem may be addressed at two levels: there is a simple algorithm which produces, for a given k, a finite list of would-be dimension k-tuples. However there are subtle Number Theoretical aspects, and some of these k-tuples are consistent only under special circumstances. The tricky point is as follows: a special geometry is, in particular, an analytic family of polarized Abelian varieties. The polarization corresponds physically to the Dirac electro-magnetic pairing Ω, which is an integral, non-degenerate, skew-symmetric form on the charge lattice. Usually one assumes this polarization to be principal, that is, that all charges which are consistent with Dirac quantization are permitted. But physics allows Ω to be non-principal [33] at the cost of introducing additional selection rules on the values of the electro-magnetic charges and fluxes (see §. 2.1.1 for details). The deep arguments of ref. [34] suggest that Ω should be principal for a N = 2 QFT which emerges from a consistent quantum theory of gravity in some decoupling limit. It turns out that only a subset of the dimension k-tuples produced by the simple algorithm are consistent with a principal polarization; the others may be realized only in generalized special geometries endowed with suitable non-principal polarization i.e. to be consistent they require additional selection rules on the electro-magnetic charges. Therefore one expects that such Coulomb dimensions would not appear in N = 2 SCFT having a stringy construction. On the other hand, the Jacobian of a genus g curve carries a canonical principal polarization; thus the special geometry of a SCFT with such "non-principal" Coulomb dimensions cannot be described by a Seiberg-Witten curve.  Table 2: Values of the ratio ̺(k) (up to four digits) for various values of the rank k ∈ N, including k = 1 and the asymptotic value for k = ∞.
To determine the dimension k-tuples which are compatible with a principal polarization is a subtle problem in Number Theory. For instance, the putative dimension list in rank 2 contains the two pairs 8 {12, 6} and {12, 8} (resp. the two pairs {10/7, 8/7} and {12/7, 8/7}) but only the first one is consistent with a principal polarization. The pair {12, 6} corresponds to rank 2 Minahan-Nemeshansky (MN) of type E 8 [35,36] (resp. {10/7, 8/7} to Argyres-Douglas (AD) model of type A 4 ); since this model has a stringy construction, the Number Theoretic subtlety is consistent with the physical arguments of [34].
The reason why four of the putative rank-2 pairs {∆ 1 , ∆ 2 } are not consistent with a principal Ω looks rather exoteric at first sight: while the ideal class group of the number field Q[ζ] (ζ a primitive 12-th root of unity) is trivial, the narrow ideal class group of its totally real subfield Q[ √ 3] is Z 2 , and the narrow class group is an obstruction to the consistency of such dimension pairs in presence of a principal polarization (a hint of why this group enters in the game will be given momentarily in §. 1.3). To see which one of the two pairs {12, 6} or {12, 8} survives, we need to understand the action of the narrow ideal class group; it turns out that Class Field Theory properly selects the physically expected dimensions {12, 6}. We regard this fact as a non-trivial check of our methods. Remark 1.2.1. Let us give a rough physical motivation for the role of Class Field Theory in our problem. It follows from the subtle interplay between the dynamical breaking of the SCFT U(1) R symmetry 9 in the supersymmetric vacua and the Dirac quantization of charge. Along the Coulomb branch M, the U(1) R symmetry should be spontaneously broken. But there are special holomorphic subspaces M n ⊂ M which parametrize susy vacua where a discrete subgroup Z n ⊂ U(1) R remains unbroken. Assuming eqn.(1.1), the locus is such a subspace M n with n the order of 1/∆ i 0 in Q/Z. To the locus M n one associates the rational group-algebra Q[e 2πiR/∆ i 0 ] of the unbroken R-symmetry Z n . The chiral ring R is then a module of this group-algebra (of non-countable dimension). Replace R by the much simpler subring I ⊂ R of chiral operators of integral U(1) R charge; I = R M where M is the quantum monodromy of the SCFT [37,38]. Since Proj I ∼ = Proj R [39] there is no essential loss of information in the process. I is a C-algebra; Dirac quantization is the statement that I is obtained from a Q-algebra I Q by extension of scalars, I ∼ = I Q ⊗ Q C. An element of I Q is simply a holomorphic function which locally restricts to an element of Q[a i , b j ], where (a i , b j ) are the periods of special geometry well-defined modulo Sp(2k, Z). E.g. if our model is a Lagrangian SCFT with gauge group G of rank k, I Q = Q[a i , b j ] Weyl(G) . I Q is a module of Q[e 2πiR/∆ i 0 ] of just countable dimension. By Maschke theorem [40] I Q is a countable sum of Abelian number fields (1.10) Being Abelian, the fields F α are best studied by the methods of Class Field Theory. On the other hand, the Coulomb dimensions ∆(φ) are just the characters of U(1) R appearing in R χ φ : e 2πitR → e 2πit∆(φ) ∈ C × , for φ ∈ R of definite dimension. (1.11) Focusing on the subspace M n ⊂ M, the characters {χ φ } induce characters of the unbroken subgroup Z n . Hence the Coulomb dimensions ∆(φ) may be read from the decomposition of R into characters of Z n . If all Coulomb dimensions are integral, R = I Q ⊗ Q C, and the last decomposition is obtained from the one in (1.10) by tensoring with C, so that we may read ∆(φ) directly from the Number Theoretic properties of the F α . The same holds in the general case, mutatis mutandis. In the main body of the paper we shall deduce the list of allowed Coulomb dimensions by a detailed geometric analysis, but the final answer is already given by eqn.(1.10) when supplemented with the obvious relation between the rank k of the SCFT and the degrees of the number fields F α .

Rank 1 and natural guesses for k ≥ 2
The case of rank one is well known [20]. The allowed Coulomb dimensions are ∆ = 1, 2, 3, 3/2, 4, 4/3, 6, 6/5, (1.12) ∆ = 1 corresponds to the free (Lagrangian) theory, ∆ = 2 to interacting Lagrangian models (i.e. SU(2) gauge theories), and all other dimensions to strongly interacting SCFTs. A crucial observation is that the list of dimensions (1.12) is organized into orbits of an Abelian group H R . For k = 1 the group is simply H R ∼ = Z 2 generated by the involution ι ι : where, for x ∈ R, x denotes the real number equal x mod 1 with 0 < x ≤ 1.
(1. 13) The Lagrangian models correspond to the fixed points of H R , ∆ = 1, 2. (1.14) There are dozens of ways to prove that eqn. (1.12) is the correct set of dimensions for k = 1 N = 2 SCFT; each argument leads to its own interpretation 10 of this remarkable list of rational numbers and of the group H R . Each interpretation suggests a possible strategy to generalize the list (1.12) to higher k. We resist the temptation to focus on the most elegant viewpoints, and stick ourselves to the most obvious interpretation of the set (1.12): Fact. The allowed values of the Coulomb dimension ∆ for rank 1 N = 2 SCFTs, eqn. (1.12), are in one-to-one correspondence with the elliptic conjugacy classes in the rank-one dualityframe group, Sp(2, Z) ≡ SL(2, Z). Lagrangian models correspond to central elements (which coincide with their class). The group H R ∼ = GL(2, Z)/SL(2, Z) permutes the distinct SL(2, Z)conjugacy classes which are conjugate in the bigger group GL(2, Z).
By an elliptic conjugacy class we mean a conjugacy class whose elements have finite order. There are several ways to check that the above Fact is true. The standard method is comparison with the Kodaira classification of exceptional fibers in elliptic surfaces [41]. Through the homological invariant [41], Kodaira sets the (multiplicity 1) exceptional fibers in one-to-one correspondence with the quasi-unipotent conjugacy classes of SL(2, Z). In dimension 1 the homological invariant of a CSG must be semi-simple. Since quasi-unipotency and semi-simplicity together imply finite order, Fact follows. The trivial conjugacy class of 1 corresponds to the free SCFT, the class of the central element −1 to SU(2) gauge theories, and the regular elliptic classes to strongly-coupled models with no Lagrangian formulation. The map between (conjugacy classes of) elliptic elements of SL(2, Z) and Coulomb branch dimensions ∆ is through their modular factor (cτ + d) evaluated at their fixed point 11 τ in the upper half-plane h. Explicitly: where τ is a solution to aτ + b = τ (cτ + d), (1.15) and log z is the branch of the logarithm such that log(e 2πix ) = 2πi x for x ∈ R (cfr. eqn.(1.13) for the notation). The action of ι ∈ H R (eqn. (1.13)) is equivalent to 12 τ ↔τ , i.e. log(cτ + d)/2πi ↔ 1 − log(cτ + d)/2πi .
The basic goal of the coarse-grained classification of N = 2 SCFT is to provide the correct generalization of the above Fact to arbitrary rank k. The natural guess is to replace the rank-one duality group SL(2, Z) by its rank-k counterpart, i.e. the Siegel modular group 10 To mention just a few: the set of Z 2 -orbits in eqn. (1.12) is one-to-one correspondence with: (i) Coxeter labels of the unique node of valency > 2 in an affine Dynkin graph which is also a star; (ii) Coxeter numbers of semi-simple rank-2 Lie algebras; (iii) degrees of elliptic curves written as complete intersections in WPS, (iv) and so on. 11 The locus of fixed points in H of an elliptic element of the modular group SL(2, Z) is not empty and connected, see Lemma 4.2.1. Note that (cτ + b), being a root of unity, is independent of the chosen τ in the fixed locus.
12τ is in the lower half-plane; to write everything in the canonical form, one should conjugate it to a point in the upper half-plane by acting with the proper orientation-reversing element of GL(2, Z).
Sp(2k, Z), and consider its finite-order conjugacy classes. Now the fixed-point modular factor, Cτ + D, is a k × k unitary matrix with eigenvalues λ i , (i = 1, . . . , k and |λ i | = 1), to which we may tentatively associate the k-tuple {∆ i } k i=1 of would-be Coulomb dimensions giving a putative 1-to-k correspondence 13 between Sp(2k, R)-conjugacy classes of elliptic elements in the Siegel group Sp(2k, Z) and would-be dimension k-tuples. The candidate correspondence (1.16) reduces to the Kodaira one for k = 1, and is consistent with the physical intuition of Remark 1.2.1. It turns out that for k ≥ 2 the guess (1.16) is morally correct, but there are many new phenomena and subtleties with no counterpart in rank 1, so the statement of the correspondence should be taken with a grain of salt, and supplemented with the appropriate limitations and specifications, as we shall do in the main body of the present paper. In particular, the same k-tuple {∆ i } k i=1 is produced by a number ≤ k of distinct conjugacy classes; in facts, the geometrically consistent k-tuples are those which appear precisely k times (properly counted).
In rank k ≥ 2 the notion of "duality-frame group" is subtle. The Siegel modular group Sp(2k, Z) is the arithmetic group preserving the principal polarization. If Ω is not principal, Sp(2k, Z) should be replaced by the arithmetic group S(Ω) Z which preserves it S(Ω) Z = m ∈ GL(2k, Z) : m t Ωm = Ω . (1.17) Sp(2k, Z) and S(Ω) Z are commensurable arithmetic subgroups of Sp(2k, Q) [42]. If a SCFT has a non-principal polarization Ω, its Coulomb dimensions are related to the elliptic conjugacy classes in S(Ω) Z which (in general) lead to different eigenvalues λ i and Coulomb dimensions ∆ i . As in the k = 1 case, the dimension k-tuples {∆ i } k i=1 form orbits under a group. The most naive guess is that this is the "automorphism group" of eqn.(1.16), Z k ⋊ Z k 2 , where the first factor cyclically permutes the λ i while Z k 2 is the straightforward generalization of ι for k = 1 : However, in general, this action would not map classes in Sp(2k, Z) to classes in the same group but rather in some other arithmetic group S(Ω) Z . The proper generalization of the k = 1 case requires to replace 14 Z k 2 by the Abelian group H R which permutes the Sp(2k, R)- 13 Since we have a k-fold choice of which eigenvalue we wish to call λ 1 . 14 Z k 2 is the group which permutes the Sp(2k, R)-conjugacy classes of elliptic elements of the real group Sp(2k, R) which are conjugate in GL(2k, C). However some real conjugacy class may have no integral conjugacy classes of elliptic elements of the Siegel modular group Sp(2k, Z) which are conjugate in GL(2k, C). H R is a subgroup of the group (1.18), i.e. we have an exact sequence for some 2-group C. In the simple case when the splitting field K of the elliptic element has class number 1, C is just the narrow class group C nar k of its maximal totally real subfield k ⊂ K. For instance, the dimension pair {12, 6} (the k = 2 E 8 MN model mentioned before) is reproduced by eqn.(1.16) for log λ 1 = 2πi/12 and log λ 2 = 14πi/12; applying naively eqn.(1.18) we would get the dimension pair ι 2 {12, 6} = {12, 8}. However in this case ι 2 ∈ H R , and the H R -orbit of {12, 6} does not contain {12, 8} which then is not a valid dimension pair for k = 2 for the duality-frame group Sp(2k, Z) but it is admissible if the duality-frame group is S(Ω) Z with det Ω = 2 or larger. Remark 1.3.1. The generalization to higher k of the k = 1 criterion (1.14) for the Coulomb dimensions to be consistent with a weakly-coupled Lagrangian description is as follows. Let ι ∈ H R be given by If a dimension k-tuple {∆ 1 , · · · , ∆ k } may be realized by a Lagrangian SCFT then it is left invariant by ι up to a permutation of the ∆ i . The inverse implication is probably false.

Springer Theory of reflection groups
The proposed dimension formula (1.16) may look puzzling at first. Being purportedly universal, it should, in particular, reproduce the correct dimensions for a weakly-coupled Lagrangian SCFT with gauge group an arbitrary semi-simple Lie group G. By the nonrenormalization and Harrish-Chandra theorems [43], in the Lagrangian case the dimension k-tuple {∆ i } is just the set {d i } of the degrees of the Casimirs of G (its exponents +1). Thus eqn. (1.16), if correct, implies a strange universal formula for the degrees of a Lie algebra which looks rather counter-intuitive from the Lie theory viewpoint. The statement that (1.16) is the correct degree formula (not just for Weyl groups of Lie algebras, but for all finite reflection groups) is the main theorem in the Springer Theory of reflection groups [44][45][46], We shall see in §. 4.3.5 that the correspondence between our geometric analysis of the CSG and Springer Theory is more detailed than just giving the right dimensions. In particular, Springer Theory together with weak-coupling QFT force us to use in eqn.(1.16) the universal determination of the logarithm we call log (see after eqn.(4.10)), which is therefore implied by conventional Lagrangian QFT.
In other words, the proposed Universal Dimension Formula (1.16) may also be obtained using the following element, and only a subgroup survives over Z. This implies that H R is indeed a subgroup of Z k 2 .
Strategy. Write the usual dimension formula valid for all weakly-coupled Lagrangian SCFTs in a clever way, so that it continues to make sense even for non-Lagrangian SCFT, i.e. using only intrinsic physical data such as the breaking pattern of U(1) R . This leads you to eqn. (1.16). Then claim the formula to have general validity.
This is the third heuristic derivation of (1.16) after the ones in §.1.2 and 1.3. The sheaftheoretic arguments of §. 4.3.4 will make happy the pedantic reader (at least we hope). It will also supplement (1.16) all the required specifications and limitations. Remark 1.4.1. Inverting the argument, we may say that our analysis of the CSG yields a (simpler) transcendental proof of the classical Springer results.

Organization of the paper
The rest of the paper is organized as follows. Section 2 contains a review of special geometry, structures on Riemannian cones, and all the basic geometric tools we need. The only new materials in this section are the implications for special geometry of the sphere theorems of comparison geometry and the relation of CSG with the theory of log-Fano varieties. In section 3 we discuss the Coulomb chiral ring and the class of CSG with constant period map. Section 4 is the core of the paper, where we deduce the dimension formulae. Section 5 is quite technical: here we discuss fine points about the elliptic conjugacy classes in Siegel modular groups and their non-principal counterparts S(Ω) Z . Section 6 contains a few sample dimension tables. Some technical material is confined in the appendix. Remark 1.5.1. The various sections of the paper are written with quite different standards of mathematical rigour. The core of the paper - §. 4.3.4 where the dimension formula is deduced -is (as far as we can see) totally rigorous once we take for granted that the chiral ring R is a (graded) free polynomial ring. The dimension formula then follow as a simple application of the Oka principle.

Special cones and log -Fano varieties
In this section we review special geometry and related topics to set up the scene. The first three subsections contain fairly standard material; our suggestion to the experts is to skip them (except for the disclaimer in §.2.1.1). Later subsections describe basic properties of conic special geometries (CSG) which were not previously discussed in the literature: we aim to extablish that a CSG is an affine (complex) cone over a special kind of normal projective manifold: a simply-connected log-Fano with minimal Hodge numbers.

Special geometric structures
In this paper by a "special geometry" we mean a holomorphic integrable system with a Seiberg-Witten (SW) meromorphic differential [33,47,48]: Definition 1. By a special geometry we mean the following data: D1: A holomorphic map π : X → M between two normal complex analytic manifolds, X and M, of complex dimension 2k and k, respectively, whose generic fiber is (analytically isomorphic to) a principally polarized Abelian variety. π is required to have a zerosection. The closed analytic set D ⊂ M at which the fiber degenerates is called the discriminant. The dense open set M ♯ ≡ M \ D is called the regular locus; D2: A meromorphic 1-form (1-current) λ on X (the Seiberg-Witten (SW) differential) such that dλ is a holomorphic symplectic form on X, with respect to which the fibers of π are Lagrangian.

Three crucial caveats on the definition
The one given above is the definition which is natural from a geometric perspective. However in the physical applications one also considers slightly more general situations which may easily be reduced to the previous one. This aspect should be kept in mind when making comparison of our findings with existing results in the physics literature. We stress three aspects: Multivalued symplectic forms. In Definition 1 X is globally a holomorphic integrable system with a well-defined holomorphic symplectic form dλ. Since the overall phase of the SW differential λ is not observable, in the physical applications sometimes one also admits geometries in which λ is well-defined only up to (a locally constant) phase, see [20] for discussion and examples. Let C be the Coulomb branch of such a generalized special geometry; there is an unbranched cover of the regular locus, M ♯ → C ♯ , on which λ is univalued. M ♯ is the regular locus of a special geometry in the sense of Definition 1. The cover branches only over the discriminant D. Dually, we have an embedding of chiral rings R C ֒→ R M . Away from the discriminant, there is little difference between the two descriptions: working in C we identify vacua having the same physics, while in M we declare them to be distinct states (with the same physical properties). In the first picture we consider nonobservable the chiral operators which distinguish the physically equivalent Coulomb vacua, that is, where G is the (finite) deck group of the covering. R C is still free iff G is a reflection group [49,50] acting homogeneously; in this case, the dimensions of its generators are multiples of the ones for R M .
The two special geometries C and M may lead to different ways of resolving the singularities along D, and hence they may correspond to physical inequivalent theories in the "same" coarse class. It may happen that we may attach physical sense only to the chiral sub-ring R C . To compare our results with those of papers which allow multivalued λ, one should first pull-back their geometries to a cover on which the holomorphic symplectic form is univalued.
Non-principal polarizations. In Definition 1 the generic fiber of X → M is taken to be a principally polarized Abelian variety. As already stressed in the Introduction, we may consider non principal-polarization. This means that not all electric/magnetic charges and fluxes consistent with Dirac quantization are present in the system [33]. This is believed not to be possible in theories arising as limits of consistent quantum theories containing gravity [34]. Every non-principally polarized Abelian variety has an isogenous principally polarized one [51,52].
We see the polarization of the regular fiber X u as a primitive, 15 integral, non-degenerate, pairing [52,53] −, − : which has the physical interpretation of the Dirac electro-magnetic. We may find generators γ i of the electro-magnetic charge lattice H 1 (X u , Z) so that the matrix Ω ij ≡ γ i , γ j takes the (unique) canonical form [54] The polarization is principal iff e i = 1, i.e. det Ω = 1. Physically, the integers e i are charge multipliers: (in a suitable duality frame) the allowed values of the i-th electric charge are integral multiples of e i .
If Ω is principal, the duality-frame group is the Siegel modular group Sp(2k, Z), while in general it is the commensurable arithmetic group S(Ω) Z , eqn. (1.17). Since, as mentioned in the Introduction, the Coulomb dimensions {∆ i } are related to the possible elliptic subgroups of the duality-frame group, there is a correlation between the set of charge multipliers {e i } and the set of dimensions {∆ i }. The simplest instance of this state of affairs appears in rank 2: the set of dimensions {12, 8} is not allowed for Ω principal, but it is permitted when the charge multiplier e 2 is (e.g.) 2 or 3.
Non-normal Coulomb branches and "non-free" chiral rings. In Definition 1 the Coulomb branch M is taken to be normal as an analytic space, that is, we see the Coulomb branch as a ringed space (M, O M ) where M is a Hausdorf topological space and O M is the structure sheaf whose local sections are the local holomorphic functions. Being normal means that the stalks O M, x at all points x ∈ M are domains which are integrally closed in the stalk M x of the sheaf of germs of meromorphic functions [55,56]. Geometrically this is the convenient and natural definition; indeed, there is no essential loss of generality since we may always replace a non-normal analytic space M 0 by its normalization M: just replace the structure sheaf O M 0 with its integral closure O M and the topological space M 0 by the analytic spectrum M of O M [56]. Roughly speaking, passing to the normalization just enlarges the ring of the holomorphic functions from global sections of O M 0 to global 15 That is, the matrix of the form Ω ij ∈ Z(2k) satisfies gcd i,j {Ω ij } = 1.
sections of O M . In facts, the normalization corresponds to the maximal extension of the ring of local holomorphic functions compatible with O M 0 -coherence. Thus, geometrically, a non-normal Coulomb branch just amounts to "forget" some (local) holomorphic function. The simplest example of a non-normal analytic space is the plane cuspidal cubic whose ring of regular functions is C[u 1 , u 2 ]/(u 2 1 − u 3 2 ). Its normalization is the affine line with ring C[t], corresponding to the parametrization u 1 = t 3 , u 2 = t 2 . In this example the normalization ring Γ(M, O M ) has the (topological) basis 1, t, t 2 , t 3 , · · · while the basis of the non-normal version, Γ(M 0 , O M 0 ), is 1, t 2 , t 3 , · · · where one "forgets" the function t.
From the physical side the situation is subtler. We define the (geometric) chiral ring R to be the Frechét ring of the global holomorphic functions R ≡ Γ(M, O M ). This geometric ring may or may not coincide with the physical chiral ring R ph , defined as the ring of holomorphic functions on M ♯ ⊂ M which may be realized as vacuum expectation values of a chiral operator. Clearly R ph ⊂ R, and we get the physical ring by "forgetting" some holomorphic function. Then R ph = Γ(M ph , O ph ) where the stalks of O ph are domains 16 which may or may not be integrally closed. In the second case the physics endows the Coulomb branch with the structure of a non-normal analytic space (M ph , O ph ). Geometrically it is natural to replace it with its normalization (M, O M ) while proclaiming that only a subring R ph of the chiral ring R is a ring of physical operators. Notice that the full geometric ring R may be a free polynomial ring, C[u 1 , · · · , u k ], while the physical ring R ph is a non-free finitely-generated ring, as the example of the cuspidal cubic shows.
The putative "non-free" Coulomb branch geometries of ref. [28] arise this way: they are non-normal analytic spaces whose normalization has a free polynomial ring of regular functions, C[u 1 , · · · , u k ]. That is, the "non-free" chiral rings are obtained from free geometric rings by forgetting some holomorphic functions of R. The physical rationale for "forgetting" functions is the unitarity bound. In a CSG R is graded by the conformal dimension ∆, and unitarity requires that a non-constant physical holomorphic function has ∆ ≥ 1. Hence one is naturally led to the proposal If 0 < ∆(φ) < 1 for some φ ∈ R, R ph defines a non-normal structure sheaf O ph and the physical ring is non-free. Is this fancy possibility actually realized?
The equations determining the dimensions ∆ i for the normalization M of a CSG (satisfying our regularity conditions), deduced in §. 4.3.4 below, always have a (unique) solution such that ∆(φ) ≥ 1 for all φ ∈ R, φ = 1, with equality precisely when φ is a free field. Indeed, with the log determination of the logarithm, the formula (1.16) expresses ∆ i as 1 plus a manifestly non-negative quantity. To produce R ph = R, we may try to replace log by some bizzarre branch of the logarithm, with the effect that ∆(φ) → ∆(φ) new = 2 − ∆(φ), so that an element with 1 < ∆(φ) < 2 would be reinterpreted as having the dimension 0 < ∆(φ) new < 1. However this is extremely unnatural and gruesome since it will spoil the universality of the prescription to compute the dimension ∆ that better be the same one for all SCFT and all chiral operators (the correct prescription should be the unique one which reproduces the correct results for Lagrangian QFT, see §. 1.4. Assuming universality, the fancy possibilities of ref. [28] cannot be realized, and we shall neglect them for the rest of this paper. If the reader is aware of physical motivations for their existence and wants to study them, he needs only to perform the non-universal analytic continuation of the relevant formulae.
For most of the paper we focus on special geometries in the sense of Definition 1, with R ph = R and principally polarized fibers. Occasionally we comment on the modifications required for non-principal Ω.

Review of implied structures
The data D1, D2 imply the existence of several canonical geometric structures. We recall just the very basic ones (many others may be obtained by the construction in S5): S1: (polarized local system) A local constant sheaf Γ on M ♯ with stalk ∼ = Z 2k equipped with a skew-symmetric form −, − : Γ × Γ → Z under which Γ ≃ Γ ∨ . Γ is given by the holomogy of the fiber Γ u = H 1 (π −1 (u), Z) with the intersection form given by the principal polarization; S4: (Hodge bundle) V → M ♯ : it is the holomorphic sub-bundle of E whose fibers are (1,0) cohomology classes, i.e. V u = H 0 (π −1 (u), Ω 1 ). The flat connection ∇ GM of E induces the sub-bundle (holomorphic) connection ∇ H on V [57,58]. Note that Γ ∨ acts by translation on V and that V/Γ ∨ ∼ = X ♯ ≡ π −1 (M ♯ ); S5: (period map and the family of homogeneous bundles) the period matrix τ ij of the Abelian variety π −1 (x) is a complex symmetric matrix with positive imaginary part well defined up to Sp(2k, Z) equivalence; hence τ defines the holomorphic map: The period map τ yields a universal construction of many other canonical geometrical objects on M ♯ . We limit ourselves to a special class of holomorphic ones. The Griffiths is an open domain in its complex Griffiths compact dual [57][58][59] Sp(2k, R)/U(k) ⊂ Sp(2k, C)/P (k), where P (k) ⊂ Sp(2k, C) is the Siegel parabolic subgroup. (2.5) By general theory, to every P (k)-module (in particular to all U(k)-modules) we associate a holomorphic vector bundle over the compact dual equipped with a unique metric, complex structure, and connection having an explicit Lie theoretic construction [58]. These bundles, metrics, and connections may be restricted to the period domain and then pulled back to M ♯ via τ to get God-given bundles, metrics, and connections on M ♯ . All the quantities of "special geometry" (including the Kähler metric) arise in this way from Lie theoretic gadgets. For instance, the Hodge bundle V (resp. the flat bundle E) is the pull back of a homogenous bundle, and the connections ∇ H and ∇ GM are the pull-back of the corresponding canonical connections on the symmetric space (2.5); S6: (periods and local special coordinates) Let U ⊂ M ♯ be simply connected. We trivialize Γ in U choosing local sections making a canonical symplectic basis The local special (holomorphic) coordinates a i and their duals b i are (in U) the holomorphic symplectic form becomes Since the holomorphic coordinates along the fiber are z i = x i + τ ij y j we get Since τ ij is symmetric, locally there exists a prepotential (holomorphic) function F (a j ) such that (2.11) S7: (the dual bundle V ∨ ≃ E/V) This is yet another bundle whose metric and connection is given by the general construction in S5. It coincides with T M ♯ , so it yields the geometry of the base. On the intersection of two special coordinate charts U, U ′ we have: so that the periods (a i , b i ) are flat sections of E ∨ ≃ E. The holomorphic tangent bundle T M ♯ is then identified with the quotient bundle E/V (again the pull-back of a homogeneous bundle). In particular, the flat connection ∇ GM induces canonically a quotient bundle connection ∇ Q on E/V, that is, on T M ♯ . Taking the differential and using db i = τ ij da i we get the modular transformation of the k × k period matrix τ S8: (the hyperKähker structures on X ♯ and V) On the total space of V, equivalently of the flat real bundle Γ ⊗ R, there is a hyperKähler structure (I a , g) invariant under translation by local sections of Γ ⊗ R; then (I a , g) descends to a hyperKähler structure on the total space of H ♯ [60]. The complex structure of V is the ζ = 0 one in hyperKähler P 1 -family of complex structures. We give the hyperKähler structure by presenting the explicit P 1 -family of local holomorphic Darboux coordinates X a (ζ) = (q i (ζ), p i (ζ)) satisfying the reality condition [61] X a (ζ) = −X a (−1/ζ ), (2.14) such that the holomorphic symplectic form in complex structure ζ ∈ P 1 is and ω α (α = 1, 2, 3) are the three Kähler forms. We have hence S9: (fiber metric and Chern connection on V) Restricting the hyperKähler metric along the fibers, we get a Hermitian metric and associated Chern connection on the holomorphic bundle V. By uniqueness of the homogeneous connection, is coincides with the subbundle connection ∇ H . The Hermitian metric is simply z 2 = y ij z izj where y ij is the inverse matrix of y ij = 2 Im τ ij ; S10: (Special Kähler metric on M ♯ and its global Kähler potential) In the same way, restricting the hyperKähler metric on V to the zero section (which is a holomorphic subspace in ζ = 0 complex structure) we get a Kähler metric on M ♯ whose Kähler form is the restriction of ω 3 . The restriction to the zero-section yields a Kähler metric on M ♯ with Kähler form We note that the assumption of the existence of a SW differential implies the existence of a globally defined Kähler potential on M ♯ : Again, by uniqueness of the homogeneous connection, the Levi-Civita connection ∇ LC of this Kähler metric is the quotient-bundle connection ∇ Q . In other words, all the relevant connections are just projections of the flat one. S11: (The cubic symmetric form of the infinitesimal Hodge deformation) This is a symmetric holomorphic cubic form of type (3,0) describing the infinitesimal deformation of Hodge structure (of the Abelian fiber) in the sense of Griffiths [57]. Locally in special coordinates it is given just by (2.21) Remark 2.1.1. For k = 1 a special structure is, in particular, a surface fibered over a curve whose general section is an elliptic curve. Hence the possible local behaviors (i.e. degenerations of fibers) are described by the classical Kodaira papers [41]. In his terminology, S3 is called the homological invariant and S5 the analytic invariant.
Example 1 (k = 1 locally flat special structures 17 ). In this paper we are interested in conic special structures. Since in real dimension 2 all metric cones are locally flat, for k = 1 we are reduced to study flat special geometries whose discriminant is a single point. The hyperKähler manifold X ♯ then is locally isometric to R 4 ∼ = H, and the singular hyperKähler geometry should be of the form C 2 /G with G a finite subgroup of SU(2). One checks that conformal invariance requires the group G to correspond via the McKay correspondence to an affine Dynkin graph which is a star, that is, D 4 , E 6 , E 7 , or E 8 . Before resolving the singularity, the spaces X sing are the well-known Du Val singular hypersurfaces 18 in C 3 [62] D 4 : y 2 − h 3 (x, u) = 0, E 6 : y 2 − 4 x 3 + u 4 = 0, That u : X sing → C is an elliptic fibration (with section) is obvious by reinterpreting Du Val singularities as the Weierstrass model of a family of elliptic curve parametrized by u. The crepant resolution X of X sing is given by the corresponding ALE space. For each of the four special geometries we have a priori two distinct special structures. Indeed, we have two dual choices for the SW differential λ: I) a holomorphic section of V with no zero in M ♯ ≡ {u = 0} which vanish in the limit u → 0 to order at most 1, or II) a holomorphic section of V ∨ with the same properties. Note that these properties fix λ uniquely up to an irrelevant overall constant. In terms of the Weierstrass model the two dual choices read: The corresponding Coulomb branch dimensions are which is the correct list of (non-free) ∆'s for k = 1. The periods can be easily computed using Weierstrass elliptic functions 19 . As an example, we write them for E 8 : from which it is obvious that the dimension of u is 6 and respectively 6/5. In the dual choice the role of a and b get interchanged, since the non-trivial element of H R inverts the sign of the polarization. Of course, the resolutions of the singularity at u = 0 are different in the two cases, the exceptional locus being Kodaira exceptional fiber of type II * and II, respectively. The periods of dx/y and x dx/y scale with opposite power of u by the Legendre relation.

Rigidity principle and reducibility
A basic trick of the trade is that global properties in special geometry fix everything. This principle is known as "the Power of Holomorphy" [64]; mathematicians call it rigidity.
Proposition 2.1.1 (Rigidity principle [58]). Two special geometries with the same compact base M, isomorphic monodromy representations, and isomorphic fibers over one point, are equivalent.
Thus the monodromy representation S3 essentially determines the special structure. In particular, if the monodromy representation splits m = m 1 ⊕ m 2 (over 20 Z) then the special geometry is a product.

Curvature properties of special geometry
Let W be a holomorphic Hermitian vector bundle with Chern connection ∇. We consider a holomorphic sub-bundle S ⊂ W and the quotient bundle Q = W/S equipped with the sub-bundle and quotient connections ∇ S and ∇ Q , respectively. The curvature of ∇ S (resp. ∇ Q ) is bounded above (resp. below) by the one of ∇, see [51] page 79 or [58]. Applying this principle to E, V and E/V we get: The curvature of the bundle V is non-positive, while the curvature of the Kähler metric on M ♯ is non-negative (in facts, positive). In particular, the Ricci curvature of M ♯ is non-negative, R i ≥ 0 and it vanishes iff M ♯ is locally flat.
Let us give an alternative proof of the last statement.
Proof. In a Kähler manifold the Ricci form is 21 ρ = −i∂∂ log det g. Thus from (2.18) where Ω is the (positive) Kähler form on the locally Hermitian space in eqn. (2.4). Note that R i = 0 only at critical points of the period map τ . By Sard theorem, the set of periods τ ij (a) at which R i = 0 has zero measure. In particular R i ≡ 0 means τ = (a constant map), so M ♯ is locally flat.
Remark 2.1.2. This result may also be understood as follows. The total space of the holomorphic integrable system, X, is hyperKähler, so carries a Ricci-flat metric. M is a complex subspace, and the Ricci curvature of its induced metric is minus the curvature of the determinant of the normal bundle whose Hermitian metric is (det Im τ ) −1 .
Sectional and isotropic curvatures. From the above Proposition it is pretty obvious that all sectional curvatures of a special Kähler metric are non-negative. A stronger property is that all its isotropic curvatures are non-negative. Indeed, we claim an even stronger statement, that is, that the curvature operators are non-negative at all points p. 20 If the splitting is over Q, the geometry is a product up to an isogeny in the fiber. 21 Cfr. [65] eqn.(2.98).
Definition 2. Let X be a Riemannian n-fold with tangent space T p X at p ∈ X. The curvature operator at p is the self-adjoint linear operator R : ∧ 2 T p X → ∧ 2 T p X, (2.26) given by the Riemann tensor. We say that X has positive (resp. weakly positive) curvature operators iff the eigenvalues of R are positive (resp. non-negative) at all p ∈ X.
The claim follows from the explicit form of the Riemann tensor where T is the cubic symmetric form of the infinitesimal Hodge deformation (structure S11).
Sphere theorems. The positivity of the curvature operators has dramatic implications for the topology of X. We collect here some results which we shall use later in the paper: Theorem 1 (Meyers [66]). Let X be a complete Riemannian manifold of metric g whose Ricci curvature satisfies R ≥ λ 2 g with λ > 0 a constant. Then X is compact with diameter d(X) ≤ π/λ. Applying the result to the Riemannian universal cover X of X, we conclude that π 1 (X) is finite.
There is a version of Meyers theorem which applies to orbifolds, see Corollary 21 in [67] or Corollary 2.3.4 in [68]. In case of Riemannian orbifolds complete should be understood as complete as a metric space. The version in [68] states that a metrically complete Riemannian orbifold X, whose Ricci curvature satisfies R ≥ λ 2 g, is compact with a diameter d(X) ≤ π/λ. Theorem 2 (Synge [69]). An even dimensional compact orientable manifold with positive sectional curvature is simply-connected.  [71]). Let X be a compact n-dimensional Riemannian orbifold. If X has positive curvature operators it is diffeomorphic to a space form S n /G, S n being the sphere and G a finite subgroup of SO(n + 1).
The special Kähler manifolds M have just weakly positive curvature operators (and are typically non-compact). However, taking Theorems 1, 3 together, one gets the rough feeling that the non-flat special Kähler manifolds are "close" to being locally spheres. The statement will become precise under the assumption that M is also a cone.

Behavior along the discriminant
We need to understand the behavior near the discriminant locus D ⊂ M where the fiber degenerates, that is, some periods (a i , b j ) vanish. Physically this means that along the discriminant locus some additional light degrees of freedom appear, so that the IR description in terms of the massless fields parametrizing M becomes incomplete and breaks down.
The singular behavior is best understood in terms of properties of the period map τ . We see the discriminant D as an effective divisor D = i n i S i , where S i are the irreducible components and M ♯ = M \ Supp D. The behavior of the period map as we approach a generic point s of an irreducible component S i is described by three fundamental results: the strong monodromy Theorem [57][58][59], the SL 2 -orbit Theorem [72], and the invariant cycle Theorem [72]. In a neighborhood U of s ∈ S i , we may find complex coordinates z 1 , · · · , z k so that, locally in U, S i is given by z 1 = 0. Then we have U ∩ M ♯ ∼ = ∆ * × ∆ k−1 where ∆ (resp. ∆ * ) stands for the unit disk (resp. the punctured unit disk). We write p for the period map τ restricted to ∆ * × (z 2 , · · · , z k ) ⊂ ∆ * × ∆ k−1 , and h for the upper half-plane seen as the universal cover of ∆ * via the map τ → q(τ ) ≡ e 2πiτ . We have the commutative diagram wherep is the lift of the (restricted) period map. Let γ be the generator of π 1 (∆ * ×∆ k−1 ) ∼ = Z and m ≡ m(γ) the corresponding monodromy element (cfr. S3). Theñ p(τ + 1) = m ·p(τ ). (2.29) Let d(·, ·) be the distance function defined by the standard invariant metric on the symmetric space Sp(2k, R)/U(k) and d P (·, ·) the distance with respect to the usual Poincaré metric in h; the inequalities on the curvatures together with the Schwarz lemma imply Proposition 2.1.3 (Strong monodromy theorem [57][58][59]). The lifted period mapp is distancedecreasing d(p(x),p(y)) ≤ d P (x, y).
All eigenvalues of m are r-th roots of unit. Since m ∈ Sp(2k, Z), its minimal polynomial M(z) is a product of cyclotomic polynomials Φ d (z) The monodromy m is semi-simple iff s = 0, that is, if s d ∈ {0, 1} for all d. We say that m is regular iff all its eigenvalues are distinct, i.e. iff s d ∈ {0, 1} and d|r s d = 2k.
The case of m semi-simple. Semi-simplicity of m has the following consequence: In other words, along an irreducible component S i of D whose monodromy element m is semisimple the period matrix τ ij is defined and regular even if the Abelian fiber itself degenerates. The Kähler metric d 2 s = 2 Im τ ij da i ⊗ dā j is singular along S i since the periods a i are not valid local coordinates at this locus. The singularity is of the mildest possible kind: just a cyclic orbifold singularity. We illustrate the situation along a semi-simple component S i of the discriminant D in a typical example. The coordinate r is globally defined, since r 2 is the momentum map of the U(1) action given by R-symmetry. On the contrary, the period of the canonically conjugate angle, θ needs not to be 2π (which corresponds to the free SCFT). The period of the angle θ is related to the Coulomb dimension ∆ by the identification θ ∼ θ + 2π/∆. Hence, if the theory is not free, ∆ = 1, at the tip of the cone we have a cyclic orbifold singularity. We note that the unitary bound ∆ ≥ 1 (with equality iff the SCFT is free) becomes (period of θ) ≤ 2π. Thus unitarity requires the curvature at the tip to be non-negative and we may smooth out the geometry by cutting away the region r ≤ ǫ and gluing back a positively curved disk. This is consistent with our discussion of the curvature in special geometry in §.2.1.4. This example shows that the curvature inequalities apply also to the δ-function curvature concentrated at orbifold points and their relation to physical unitarity.
We state this as a Physical principle. The unitarity bounds guarantee that the δ-function curvatures associated to the angular deficits at orbifold points are consistent with the positivity of curvatures required by special Kähler geometry.
We quote another useful result: 58]). Suppose that the period map τ factors trough a quasi-projective variety K M and that the discriminant of p is a snc 23 divisor with semi-simple monodromies. If p is not the constant map, p is proper. Its image is a closed analytic subvariety containing p(K ♯ ) as the complement of an analytic set.
m non semi-simple. We now turn to the case in which the monodromy m is not semisimple. Again we consider the neighborhood U ∼ = ∆ * × ∆ k−1 considered around eqn.(4.51) and pull-back all structures to its universal cover U uni ∼ = h×∆ k−1 . By the strong monodromy theorem there exist minimal integers r ≥ 1, s ≥ 0 such that (m r − 1) s+1 = 0. In the non semi-simple case s ≥ 1. Then, N s = 0 and N s+1 = 0. The nilpotent operator N defines the weight filtration of a mixed Hodge structure in the sense of Deligne [73] to which we shall return momentarily; more elementarily, by the Jacobson-Morozov theorem [42,74,75] the rational matrix N defines a polynomial homomorphism φ : SL(2, Q) → Sp(2k, Q) such that N is the image of the raising operator of the sl(2, Q) Lie algebra. φ induces a period mapp : h → Sp(2k, R)/U(k) which is the simplest solution to the functional equation (2.35): The SL 2 -orbit theorem [72] states that the actual period map p differs from the Lie-theoretic mapp(τ ) by exponentially small terms O(q 1/r ) as τ → i∞ (q ≡ e 2πiτ ). A physicists studying the corresponding (2,2) supersymmetric σ-model states the theorem saying thatp(τ ) is the perturbative solution, valid asymptotically as the coupling 4π/Im τ → 0, and this perturbative solution receives corrections only by instantons which are suppressed by the 23 snc = simple normal crossing.
exponentially small (fractional) instanton counting parameter q 1/r . To physicists working in 4d N = 2 QFT , the SL 2 -orbit theorem is familiar as a fundamental result by Seiberg [76].
Let 0 = x ∈ U r ; φ decomposes Γ x ⊗ Q into irreducible representations of SL 2 (Q); the highest weight Q-cycles ψ are defined by the condition Nψ = 0; all other Q-cycles are obtained from these ones by acting on them with the SL 2 lowering operator. Since τ is the period map of a degenerating weight 1 Hodge structure, it follows from the Deligne weight filtration (or by the Clemens-Schmid sequence, see Corollary 2 in [77]), that only spin 0 and spin 1/2 representations are presents, that is, N 2 = 0. More precisely, we have a weight filtration of Q-spaces 39) such that N : W 2 /W 1 → W 0 is an isomorphism, and the polarization −, − induces a perfect pairing between W 2 /W 1 and W 0 as well as of W 1 /W 0 with itself [72] (of course, these statements are just the usual selection rules for angular momentum). We pull-back the local family of Abelian varieties X| U to the r-fold cover U r ; we get the family π : σ * X| U → U r . By construction, the monodromy of the pulled back family is m σ ≡ m r = e N , so the monodromy invariant Q-cycles are precisely the ones in W 1 . The invariant cycle theorem guarantees that all 1-cycle γ x ∈ Γ x invariant under the monodromy there is a homologous 1-cycleγ in σ * X| U in the total space of the (local) family and all 1-cycles in the total space are of this form. Then W 2 /W 1 consists of vanishing cycles, so that the corresponding periods vanish as q 1/r → 0 i.e. a van ∝ q 1/r and then eqn.(2.37) says that the periods along the "spin-0" cycles W 1 /W 0 are regular as q → 0 while the ones in W 0 (which are dual to the vanishing ones under the Dirac pairing) go as The conclusion we got is totally trivial from the physical side. The special geometry along the Coulomb branch is the IR description obtained integrating out the massive degrees of freedom; the singularities arise because at certain loci in M some additional degree of freedom becomes massless. One gets the leading singularity by computing the correction to the low energy coupling by loops of light fields, see the discussion in §. 5.4 of the original paper by Seiberg and Witten [2]. The mixed Hodge variation formula (2.40) is just their eqn.(5.10). Thus, a part for the need to go to the local r-fold cover U r , at a generic point of an irreducible component S i of the discriminant D we do not get singularities worse than physically expected. The singularity in eqn.(2.40) is mild: R1 the squared-norm of the SW differential while not smooth along the discriminant, extends continuously to D; R2 Its differential dΦ, while singular in U, becomes continuous (non-smooth) when pulled back to the local r-fold cover U r ; R3 the points on the discriminant, q = 0, are at a finite distance from smooth points. Indeed, on the local cover U r the metric is modeled on ds 2 = (− log |q|)|dq 1/r | 2 which is length decreasing with respect the flat metric |dz| 2 , z = (− log |q|) 1/2 q 1r . On U the metric is asymptotically conical. In particular, M remain complete as a metric space; R4 the integral of the Ricci curvature on the r-fold covering disk |q 1/r | < ǫ vanishes as ǫ → 0, i.e. there is no δ-like curvature concentrated on the discriminant, except for the obvious Z r -orbifold singularity implied by the covering quotient Thus all arguments based on curvature bounds work as in the semi-simple case. We have already remarked that orbifold singularities do not spoil the curvature bounds (cfr. Physical principle).
All the above statements hold at all points of the discriminant (and not just at generic points along a smooth component) when D is a snc divisor [72]. While this is generically the situation, the special geometry describing a particular SCFT with the mass deformations switched off may well be non generic. If the SCFT admits "enough" mass/relevant deformations, we can make D to be snc by an arbitrarily small perturbation which cannot change the qualitative aspects of the physics. Even in SCFTs without (enough) deformations, it is very likely that -while the singularities may be more severe than the snc ones -the four regularity conditions R1-R4 still hold. Indeed, R3 has been advocated by Gukov, Vafa and Witten as a necessary condition for a sound SCFT [6]. In the rest of the paper we shall make the Mild assumption. Our special geometry satisfies R1-R4. Remark 2.1.5. We may look at the singularities also from the point of view of the hy-perKähler geometry of the total space X. Since hyperKähler manifolds are in particular Calabi-Yau, the discussions of refs. [6,78] directly apply with similar conclusions. Note that the statements hold also for hyperKähler orbifolds.

Some facts about complex orbifolds
In the last subsection we found that the analytic space M typically has cyclic orbifold singularities. Here we collect some well known facts about complex orbifolds that we shall need below.
Proposition 2.2.1 (See e.g. [79]). The locally ringed space (Z, O Z ) associated to a complex orbifold has the following properties: ii) the singular locus Σ(Z) is a closed reduced complex subspace of Z and has complex codimension at least 2 in Z; iii) the smooth locus Z reg is a complex manifold and a dense open subset of Z; iv) Z is Q-factorial. 24 In particular, under our mild assumption, the Coulomb branch M is a Q-factorial reduced normal analytic space.
We stress that the singular set in the orbifold sense of Z, S(Z), may be actually larger than the singular locus of the underlying analytic space, Σ(Z), see the discussion in ref. [79]. The case of maximal discrepancy between the two sets is given by the following: Let G be a Shephard-Todd group (≡ a finite complex reflexion group [45,49,50]). Then the analytic space underlying the orbifold C n /G is smooth, in fact isomorphic to A n .
We shall also need the orbifold version of the Kodaira embedding theorem: Theorem 4 (Kodaira-Baily [80]). Let Z be a compact complex orbifold and suppose Z has a positive orbi-bundle L. Then Z is a projective algebraic variety.

Structures on cones
We review the geometry of metric cones in a language suited for our purposes.

Riemannian cones
A metric (Riemannian) cone over the (connected, Riemannian) base B is the warped product where ds 2 y = γ ab (y) dy a dy b is a metric on B (y a being local coordinates in B). We shall write C(B) for the singular space obtained by adding the tip of the cone r = 0 to γ(B), endowed with the obvious topology. Note that the radial coordinate r is a globally defined continuous real function on C(B) taking all non-negative values. A cone C(B) possesses the following 24 An analytic space is Q-factorial if all Weil divisor has a multiple which is a Cartier divisor. canonical (global) structures: the plurisubharmonic function r 2 and the concurrent vector field E = r∂ r (Euler field ) which satisfy the following properties ). Conversely, if the Riemannian manifold (C, g) has a vector field E whose dual form is closed and £ E g = 2g, there exist coordinates such that the metric takes the conical form (2.43).
Definition 3. By a good cone we mean a cone C(B) = R >0 × r 2 B with B smooth and complete. For a good cone, the only possibly singular point is the tip of the cone r = 0. Note that a non-trivial product of metric cones is never good unless one of the factors is R k with the flat metric.
On a smooth Riemannian manifold the two notions of geodesic completeness and metric-space completeness coincide (Hopf-Rinow theorem [65]). This is not longer true in presence of singularities. Example 2 illustrates the point: the Minahan-Nemeshanski geometry is complete in the metric space sense, but certainly not in the geodesic one. The singular Riemannian spaces which are "physically acceptable" better be complete as metric space. This is part of regularity assumption R3.
For later use, we give the well-known formulae relating the curvatures of C(B) and its base B. We write R ijkl (resp. R ij ) for the Riemann (Ricci) tensor of C(B) and B abcd (resp. B ab ) for the Riemann (Ricci) tensor of B.

Singular Kähler cones: the Stein property
Now suppose the Riemannian manifold M is both Kähler (with complex structure I) and a (metric) cone, M ∼ = C(B). Eqn.(2.45) implies that Φ = r 2 is a globally defined Kähler potential assuming all values 0 ≤ Φ < ∞.
In the applications to N = 2 SCFT we have in mind, the Kähler metric on the cone M is singular even away from the tip r = 0. We specify the class of geometries we are interested in.
Definition 4. By a singular Kähler cone M we mean the following: 1) M is an analytic space (which we may assume to be normal 25 ) with an open everywhere dense smooth complex submanifold M ♯ = M \ D. 2) On M ♯ there is a smooth conical Kähler metric (in particular, M ♯ is preserved by the C × action generated by the holomorphic Euler vector E, see eqn.(2.52)).
3) The global Kähler potential Φ ≡ r 2 on M ♯ extends as a continuous function to all M (cfr. regularity condition R1). Then the Kähler form i∂∂r 2 extends to M as a positive (1,1) current. The continuous function r 2 is then plurisubharmonic in the sense of ref. [82]. Indeed, r 2 is a continuous plurisubharmonic function which is an exhaustion for M. The statement is then the Narasimhan singular version of Oka theorem (see e.g. page 48 of [82]).
Then the singular cone M is Stein and Cartan's Theorem A and Theorem B apply [51,83,84]. Below we shall exploit this fact in several ways.
In the physical applications we have in mind, the Fréchet ring [85] of global holomorphic functions, R = Γ(M, O M ), is the Coulomb branch chiral ring, our main object of interest. M being Stein implies that R contains "many" functions: around all points of M we may find local coordinate systems given by global holomorphic functions, and R separates points, i.e. given two distinct points we may find a global holomorphic function which takes on these two points any two pre-assigned complex values. Affine varieties over C are in particular Stein [84]. The converse is not true in general, but it holds under some mild additional conditions [86]. We shall see that that the M's which are Coulomb branches of N = 2 SCFTs are always affine.
In particular, the real vector R = IE, or in components is a Killing vector. The physical interpretation of this geometric result is as follows: a Kählerian cone may be used as a target space of a (classical) 3d supersymmetric σ-model. The fact that it is Kähler means the model is N = 2 supersymmetric, while the fact that it is a cone means that is classically conformally invariant [87]; the two statements together imply that the model has classically N = 2 superconformal symmetry hence a U(1) R R-symmetry which is part of the algebra. The action of U(1) R on the scalars is given by a (holomorphic) Killing vector which is R. We note that so that R = R a (y)∂ ya is in facts a Killing vector for the metric ds 2 y on the base B whose norm is 1, i.e. R a R a = 1. For a holomorphic function h on a conic Kähler manifold the actions of E and R (in physical language: their dimension and R-charge) are related by which physically says that these two quantum numbers should be equal for a chiral superconformal operator. We refer to E as the holomorphic Euler vector.
Sasaki [79]. The base B is in particular a K-contact manifold whose Reeb vector is R.

Geometric "F -maximization"
We pause a second to digress on a different topic, namely F -maximization in 3d [88]. A problem one encounters in studying SCFT is the exact determination of the R-charge which enters in the superconformal algebra. For (classical) σ-models with conic Kähler target spaces [87], this is the problem of identifying the Reeb Killing vector R between the family of Killing vectors with the appropriate action on the supercharges £ V Q = ± 1 2 Q. The general such Killing vector has the form V = R + F with F a 'flavor' Killing symmetry. One has the following: Claim. Let M be a Kähler cone and V a Killing vector on M which acts on supercharges as £ V Q = ± 1 2 Q. Then (point-wise) with equality iff V is the Reeb vector R. That is, the true superconformal R-charge extremizes the square-norm.
The reader may easily check that this purely geometric fact is really Fmaximization for the partition function on S 3 of the corresponding 3d σ-model in the classical limit → 0. By considering the low-energy effective theory on the moduli space of susy vacua of a (quantum) 3d N = 2 SCFT, one reduces the general case to this statement.

Quasi-regular Sasaki manifolds
In the physical applications we are mainly interested in Kähler cones with quasi-regular Sasaki bases B, that is, with compact Reeb vector orbits, so that R generates a (locally free) compact group of isometries U(1) R which we identify with the R-symmetry group which must be compact on physical grounds. B is regular if in addition, the U(1) R isometries act freely. We collect here some useful results.
Proposition 2.3.3 (see e.g. [79]). B is a quasi-regular Sasakian manifold. Then i) The Reeb leaves are geodesic; If the flow is regular K is a Kähler manifold and B a principal S 1 -bundle.
Indeed, K is just the symplectic quotient of M with respect to the Hamiltonian U(1) R flow, r 2 being its momentum map, as eqn.

Conical special geometries
We may introduce distinct notions of "conical special geometry". The weakest one corresponds to a holomorphic integrable system X → M whose Kählerian base M (with Kähler form (2.18)) happens to be a metric cone. A slightly stronger notion requires M to be a cone and the full set of geometric structures S1-S11 to be equivariant with respect to the Euler action £ E (or £ E ). An even stronger notion requires in addition that the base of B of M is a quasi-regular Sasaki manifold. 27 By a conical special geometry (CSG) we shall mean the strongest notion together with the regularity conditions R1-R4.

Weak special cones
Locally in the good locus M ♯ ⊂ M we may write the special structure in terms of special complex coordinates and holomorphic prepotential F (a). By the assumption of the existence of a SW differential, we know that there is a globally defined Kähler potential If M is also a cone, so r 2 is also a globally defined Kähler potential, and for some local holomorphic function h with global real part. The metric is conic iff the vector dual to the (0, 1) form∂(iΦ +h) is holomorphic. Locally this happens if there are constants The slighter stronger notion of special cone corresponds to the case (c i , d j ) = 0 (so r 2 ≡ Φ); in other words, to get a slightly stronger special cone out of a weak one we simply absorb the constants in the definition of the periods a i , b i by a shift δλ = c i dx i + d i dy i of the SW differential. Then which means that locally we can find a prepotential F (a) which is homogeneous of degree 2 in the special coordinates a i . Indeed, in the slightly strong conic case, from eqn.(2.57) the Euler (Reeb) vector have the local expression The vector E agrees in the overlaps between two special coordinate patches if the condition £ E F = 2F holds in one of the two patches (and then also in the other, up to a constant). Indeed, the second eqn.
so that   [48]. By definition, projective special Kähler manifolds are the geometries appearing in N = 2 supergravity (as contrasted to N = 2 gauge theory); "wrong sign" means that K corresponds to supergravity with an unphysical sign for the Newton constant.

Properties of the Reeb flow/foliation: the Reeb period
As already anticipated, the special cones M which arise as Coulomb branches of N = 2 SCFTs have the property that the flow of the Reeb vector field R yields a U(1) R action on M, i.e. the Reeb leaves [79] are compact in M. This must be so because the exponential map 28 t → exp(2πtR) should implement the superconformal U(1) R symmetry which is compact in a regular SCFT. The action is automatically locally free, since the Reeb vector has constant norm 1 and hence does not vanish anywhere. 29 The statement that R generates a locally-free U(1) R action is equivalent to the statement that its basis B is a quasi-regular Sasaki orbifold.
We now give our final definition: Definition 5. By a conical special geometry (CSG) we mean a complex analytic integrable system X → M with SW differential λ such that the base M (with the Kähler metric S10) is a singular Kähler cone (satisfying R1-R4) such that the restriction to M ♯ of its Euler vector E satisfies while its base B is a quasi-regular Sasaki orbifold.
Note that this means (on M ♯ ) Exponential action of the Reeb field R. Quasi-regularity requires the existence of a minimal positive real number α > 0 such that the Reeb exponential map exp(2παR) : M → M (2.69) 28 By exp(2πtR) we always mean the finite isometry of M or B generated by the Reeb Killing field of parameter 2πt. 29 Of course, this geometric statement also follows from unitarity of the SCFT.
is the identity diffeomorphism. If the base B is compact, the map exp(2παtR) may have fixed points in M \ {r = 0} only for a finite set of rational values 0 < t < 1.
Definition 6. The real number α > 0 is called the Reeb period of the CSG.
The Reeb period is a basic invariant for a 4d N = 2 SCFT.
Reeb period and Coulomb dimensions. The chiral ring R = Γ(M, O M ) of a CSG is (the Fréchet closure of) a graded ring, the grading of a (homogeneous) global holomorphic function h being given by its dimension In the chiral ring of a unitary SCFT, except for the constant function 1 which has dimension zero, all other dimensions ∆(h) should be strictly positive for h to be regular at the tip. The Reeb exponential map then yields and the definition (2.69) implies so R is graded by the semigroup N/α. We claim that α is a positive rational number ≤ 1 (so that the dimensions of all chiral operators, but the identity, are rational numbers ≥ 1). Indeed, the map e 2παR acts on the periods as (a i , b j ) → e 2πiα (a i , b j ). From (2.69) we deduce that the initial and final periods are equivalent up to the action of an element m of the monodromy group. Then e 2πiα ≡ λ is an eigenvalue of a monodromy, hence a root of unity. Therefore α ∈ Q >0 and α = log(λ)/2πi. The requirement that the curvature at the tip of the cone is non-negative forces us to use the log determination of the logarithm, see below eqn.(4.10). The order r of 1/α in Q/Z coincides with the order of the quantum monodromy M which is well-defined in virtue of the Kontsevitch-Soibelman wall-crossing formula, see discussion in [37,38].

Local geometry on K ♯
In a special cone, the discriminant locus is preserved by the C × action generated by the vector fields E and R. Hence the singular locus on its base B is preserved by the R-flow, and so is its smooth locus B ♯ . By our definition, B ♯ must be Sasaki quasi-regular. Then the Hamiltonian quotient yields a Kähler orbifold (manifold if B ♯ is regular) where the SW differential λ is seen as a section of a holomorphic line sub-bundle L ⊂ V. The period matrix τ ij (a) is homogeneous of degree zero, £ R τ ij (a) = 0 so the (restriction of the) period map τ | M ♯ factors trough K ♯ . We write p : K ♯ → Sp(2k, Z)\H k for the period map so defined.
In view of Meyer theorem (Theorem 1) we conclude that for k ≡ dim C M > 1 the base B of the cone is compact, π 1 (B) is finite, and the diameter of B is bounded In other words: if M is a special cone of dimension k > 1 over a smooth base B, with M a simply-connected Kähler cone with compact base of diameter ≤ π/ 2(k − 1) and G is a finite group acting freely. Indeed, K is compact, smooth, and, being complex, oriented of even real dimension. Its sectional curvatures are positive. Then the last statement follows from Synge theorem (Theorem 2). In facts, since B is compact by Meyer theorem, Theorem 3 yields an even stronger statement:

Relation to Fano manifolds
Recall that a Fano manifold X is a smooth projective variety whose anticanonical line bundle −K X is ample. We have: Under the assumptions of Proposition 2.4.1, the Kähler manifold K is a Fano projective manifold.
Proof. K is smooth by assumption and compact by Meyers theorem. By Lemma 2.4.2 the Ricci form is ≥ (dim K + 1)ω, ω being the (positive) Kähler form. Thus the anti-canonical is ample, hence K is projective (by Kodaira embedding theorem [51]) and Fano. ii) the rank of the Picard group ( Picard number) of the Fano manifold K is 1: Proof. B is compact and simply connected by Meyers' theorem. From Lemma 2.3.1 and eqn.(2.27) we see that the eigenvalues of the curvature operators are bounded below by 1.
Using Theorem 3 we get i). Then B ≡ B/π 1 (B) has real cohomology To this fibration we apply the Leray spectral sequence of de Rham cohomology. 30 We get This shows iii). To get ii) note that the exponential exact sequence yields the implication and then ii) follows from iii).
We have a much stronger statement: Under the assumptions of Proposition 2.4.1, the Fano manifold K ∼ = P k−1 and the period map p is constant.
Before proving this Proposition we define the index ι(F ) of a Fano manifold F whose canonical divisor we write K F . The index ι(F ) is the largest positive integer such that −K F /ι(F ) is a (ample) divisor [26]. Under the assumptions of Proposition 2.4.1 we have a line bundle L → K such that ω is its Chern class (up to normalization). Thus from Lemma 2.4.2 we get: Proof. The Picard number of K is 1, and we have with equality iff p =constant.
Then Proposition 2.4.5 follows from this Lemma and a basic fact from the theory of Fano varieties: the index of a Fano manifold F cannot exceed dim F + 1, and if ι(F ) = dim F + 1 then F ∼ = P dim F [26].
Corollary 2.4.2. Suppose we have a special cone of dimension k > 1. If its base B is a smooth, complete, regular Sasaki manifold, the period map τ is constant and M = C k with the a Kähler metric. That is: if the geometry is regular except for the singularity at the vertex of the cone, then the SCFT is free, as expected.
Remark 2.4.3. The result does not hold for k = 1. All k = 1 Minahan-Nemeshanski geometries satisfy the other assumptions, yet the SCFT is not free. The essential point is that for k > 1 the assumptions imply regularity in codimension 1, whereas the tip of the cone is automatically a codimension 1 singularity for k = 1.
Remark 2.4.4. The last result also follows from rigidity. If K is smooth and compact, the monodromy group is trivial, hence (by uniqueness) the period map should be the constant one.

Properties of non-smooth CSG
In the previous sub-section we considered the case in which the geometry of the special cone is totally regular away from its vertex. We got only the (known) k = 1 geometries and free theories for k > 1. In all other cases there are singularities in M \ {0} of the kind consistent with R1-R4.
Since B ♯ is only quasi-regular, we have a locus N ⊂ B ♯ on which U(1) R does not act freely. We writeB = B ♯ \N. For k > 1,K =B/U (1) is a open dense Kähler sub-manifold of the singular space K, in factsK = K \ D, for some divisor D. Since the period matrix τ ij is homogeneous of degree zero, the period map τ factors through K. We consider its restriction to the regular subspace, τ :K → Sp(2k, Z)\H k . All the considerations in §. 2.1.5 apply to this map; in particular, we have a monodromy representationm : π 1 (K) → Sp(2k, Z). Along an irreducible component D i of D such that the monodromy is semi-simple, we may extend τ holomorphically and K has along D i only a Z r cyclic orbifold singularity: indeed, the geometry becomes smooth after the local base change U → U r , cfr. §. 2.1.5. Otherwise, the monodromy along D i satisfies (m r −1) 2 = 0; the base change U → U r sets the local geometry in the form discussed around eqn. (2.40). Locally U r has the form ∆ * × ∆ k−2 and we may choose local coordinates so that the Kähler form take the form (here |q| < 1) (2.85) We may modify this metric to where f C (x) : R → R is a smooth function which equals 1 for x ≤ 0 and has all derivatives vanishing as x → +∞. The new Kähler form (2.86) is smooth in U r and agrees with the original one for |q| ≥ ǫ; one checks that one may choose the local deformation so that the metric and the curvatures remain positive. Of course, it is no longer a special Kähler metric. The point we wish to argue is that K admits a non-special orbifold Kähler metric, with only Z r orbifold singularities, whose Ricci form satisfies a bound of the form R i ≥ kg i . The same conclusion applies to M (the Ricci tensor being non-negative in this case); then the basis B is also regular except for cyclic orbifolds singularities. We may apply to B the orbifold version of Meyers [67,68]): a metrically complete Riemannian orbifold X, whose Ricci curvature satisfies R ≥ λ 2 g, is compact with a diameter d(X) ≤ π/λ. Once we are assured that B, albeit singular, is compact we conclude that M is Stein. Moreover, from the Synge theorem for orbifolds [68], we see that K, albeit no smooth, is still compact and simply-connected.
L is now a line orbi-bundle which is still positive by eqn.(2.74), Kodaira-Baily embedding theorem (Theorem 4) guarantees that K is a normal projective algebraic variety with at most cyclic orbifold singularities. The anticanonical divisor −K K is now a Weyl divisor which is ample as a Q-Cartier divisor. A normal projective variety with ample anti-canonical Qdivisor having only cyclic singularities is a log-Fano variety, to be defined momentarily. The Picard number ̺ is 1 as in the smooth case. Indeed, B is still diffeomorphic to a generalized Lens space S 2k−1 /G (see Proposition 2.4.2) and the finite group G centralizes the Reeb U(1) action, so that the orbifold K is homeomorphic to a finite quotient of P k−1 , and hence rank H 2 (K, Z) = 1. The index ι(F ) of a log-Fano variety F is the greatest positive rational such that −K F = ι(F ) H for some Cartier divisor H (called the fundamental divisor ). By a theorem of Shokurov [26], the index of a log-Fano F is at most dim F + 1. This is not necessarily in contradiction with eqn.(2.74) since now L is just a Q-divisor. Let σ be the smaller positive rational number so that L σ is a bona fide line bundle. Then The theorem of Shokurov then yields 1 ≤ σ with equality if and only if our special cone M is C k with the flat metric, i.e. we are talking about the free SCFT. Shokurov inequality is one of the many geometrical properties underlying the physical fact that saturating the unitarity bound means to be free. We see that the main difference between the smooth case of the previous subsection and the general is that in the second case G acts properly discontinuously on S 2k−1 but not freely. The previous argument giving ̺ = rank Pic(K) = 1 extends to this (slightly) more general case (indeed, K is simply-connected and topologically a finite quotient of P k−1 ).

log -Fano varieties
A singular Fano variety whose only singularities are cyclic orbifold ones is a log-Fano variety. We recall the relevant definitions [26]: 3) A normal variety X is said to have terminal, canonical, log terminal, or log canonical singularities iff its canonical divisor K X is Q-Cartier and there exists a projective birational morphism (whose exceptional locus has normal crossing 32 ) f : V → X from a smooth variety V such that: (2.88) 31 Indeed, X is non-singular in codimension 1, so a canonical divisor over the smooth open set X smooth ⊂ X can be extended as a Weil divisor to X. 32 Such resolutions exists by Hironaka theorem [91]. and, respectively:

4)
A normal projective variety X with only log-terminal singularities whose anti-canonical divisor −K X is an ample Q-Cartier divisor is called a log-Fano variety. 5) The greatest rational ι(X) > 0 such that −K X = ι(X) H for some (ample) Cartier divisor H is called the index of X and H is called a fundamental divisor. 6) The degree of a Fano variety X is the self-intersection index d(X) = H dim X .
Comparing our discussion in §.2.4.6 with the above definitions, we conclude (cfr. Proposition 7.5.33 of [79]): The symplectic quotient K = M/ /U(1) of a CSG is a log-Fano variety with Picard number one and Hodge diamond h p,q = δ p,q . Moreover, in the smooth sense, Thus a CSG M is a complex cone over a normal projective variety of a very restricted kind; in particular, M is affine and a quasi-cone in the sense of §.3.1.4 of [29].

Example 3. A large class of examples of log-Fano varieties with the properties in the
Corollary is given by the weighted projective spaces (WPS) [29]. Il w = (w 1 , w 2 , · · · , w k ) ∈ N k is a system of weights, P(w) is (2.90) All such spaces are log-Fano with just cyclic orbifold singularities, simply-connected, have Hodge numbers h p,q = δ p,q , and are isomorphic to P k−1 /G for some finite Abelian group G.
More generally, all quasi-smooth complete intersections in weigthed projective space of degree d < i w i and dimension at least 3 are log-Fano, simply-connected, and have ̺ = 1 [29] but typically their Hodge numbers h p,q ≡ δ p,q . A slightly more general class of examples of projective varieties satisfying all conditions above is given by the fake weighted projective spaces [30]. A fake WPS is canonically the quotient of a WPS by a finite Abelian group. One shows that a log-Fano with Picard number one which is also toric is automatically a fake WPS.
3 The chiral ring R of a N = 2 SCFT

General considerations
In the previous section we reviewed the general properties of the CSG describing the Coulomb branch M of a 4d N = 2 SCFT in which we have switched off all mass and relevant deformations so that the conformal symmetry is only spontaneously broken by the non-zero expectation values of some chiral operators φ i x = 0 in the susy vacuum x ∈ M. Although the Kähler metric on M has singularities in codimension 1, due to additional degrees of freedom becoming massless or symmetries getting restored, these singularities are assumed to be mild and the underlying complex space M is regular 33 . The analysis of local special geometry on the open, everywhere dense, regular set M ♯ = M \D only determines the special geometry up to "birational equivalence" since the singular fibers over the discriminant locus D may be resolved in different ways; different consistent resolutions give non-isomorphic CSG which correspond to distinct physical models. For instance, SU(2) SQCD with N f = 4 and SU(2) N = 2 * have "birational equivalent" CGS whose total fiber spaces X are certainly different in codimension 1; other examples of such "birational" pairs in rank one are given by Minahan-Nemeshansky E r SCFTs and Argyres-Wittig models [90] with the same Coulomb branch dimension but smaller flavor groups. The basic invariant of an equivalence class of SCFTs is the chiral ring R = Γ(M, O M ).
The common lore is that R is (the Fréchet completion of) a free graded polynomial ring R = C[u 1 , · · · , u k ] whose grading is given by the action of the holomorphic Euler field, £ E u i = ∆ i u i . The set of rational number {∆ 1 , · · · , ∆ k } are the Coulomb dimensions. This lore may be equivalently stated by saying that the Hamiltonian reduction K = M/ /U(1) R is (birational to) the weighted projective space (WPS) P(∆ 1 , · · · , ∆ k ). In the previous section we deduced from special geometry several detailed properties of M and K which are consistent with this idea: M is affine while K is a normal projective log-Fano variety, which is simply-connected and has Hodge numbers h p,q = δ p,q . These are quite restrictive requirements on an algebraic variety, and they hold automatically for all WPS; indeed they almost characterize such spaces. For instance, if one could argue that K (or M) is a toric variety then these properties would imply that K is a fake WPS [30], and in particular a finite Abelian quotient of a WPS.
We sketch some further arguments providing additional evidence that K is (birational to) a finite quotient of a WPS (possibly non-Abelian). The monodromy and SL 2 -orbit theorems describe the asymptotical behavior of the special Kähler metric on M as we approach the discriminant D (at least for D snc). As argued in §. 2.4.6, we can modify the metric to a Kähler metric which is smooth in M \ {0}, agrees with the original one outside a tubular neighborhood of D of size ǫ, is conical and Kähler with non-negative curvatures. Of course the new metric is no longer special Kähler and cannot be written in terms of a holomorphic period map; nevertheless it is a nice regular metric on the complex manifold M \ {0} with all the good properties. Working in the smooth category, and using the sphere theorems (cfr. §.2.1.4), we conclude that the Riemannian base B of the cone is diffeomorphic to a space form S 2k−1 /G for some finite G acting freely, hence to S 2k−1 up to a finite cover. The finite cover M ′ of M is a Riemannian cone over S 2k−1 is diffeomorphic to R 2k ∼ = C k . Moreover we know that M ′ is Stein, in fact a normal affine variety. If we could say that M ′ , being an affine variety diffeomorphic to C k is biholomorphically equivalent to C k , we would be done. Unfortunately, for k ≥ 3 there are several examples of exotic C k , that is, affine algebraic spaces which are diffeomorphic to C k but not biholomorphic to C k [96]. However, no example is known of an affine variety over C which is diffeomorphic to C k but not birational to C k ; the conjecture which states that there are none being still open [96]. Hence, assuming the conjecture to be true, we conclude that M ′ is birational to C k . On the other hand, M ′ is not just a complex space diffeomorphic to C k , it has additional properties as (e.g.) a holomorphic Euler vector E whose spectrum is an additive semigroup spec(E) ⊂ {0} ∪ Q ≥1 , and the condition of non-negative bisectional curvatures. Were not for the singularity at the tip of the cone, this last property by itself would guarantee 34 that M ′ is analytically isomorphic to C k .
There is still the question of the relation between the Coulomb branch M and its finite cover M ′ . Since M ∼ = M ′ /G, the statement M = M ′ is equivalent to G = 1, that is (for k > 1) that after smoothing the discriminant locus by local surgery M becomes simplyconnected. This holds in the sense that the monodromy along a cycle not associated to the discriminant is trivial (since the period map factors through K).
All the above results and considerations, while short of a full mathematical proof, provide convincing evidence for the general expectation that M is birational to C k , so for the purposes of our "birational classification" we may take M to be just C k . C × acts on M through the exponential action of the Euler field E. Again, an action of C × on C k is guaranteed to be linear only for k ≤ 3 [96]; in larger dimensions exotic actions do exist. However, in the present case the action is linear in the smooth sense; under the assumption that the complex structure is the standard one for C k , we have a linear action, that is, the chiral ring has the form R = C[u 1 , · · · , u k ]. (3.1) where the u i can be chosen to be eigenfunctions of £ E , i.e. homogeneous of a certain degree ∆ i . Then K is the WPS P(∆ i ). Thus, under some mild regularity conditions and modulo the plausible conjecture that a smooth affine variety M whose underlying C ∞ -manifold is diffeomorphic to C n is actually bimeromorphic to C n , we conclude that the common lore is correct up to birational equivalence (and, possibly, finite covers). For the rest of this paper we shall assume this to be the case without further discussion.
Remark 3.1.1. It is interesting to compare the above discussion with the one in ref. [28].
The previous argument is based on the idea that the unitary bound ∆(u) > 1 implies that 34 The following theorem (a special case of Yau's conjecture [93,94]) holds: Theorem (Chau-Tam [95]). N a complete non-compact Kähler manifold with non-negative and bounded bi-sectional curvature and maximal volume growth. N is biholomorphic to C n .
M with the modified metric satisfy non-negativity of bisectional curvatures and maximal volume growth, but the curvatures (if non zero) blow up as r → 0, and geodesic completeness fails at the vertex.
we may smooth out the metric while preserving the curvature inequalities. In ref. [28] they find that if the chiral ring is not free there must be a local parameter u with ∆(u) < 1, so if there exists such an u the argument leading to (3.1) cannot be invoked. Clearly u should not be part of the physical chiral ring R ph . However, in a Stein space the local parameter may always be chosen to be a global holomorphic function, so u ∈ Γ(M, O M ), which is our definition of the "chiral ring". Going through the examples of ref. [28] one sees that their Coulomb branches are non-normal analytic spaces, that is, they have the same To fully determine the graded rings R which may arise as Coulomb chiral rings of a CSG, it remains to determine the allowed dimension k-tuples {∆ 1 , . . . , ∆ k }. We shall address this question in section 4. Before going to that, we consider the simplest possible CSG just to increase the list of explicit examples on which we may test the general ideas.

The simplest CSG: constant period maps
For k = 1 all CSG have constant period map τ = const. For k > 1 constant period maps are still a possibility, although very special. For instance, this happens in a Lagrangian SCFT in the limit of extreme weak coupling (classical limit). This is good enough to compute the Coulomb dimensions ∆ i , since they are protected by a non-renormalization theorem (or just by the fact that they are U(1) R characters and U(1) R is not anomalous).
The Kähler metric is locally flat, i.e. has the local form is a subgroup of the isotropy group Fix(τ ) of τ . Fix(τ ), being discrete and compact, is finite. Then We stress that (in general) we have just an inclusion not an equality: in Example 2 the three rank-1 Argyres-Douglas SCFT (respectively of types A 2 , A 3 and D 4 ) have with m = 5, 4, 2 respectively. Note, however, that we have still 3.5) in these cases. The point is that K does not fix uniquely the chiral ring, unless we specify its orbifold structure, see the discussion in [97]. We shall study the orbifold behavior in more generality in the next section. Here we limit ourselves to the very simplest case in which as graded C-algebras. The Shephard-Todd-Chevalley theorem [49,50] states that C[a 1 , · · · , a k ] G is a graded polynomial ring if and only if G is a finite (complex) reflection group 35 . In such a case the Coulomb branch dimensions ∆ i coincide with the degrees of the fundamental invariants u i of the group G such that C[a 1 , · · · , a k ] G = C[u 1 , · · · , u k ]. However, not all finite reflection groups can appear in special geometry, since only subgroups of the Siegel modular group, G ⊂ Sp(2k, Z), are consistent with Dirac quantization. 36 In §.4.2 we review the well known: Fact. Let G ⊂ Sp(2k, Z) be a finite subgroup of the Siegel modular group. The subset Fix(G) ⊂ H k of points in the Siegel upper half-space fixed by G is a non empty, connected complex submanifold.
Thus, if G is a finite reflection subgroup of Sp(2k, Z) there is at least one period τ with τ ⊂ Fix(G) and the quotient by G of C k with the flat metric (3.2) makes sense (as a orbifold). All such metrics are obtained by continuous deformations of a reference one, and so they belong to a unique deformation-type. The dimension of the fixed locus, d = dim C Fix(G), is given by eqn. (4.40).
The finite complex reflection groups (and their invariants u i ) have been fully classified by Shephard-Todd [49,98,99]. They are direct products of irreducible finite reflection groups. The list of irreducible finite complex reflection groups is 37 [49,98,99]: 1) The cyclic groups Z m in degree 1; 1) the symmetric group S k+1 in the degree k representation (i.e. the Weyl group of A k ); 2) the groups G(m, d, k) where k > 1 is the degree, m > 1 and d | m; 3) 34 sporadic groups denoted as G 4 , G 5 , · · · , G 37 of degrees ≤ 8.
For our purposes, we need to classify the embeddings of the Sphephard-Todd (ST) groups of rank-k into the Siegel modular group Sp(2k, Z) modulo conjugacy. We shall say that a ST group is modular iff it has at least one such embedding. A degree-k ST group G which preserves a lattice L ⊂ C k is called crystallographic [99,100]; clearly a modular ST group is in particular crystallographic. 35 A degree-k reflection group G is a concrete group of k × k matrices generated by reflections, i.e. by matrices g ∈ G such that dim coker(g − 1) = 1. In particular, a reflection group comes with a defining representation V , whose dimension k is called the degree of the reflection group. 36 More generally, subgroups of duality-frame groups S(Ω) Z . 37 In the classification one does not distinguish a group and its complex conjugate since the two are conjugate in GL(V ).
Let K be the character field of an irreducible reflection group 38 and O its ring of integers. One shows that G ⊂ GL(k, O) [98], so G is crystallographic iff K is either Q or an imaginary quadratic field Q[ Hence it preserves the skew-symmetric form with rational coefficients Clearing denominators, we get a non-degenerate integral skew-symmetric form Ω on Z 2k which is preserved by G whose entries have no non-trivial common factor.
det H = 2 is not a square in Q, and hence the polarization is non-principal with charge multipliers (e 1 , e 2 ) = (1, 2) (cfr. eqn. (2.2)). Thus G 8 is no contradiction to our claim. We shall return to this in section 5.
For the groups G(m, d, k) with m = 3, 4, 6 the usual monomial basis is both defined over O and orthonormal, so det H = 1 and they are all subgroups of Sp(2k, Z). However, for special values of (m, d, k) we may have more than one inequivalent embeddings in Sp(2k, Z):    is a single conjugacy class of embeddings G(m, d, k) ֒→ Sp(2k, Z) except for m = 3, 4 ≡ p s (p = 3, 2 is a prime), 1 < d | p s , and p | k in which case we have p + 1 inequivalent embeddings.
Remark 3.2.1. This statement is the complex analogue of the usual GSO projection of string theory [102]. Indeed, consider the real reflection group G(2, 2, k) ≡ Weyl(D k ) and count the number of inequivalent embeddings 41 G(2, 2, k) ֒→ SO(k, Z) ⊂ Sp(2k, Z), or equivalently the number of maximal local subalgebras of the Spin(2k) k chiral current algebra. We have just one, generated by the free fermion fields ψ a (z) unless p ≡ 2 | k in which case we have 3 ≡ 2 + 1 of them, the additional ones being the two GSO projections of opposite chirality. In the complex case, chirality is replaced by a Z p symmetry. The proof is essentially the same as in the GSO case [101].
Example 6. A Lagrangian SCFT with gauge group i G i at extreme weak coupling has a CSG which asymptotically takes this constant period form with G = i Weyl(G i ) in the standard reflection representation.
Example 7. The model associated to the group G(m, 1, k) (m = 3, 4, 6) has the simple physical interpretation of representing (the birational class of) the rank-k MN E r SCFT for r = 6, 7, 8, respectively. Indeed, geometrically the quotient of C k by G(m, 1, k) is the same as taking the k-fold symmetric product of quotient of C by Z m . Correspondingly, the isotropy group of the diagonal period matrix, Fix(e 2πi/m 1 k ) ⊂ Sp(2k, Z), is G(m, 1, k) (see [104] for a discussion in the k = 3 case). The dimensions are {∆, 2∆, 3∆, · · · , k∆} with ∆ = m. 41 In addition there is an embedding G(2, 2, k) ֒→ Sp(2k, Z) which does not factor through SO(k, Z). Example 7 may be generalized. One has a CSG M and takes the n-th symmetric power. This works well if M has dimension 1, but in general the resulting geometry may be more singular than permitted.
In general, when we have a CSG of the form C k /G, where G = a G a with G a irreducible reflection groups, and, in addition, R = C[a 1 , · · · , a k ] G , the Coulomb dimensions are equal to the degrees of G. The period τ , being symmetric, transforms in the symmetric square of the defining representation V of the reflection group G ⊂ Sp(2k, Z) (see §.4.2 below). Hence the dimension of the space of allowed deformations of τ , that is, the dimension d of the conformal manifold of a constant-period CSG is given by the multiplicity of the trivial representation in ⊙ 2 V . By Schur-Frobenius, which are Weyl groups, (3.10) since a reflection group has a degree 2 invariant iff it is defined over the reals (and hence, if crystallographic, should be a Weyl group). The physical interpretation of this result is that such a CSG represents a gauge theory with gauge group the product of all simple Lie groups whose Weyl groups are factors of G coupled to some other intrinsically strongly interacting SCFT associated to the complex factor groups G a (as well as hypermultiplets in suitable representations of the gauge group). The d marginal deformations of the geometry are precisely the d Yang-Mills couplings which are associated to d chiral operators with ∆ = 2. Note that the Yang-Mills couplings may be taken as weak as we please.
Thus the Lagrangian models and higher MN SCFTs already account for all CSG with constant period map up (at most) to finitely many exceptional ones.

The Coulomb dimensions ∆ i
Now we come to the main focus of the paper, namely to get geometric restrictions on the spectra of Coulomb branch dimensions ∆ i .
We may address two different problems: Problem 1. For k ∈ N specify the set such that: there is a CSG M with dim M ≤ k and a generator The solution to the Problem 2 contains vastly more information than the answer to Problem 1, since there are strong correlations between the dimensions ∆ i of a given CSG and Λ(k) is a rather small subset of Ξ(k) k . However Problem 1 is much simpler, and its analysis is a first step in answering Problem 2. We give a solution in the form of a necessary condition: Ξ(k) ⊂ Ξ(k), where Ξ(k) is a simple explicit set of rational numbers. There are reasons to believe that the discrepancy between the two sets Ξ(k) and Ξ(k) is small and vanishes as k → ∞. In facts Ξ(k) = Ξ(k) for k = 1, 2, and the equality may hold for all k.
To orient our ideas, we discuss a few special instances as a warm-up for the general case.
Our basic strategy is to mimic this analysis of the k = 1 for general k. Before doing that, we discuss another special case in two different ways: first we review the conventional approach and then recover the same results by reducing the analyis to the k = 1 case. This will give the first concrete example of the basic strategy of the present paper.

Hypersurface singularities in F -theory and 4d/2d correspondence
There is a special class of simple (typically non-Lagrangian) 4d N = 2 SCFTs which are engineered in F -theory out of an isolated quasi-homogeneous dimension-3 hypersurface singularity withĉ < 2 (see eqn.(4.8)) [6,37]. Their SW geometry (in absence of mass deformations) is given by the hypersurface F ⊂ C 4 of equation where W (x i ) is a quasi-homogeneous polynomial The regularity condition of ref. [6],ĉ < 2, ensures that the u a 's have positive degree. The SW 3-form is the obvious one Ω = P R dx 1 ∧ · · · ∧ dx 4 F , (4.10) where P R stands for "Poincaré residue" [51]. At the conformal point, u a = 0, Ω has degree Since Ω (by definition) has U(1) R charge q equal 1, for all chiral object φ with a definite degree, we have in particular, To get the Coulomb branch dimensions ∆(u a ), it remains to determine the k rational numbers (4.14) There are many ways of doing this, including pretty trivial ones. Here we shall compute the t a 's in a way that seems unnaturally complicate: but recall that we are doing this computation as a warm-up, meaning that we wish to perform this elementary computation in a way which extends straightforwardly to the general case where simple minded methods fail.
Picard-Lefschetz analysis [103]. One convenient viewpoint is the 4d/2d correspondence of ref. [37]. One considers the 2d (2,2) Landau-Ginzburg model with superpotential F (x i ) and uses the techniques of tt * geometry [105,106] to compute 2d quantities which are then reinterpreted in the 4d language. In 2dĉ is one-third the Virasoro central charges and deg φ is the R-charge (in the 2d sense) of the chiral object φ.
In the 2d approach, the t a 's are computed using the Picard-Lefschetz theory [62] (see [106] for a survey in the present language). Consider the family of hypersurfaces F z = {F (x i ) = z} parametrized by z ∈ C; we have H 3 (F z , Z) ∼ = Z µ , where µ is the Milnor number of the singularity W = 0 (i.e. the dimension of the (2,2) chiral ring R). The classical monodromy H of the quasi-homogeneous singularity W is given by the lift on the homology of the fiber, H 3 (F z , Z), along the closed loop in the base z = ρ e 2πit (t ∈ [0, 1] and ρ ≫ 1). Concretely, H is a µ × µ integral matrix acting on the lattice Z µ whose action on C µ ≡ Z µ ⊗ C is semisimple of spectral radius 1 [106]. Let Φ ⊂ Z µ be the sublattice fixed (element-wise) by H, and consider the quotient lattice Γ = Z m /Φ which has rank 2k. H induces an automorphism H of Γ. The intersection form in the homology of the hypersurface F z , induces a non-degenerate, integral, skew-symmetric pairing −, − on Γ, preserved by H. Then In simple examples the induced polarization −, − is principal, and one has H ∈ Sp(2k, Z); in the general case we reduce to this situation by a suitable isogeny in the intermediate Jacobian of F z . Moreover, H is semi-simple of spectral radius 1 so (by Kronecker's theorem) it has a finite order ℓ. Its eigenvalues are of the form {exp(2πiα a ), exp(2πi(1 − α a ))} for some 0 < α a ≤ 1, a = 0, 1, . . . , k − 1 with ℓα a ∈ N. These eigenvalues are related to the t a by the 2d spectral-flow relation [106] Spectrum H = e 2πiαa , e 2πi(1−αa) ≡ e ±2πita . (4.16) Since 0 < t a <ĉ/2 < 1, for each index a = 0, 1, . . . , k − 1 we have two possibilities where we relabel the indices of t a so that t 0 = max t a . Thus knowing the spectrum of H fixes the t a 's up to a 2 k -fold ambiguity corresponding to choosing for each a one of the two possible values (4.17). Let us explain the origin of this ambiguity. For simplicity of illustration we assume the characteristic polynomial P (z) of H to be irreducible over Q; in this case the spectrum uniquely fixes H up to conjugacy in GL(2k, Q). However two physical systems described by monodromy matrices H 1 , H 2 ∈ Sp(2k, Z) are physically equivalent iff they are related by a change of duality frame i.e. iff the corresponding reduced monodromies H 1 , H 2 are conjugate in the smaller group Sp(2k, Z): the unique GL(2k, Q)-conjugacy class of H decomposes 42 into 2 k distinct Sp(2k, Q)-conjugacy classes in one-to-one correspondence with the inequivalent choices of the t a 's. (The conjugacy classes over the integral group, Sp(2k, Z), are trickier, and will be discussed in section 5). The Picard-Lefschetz theory has a canonical symplectic structure (i.e. the intersection form in homology) and hence a canonical choice of the {t a }. The spectrum of Coulomb branch dimension for the SCFT engineered by the singularity is given by plugging these canonical {t a } in the expressions The ray analysis. Let us rephrase the above Picard-Lefschetz analysis in a different language. We return to eqn.(4.7) and consider the Coulomb branch M of the associated N = 2 SCFT; we see M as the affine cone over the WPS with homogeneous coordinates (u 0 , u 1 , · · · , u k−1 ) the coordinate u a having grade ∆(u a ). We focus on the closed one-dimensional sub-cone M 0 parametrized by the vev u 0 of the chiral operator of largest dimension ∆(u 0 ), b) is not democratic in the following sense: we have many sub-loci in the Coulomb branch M over which some discrete R-symmetry is restored, but the classical monodromy (when applicable) applies to just one of them (the one with the largest unbroken symmetry).
In order to solve Problems 1, 2 we need a generalization of the classical monodromy construction which may be applied uniformly to all loci in the Coulomb branch with some unbroken discrete R-symmetry, and to all SCFT, while reducing to the classical Picard-Lefschetz theory when we consider the locus of largest unbroken U(1) R symmetry of a SCFT engineered by a F -theory singularity.
Suppose such a generalization exists. Along the Coulomb branch of a typical SCFT we have several loci with enhanced (discrete) R-symmetry; each such locus produces a list of ∆ a . Then we get the highly non-trivial constraint that the dimension set {∆ a } should be the same independently of which special locus we use to compute it. On the other hand, the agreement of the dimensions computed along different loci in M is convincing evidence of the correctness of the method.
In the remaining part of this note we describe the generalized method, and check its consistency is a variety of examples. Although we could present the algorithm already at this stage in the form of an educated guess inspired by the classical Picard-Lefschetz formulae, we prefer to deduce it mathematically from scratch. Before doing that, we need some elementary preparation.

Cyclic subgroups of Siegel modular groups I
We saw already in the k = 1 case that an important ingredient in the classification of all possible dimension sets {∆ a } is the list of all embeddings of the cyclic group Z n (or more generally of a finite group G) into the Siegel modular group Sp(2k, Z) modulo symplectic conjugacy in Sp(2k, Z) (cfr. discussion around eqn.(4.17)).
We start by establishing a fact, already mentioned in §.3.2, which applies to all subgroups of the Siegel modular group, cyclic or otherwise.
is non-empty and connected. Proof. Being finite, G is compact. Hence G ⊂ K for some maximal compact subgroup K ⊂ Sp(2k, R). All maximal compact subgroups in Sp(2k, R) are conjugate to the standard one, the isotropy group of i1 k ∈ H k , that is, there is R ∈ Sp(2k, R) The Cayley transformation C maps biholomorphically the Siegel upper half-space H k into the Siegel disk [115,116] D k = w ∈ C(k) w t = w, 1 − ww * > 0 , (4.24) taking τ = i1 k to the origin w = 0 and conjugating the standard maximal compact subgroup into the diagonal subgroup. Then Consider the embedding U : G ֒→ U(k) sending g into the upper-left block of CRgR −1 C −1 ; we write V for the corresponding degree-k unitary representation. The action of G on the Siegel disk D k is linear i.e. the Cayley-rotated period w transforms in the symmetric square representation ⊙ 2 V . The fixed locus of G is the intersection of D k ⊂ ⊙ 2 V with the linear subspace of trivial representations whose dimension d is as in eqn.(4.40); in particular Fix(G) is nonempty and connected. In the special case that ⊙ 2 V does not contain the trivial representation, the fixed locus reduces to the origin in D k , and hence is an isolated point. Mapping back to the Siegel upper half-space H k , the fixed point is In the general case (4.30) and the embedding U : G ֒→ U(k) is given by the modular factor Cτ + D ∈ U(k), cfr. the discussion of the k = 1 case around eqn. (4.4). This is the same factor appearing in the transformation of the a-periods, a → (Cτ + D)a, and is the one which controls the Coulomb dimensions, as we saw in the k = 1 case. Implicitly we have already used these facts in the discussion of the CSG with constant period map, §. 3.2.
Let us specialize to the case in which G is a cyclic group Z n generated by a matrix m ∈ Sp(2k, Z). m is called regular iff Fix(m) is an isolated point, i.e. iff ⊙ 2 V does not contain the trivial representation. Note that this is a weaker notion than m being regular as an element of the Lie group Sp(2k, R) which requires the characteristic polynomial of m to be square-free; we shall refer to the last situation as strongly regular. The spectrum of the unitary matrix U(m) = Cτ + D is a set of k n-th roots of unity, {ζ 1 , . . . , ζ k }, and m is regular iff The eigenvalues of the 2k ×2k matrix m are {ζ 1 , . . . , ζ k }∪{ζ −1 1 , . . . , ζ −1 k } the union being disjoint iff m is regular. Let ψ ζ be the normalized eigenvector of m associated to the eigenvalue ζ. The symplectic matrix Ω of Sp(2k, Z) corresponds to the 2-form Thus, for m regular, the splitting of the spectrum of m in the two disjoint sets Spectrum U(m) and Spectrum U(m) may be read directly from Ω: the eigenvalue ζ belongs to the spectrum of U(m) iff the term ψ ζ ∧ψ ζ −1 appears in −iΩ with the + sign, otherwise ζ −1 ∈ Spectrum U(m).
We shall present a much more detailed discussion of the cyclic subgroups of the Siegel modular group in section 5 below.

Normal complex rays in CSG
As anticipated at the end of §.4.1.1, our goal is to reduce the determination of the ∆ a 's to the analysis of one-dimensional conic complex geometries. In this subsection we introduce the basic construction. We start with a definition:  Proof. Since M is a Kähler cone, we have the holomorphic field E ≡ E + iR such that ∇īE j = 0 while ∇ i E j = δ j i . With respect to the induced Kähler metric, M * is also a Kählerian cone with a holomorphic Euler vector E * = E| M * . E * ,Ē * span the real tangent bundle T M * . Let ∇ * be the Levi-Civita connection of the induced metric on M * . The second fundamental form of M * in M is The chiral ring of M is the ring of global homomorphic functions, R = Γ(M, O M ). The linear differential operator E acts on R and we write S for its spectrum, i.e. the set of dimensions of chiral operators. R is the Frechét completion of a finitely generated graded ring. If φ ∈ R is a homogeneous element of degree ∆(φ), we have Eφ = ∆(φ)φ. In the same way we write R * for the ring of holomorphic functions on the ray M * and S * for the spectrum of E * in R * . Proof. M is Stein, and M * ⊂ M is a closed analytic subspace. Hence, by Cartan's extension theorem [83,84] we have an epimorphism of chiral rings Then, although the result is valid in general, to apply it for M * ⊂ M not normal we need to be able to distinguish holomorphic functions in the two different senses. If M * is normal, it coincides with its own normalization, and the two notions coincide.
Since M * is one-dimensional and normal, it is smooth. 43 On M * we have a C × action ζ → ζ E , acting transitively on M * \ {0}. Then M * is analytically (hence algebraically) a copy 43 As a complex space; the Kähler metric has a conical singularity at the tip.
of C on which ζ E acts by automorphisms fixing the origin. Let u * be a standard coordinate on this C; the polynomial ring C[u * ] is dense in Γ(M * , O M * ) and graded by deg u * = Eu * > 0.
If w ∈ R is a homogeneous holomorphic function with ∆(w) ∈ S * we must have w| M * ≡ 0. Let {v i } k i=1 be a set of homogeneous generators of (a dense subring of) the global chiral ring R. Not all restrictions {v i | M * } k i=1 may vanish identically since in a Stein manifold the ring of homolorphic functions separates points [83]. Let u ∈ {v i } be a generator of R with u| M * ≡ 0 of smallest degree d * . All (non constant) homogenous elements f ∈ R either restrict to zero f | M * ≡ 0 or have a degree ≥ d * . Since R * ∼ = C[u * ], for some u * having minimal positive degree in R * , and u * is the restriction of a function in R, we conclude that we may choose u * = u| M * . Therefore the dimension d * ≡ ∆ * of the generator u * of R * , is equal to the dimension ∆(u) of the generator u of the full chiral ring R.
Example 9. We illustrate the subtle point in the proof of the above Proposition. We consider SU(3) SQCD with N f = 6 at weak coupling and zero quark masses. In this case the Coulomb branch ring is C [u, v] where the generators u and v have dimensions 2 and 3. The (reduced) complex rays are ; we see that the generators of the two normal ray rings, u and v, are generators of the full chiral ring R. Instead, M (α) is a plane cubic with a cusp, which is the simplest example of a non-normal variety (see e.g. §.4.3 of [107]). The ring of holomorphic functions in the subspace sense is not integrally closed. The integral closure of the ray ring contains the function v/u (indeed, (v/u) 2 = αu) which is holomorphic in the normalization and has dimension 1, a dimension ∈ S. Remark 4.3.1. Note that the non-normal rays M (α) in the above example correspond to the "non-free" geometry discussed in ref. [28].

Unbroken R-symmetry along a ray
The statements in §.4.3.1 reduce the computation of the Coulomb dimensions of the generators of R to local computations at normal complex rays in the conical Coulomb branch M. This leads to the following two questions: 1) are there enough normal complex rays to compute all Coulomb dimensions? 2) how we characterize the normal complex rays? In Example 9 we see that the two normal complex cones in the Coulomb branch M of SU(3) SQCD are precisely the loci of Coulomb vacua with an unbroken discrete subgroup of U(1) R , namely Z 3 for M v and Z 2 for M u . All non-normal rays consist of vacua which completely break U(1) R (except at the tip where the full U(1) R is restored). This characterization of normal rays by unbroken R-symmetry holds in general.
To a complex ray M * there is associated a rational positive number α * , namely the smallest positive number such that exp(2πα * R) = Id M * . (4.37) α * ∈ Q >0 since the R-symmetry group U(1) R is compact. In the physical language this means that a subgroup of U(1) R is unbroken: n is the order of α * in Q/Z. We say that M * is an elliptic ray iff n > 2.
In section 3 we presented some evidence that the chiral ring is a free polynomial ring (or simply related to such a ring). In this case M is parametrized by weighted homogeneous coordinates u i having weights ∆ i ∈ Q ≥1 . Then the complex ray along the i-th axis is normal and has α i = 1/∆ i . If ∆ i = r i /s i > 1 with (r i , s i ) = 1, the order of the residual R-symmetry is n i = r i ≥ 2 with equality iff ∆ i = 2, i.e. iff u i is the vev of a (superficially) marginal operator. In particular, if R is a free polynomial ring we do have enough normal rays.
Let M * be a normal ray and x ∈ M * \ {0}. We consider the closed R-orbit The monodromy of this path is an element m * of the modular group Sp(2k, Z) which is independent of x modulo conjugacy. We have (4.40) since exp(2πnα * R) acts trivially on the periods. We say that M * is regular if its monodromy m * is strongly regular. Regularity is equivalent to s(ℓ) ∈ {0, 1} (so s(1) = s(2) = 0 in the regular case). Regularity implies that the fixed period τ is unique. More crucially, it means that the ray is not part of the "bad" discriminant locus, M * ⊂ D bad . Here D bad is the union of the irreducible components of D with a non-semi-simple monodromy; along D bad the period matrix τ degenerates in agreement with the SL 2 -orbit theorem, see §. 2.1.5. Indeed, strong regularity implies that all the eigenvalues of m * are distinct, and no non-trivial Jordan block may be present. More generally, we split the product in (4.40) in the square-free factor s(ℓ)=1 Φ ℓ (z) and the complementary factor s(ℓ)>1 Φ ℓ (z) s(ℓ) . m is conjugate in Sp(2k, Q) to a block-diagonal matrix of the form m reg ⊕ m comp with m reg ∈ Sp(2k reg , Z) strongly regular.
Then, up to isogeny, 44 τ = τ reg ⊕ τ comp for a unique τ reg . The regular rank k reg of M * is Thus, locally at the ray, the family of Abelian varieties X → M * splits (modulo isogeny) in a product X 1 × X 2 → M * , and M * is not in the "bad" discriminant of the first factor (but it may be for the second one).
Let M * be a normal ray which is also regular. The real function r 2 = Im τ ij a iāj is smooth and non-zero on M * ; since Im τ ij is the unique fixed period, which is non singular on M * (in particular, bounded), it means that there exists a C-linear combination a * of the periods a i which does not vanish on M * (more precisely, a * is well defined on a finite cyclic cover of M * branched at the tip). Applying m * to a * , we see that e 2πiα * ≡ e 2πi/∆(u * ) belongs to the spectrum of m * . Since m * is the lift of the generator of the unbroken subgroup Z n ⊂ U(1) R , we conclude that e 2πiα * is a n-root of unity. In facts, it should be a primitive root, otherwise the unbroken symmetry would be smaller. Comparing with the k = 1 case, and using Proposition 4.3.2 we learn that 1/α * is the dimension of a generator of the full chiral ring R. Now suppose that M * is not regular. Taking into account only the "good" block, that is, focusing on the first local family, X 1 → M * , we reduce to the regular situation. The argument applies to the irregular block too, as long as Im τ ij is not singular on M * . In general we may decompose Im τ ij in a regular block and one which is in the modular orbit of i∞. The argument works as long as the regular block is non-trivial, that is as long as along M * not all photons decouple. Then Fact. If in M there is a normal ray with residual R-symmetry Z n , there should be a generator of R with ∆ = n/s, s ∈ (Z/nZ) × i.e. e 2πi/∆ is a primitive n-th root of unity. e 2πi/∆ is an eigenvalue of a quasi-unipotent element m * ∈ Sp(2k, Z), and hence φ(n) ≤ 2k.

Remark 4.3.2.
Above we assumed the polarization Ω to be principal. In the general case, we replace the Siegel modular group with the relevant duality-frame group S(Ω) Z . The conclusion φ(n) ≤ 2k being still valid.

Tubular neighborhoods and the Universal Dimension Formula
Identifing the tubular neighborhood M • of the normal ray M * with a neighborhood of the zero-section in the normal bundle, we may introduce homogeneous complex coordinates such that M * ⊂ M • is given by the analytic set v 1 = v 2 = · · · = v k−1 = 0, while u = u * is the coordinate along the normal ray M * . Indeed, the additional coordinates v i are just linear coordinates along the fibers of the holomorphic normal bundle. The v i are globally defined in M • since the holomorphic normal bundle of M * is holomorphically trivial. This follows from a result of Grauert (cfr. Theorem 5.3.1(iii) of [84]) since dim C M * = 1. We saw in the previous subsection that u is homogeneous of degree 1/α * . Along the ray M * only a complex-linear combination of the a-periods, a * , does not vanish. In the tubular neighborhood M • all k linear combinations of the a-periods are not (identically) zero. In a conical special geometry the a periods transform through the modular factors a ′ = (Cτ + D)a, (4.44) where τ is the (constant) period matrix on M * . Hence, if m * is a regular elliptic element of Sp(2k, Z) and (e 2πiα * , e 2πiβ 1 , · · · , e 2πiβ k−1 ) is the spectrum of Cτ + D, with e 2πiα * a primitive n-th root of unity, we may find complex-linear combinations of a-periods a * , a i in M • which diagonalize the action of m * a ′ * = e 2πiα * a * , 0 < α * ≤ 1 a ′ s = e −2πiβs a s s = 1, . . . , k − 1, 0 ≤ β s < 1, where a * is the linear combination non-zero along the ray M * . Then the k dimensions of the generators of R are where we set v k ≡ u * and β k = 1 − α * .
If M * is normal but non regular, we cannot determine all dimensions from an analysis in the neighborhood M • of M * but only as many as its regular rank k reg . Remark 4.3.3. These formulae have the following natural property. Let m * be weakly regular, that is, some eigenvalue ζ of U(m * ) have multiplicity s > 1. Assume a * is an eigenperiod associated to ζ; then α * = α while (s − 1) β's are equal 1 − α. The dimensions of the s operators associated to the eigenvalue ζ are all equal to 1/α, without distinction between the operator parametrizing the ray and the operators parametrizing its normal bundle. This property guarantees that we get the correct dimension spectrum for SCFT whose Coulomb branch is birational to a product of identical cones, so that the largest dimension ∆ max is degenerate. This happen e.g. in class S The third requirement is quite strong, and it seems a priori quite unlike that such a strong property may be acutally true. We perform the three checks in turn.

Relation to Springer Theory
We have to check that the "abstract" dimension formula (4.51) reproduces the obvious dimensions for a weakly-coupled Lagrangian SCFT and more generally for all CSG with constant period maps of the form M = C k /G for a degree-k ST group G whose chiral ring R coincides with the ring of polynomial invariants (see §.3.2). That eqn.(4.51) correctly reproduces the ST degrees d i as dimensions ∆ i of the generators of R is a deep result in the Springer Theory of regular elements in finite reflection groups [44][45][46].
We recall the definitions: let the finite group G act as a reflection group on the C-space V . A vector v ∈ V is said to be regular iff it does not lay in a reflection hyperplane. An element g ∈ G is said to be regular if it has a regular eigenvector v. The regular degrees of G are a (minimal) set of integer numbers such that the order of all regular elements of G is a divisor of an element of the set and conversely all divisors of these numbers are the order of a regular element. Then Theorem 5 (see [44,45]). Let ζ be a primitive d-root of unity. Let g ∈ G be regular with regular eigenvector v ∈ V and related eigenvalue ζ. Denote by W the ζ-eigenspace (4.52) Then: (i) d is the order of g, and g has eigenvalues ζ 1−d 1 , ζ 1−d 2 , · · · , ζ 1−d k , where d i are the degrees of G; (ii) dim W = #{i | d is a divisor of d i }; (iii) the restriction to W of the centralizer of g in G defines an isomorphism onto a reflection group in W whose degrees are the d i divisible by d and whose order is d|d i d i ; (iv) the conjugacy class of g consists of all elements of G having dim W eigenvalues ζ.
One can show that an integer is regular iff it divides as many degrees as co-degrees [99].  [45,99], one sees that part (ii) of the Theorem implies that for an irreducible crystallographic Shephard-Todd group dim W = 1 for all regular degrees d. In other words, in the crystallographic case, the only degree which is an integral multiple of the regular degree d is d itself.
Let us re-interpret Theorem 5 in the context of the constant-period class of CSG's discussed in §. 3.2, with Coulomb branch M = C k /G, G a degree-k crystallographic ST group, and chiral ring R = C[a 1 , · · · , a k ] G . For simplicity, we take G irreducible.
By definition, v ∈ C k is a regular vector iff it does not lay on a reflection hyperplane, i.e. if its projectionṽ in the Coulomb branch M = C k /G does not belong to the discriminant locus D ⊂ M, i.e. iffṽ ∈ M ♯ , the smooth locus.
Let d be a regular weight of G (so d ≡ d i 0 for some i 0 ). By definition, there is an element m i 0 ∈ G of order precisely d i 0 . Let v be a regular eigenvector of m i 0 corresponding to the primitive d i 0 -th root 45 ζ = e 2πi/d i 0 . The (closure of the) C × -orbit ofṽ ∈ M ♯ , M v ⊂ M, is a complex ray not lying in the discriminant D (more precisely, intersecting D only at the tip).
We claim that M v is also normal. Recall that R = C[a 1 , · · · , a k ] G ≡ C[u 1 , · · · , u k ] by the Shephard-Todd-Chevalley theorem. Homogeneity implies for some constants c i . Applying m i 0 on both sides, and using Corollary 4.3.1 i.e.
we conclude that is automatically a normal right of the form in eqn. (4.38). This also shows that m i 0 is the monodromy along the normal ray M v , which is then regular. This is exactly the set up in which we deduced the universal dimension formula (4.51). From item (i) in the Theorem 5 we see that β i = (d i − 1)/d and α * = 1 − β i 0 = 1/d. The universal formula (4.51) then yields which is the correct result. Thus the formula (4.51) is nothing else than the plain extension to the non-Lagrangian SCFT of the usual formula, valid for all Lagrangian SCFT, following from the standard supersymmetric non-renormalization theorems.

Recovering Picard-Lefschetz for hypersurface singularities
We consider the hypersurface where u a has dimension ∆ 0 (1 − deg φ a ), ∆ 0 being the relative normalization of the Rcharges in the 4d and 2d sense under the 4d/2d correspondence [37]. We consider the ray parametrized by the coupling u 0 of the 2d identity operator The corresponding monodromy m 0 , along the path u 0 → e 2πit u 0 , t ∈ [0, 1] is, by construction, the one induced on H 1 (F, Z)/rad −, − by the classical monodromy H of the hypersurface singularity, that is, (f ≡ rank rad −, − ) so that, which precisely yields back the Picard-Lefschetz formula (4.13) (q a ≡ deg φ a by definition).
Example 10 (Complete intersections of singularities). The previous argument applies to all SCFT which have a 4d/2d correspondent in the sense of [37]; in the general case the classical monodromy of the singularity H should be replaced by the (2,2) quantum monodromy as defined in [106]. The eigenvalues of the (2,2) quantum mondromy have the form e 2πi(qa−ĉ/2) where q a are the U(1) R charges of the 2d chiral operators [106]. For instance, this result applies to the models engineered by the complete intersection of two hypersurface singularities in where W a (x i ) are quasi-homogeneous of degree d 1 ≡ 1 and d 2 (we assume is a basis of the admissible deformations of the equations. 46 This correspond to a 2d model with superpotential [17] W( where Λ is an extra chiral field of U(1) R charge q Λ = 1 − d 2 . The scaling dimension of the parameters u a,α is 1 − q a,α where q α,α is the charge of the corresponding operator perturbing W, i.e. φ 1,α and Λφ 2,α , respectively. One has 1 −ĉ/2 = i w i − d 1 − d 2 and so [17] ∆ (4.66) Example 11 (The DZVX models). A third class of 4d SCFT with a nice 2d correspondent is the one constructed in [13] parametrized by a pair (affine star, simply-laced Lie algebra). Our formulae yield the correct dimension spectrum by construction.

Consistency between different normal rays
Let us consider a simple (but instructive) class of examples, the Argyres-Douglas (AD) models of type A N −1 with N odd. The special geometry of the A N −1 AD model corresponds to the intermediate Jacobian of the following family of hypersurfaces in C 4 specified by the spectrum of (Cτ + D) reg.block which consists of k a,reg out of the 2k a,reg roots of the characteristic polynomial of m a : it is a set of 2(N − a)-th roots of unity satisfying the condition in equation (4.31). We also need to specify which one of the k a,reg roots corresponds to the non-trivial period a a on the ray M a . There is a "canonical" Picard-Lefschetz choice. We have (here 0 ≤ a ≤ (N − 1)/2) we see that the several rays M a yield mutually consistent results for the spectra of dimension. M 0 and M 1 yield the full set of dimensions (which agrees between the two rays), while M 2ℓ and M 2ℓ+1 yield a partial list of k − ℓ out of the k dimensions. (But the formal "analytic continuation" gives the full correct set of dimensions at all normal rays).

Remark 4.3.5.
Note that the list of the k reg dimensions computed from a M a is a k regtuple of dimensions which is allowed for a rank k reg SCFT. In particular, the dimension of the operator parametrizing a ray with k reg = 1 should be in the one-dimensional list {1, 2, 3, 4, 6, 3/2, 4/3, 6/5}.

Example 12.
Consider the Argyres-Douglas models of type D 5 and D 6 which have k = 2. The ray parametrized by the operator of the largest dimension corresponds to Picard-Lefschetz theory and is regular, while the one associated to the operator of lesser dimension is non-regular. We deduce by the previous Remark that for these models the smaller of the two Coulomb dimensions should belong to the k = 1 list. Indeed it is 6/5 for type D 5 , and 4/3 for type D 6 . This statement may be generalized in the form of inter-SCFT consistency conditions relating the spectrum of dimensions in different SCFT. For g ∈ ADE we write {∆} g for the set of Coulomb branch dimensions of the Argyres-Douglas model of type g and we write {∆} We present a more complicated example of such consistency condition between Coulomb dimensions in different ranks.

A comment on the conformal manifold
As further evidence of the correctness of the universal dimension formula (4.51), let us consider the conformal manifold, that is, the space of moduli (deformations) of the CSG. We have already discussed the special case in which the period map is constant while the chiral ring is the invariant subring C[a 1 , . . . , a k ] G (see §. 3.2). In that case the dimension of the conformal manifold was given by the number of generators of the chiral ring of dimension 2. In the general case, the allowed continuous deformations of the CSG are described by the rigidity principle (Proposition 2.1.1). Since the monodromy representation is a discrete datum, a continuous deformation of the CSG is uniquely determined by the deformation it induces on a single Abelian fiber X u . Suppose that the CSG under considerations has a normal ray with semi-simple monodromy m * . We focus on a fiber over a point in the ray.
If the monodromy is regular, by definition there is no deformation of the fiber, that is, the conformal manifold reduces to an isolated point. On the other hand, m * regular implies that no chiral operator has dimension 2. Indeed, in an interacting theory a chiral operator of dimension 2 is necessarily a generator of R. Suppose the dimension of the i-th generator is 2. Then from eqn.(4.51) contradicting the assumption that m * is regular. If m * is semi-simple, but not regular, the same argument shows that the dimension d of the conformal manifold is ≤ the number of generators of R having dimension 2. This is the physically expected result: the number d of exactly marginal operators is not greater than the number of chiral operators of dimension 2. Note that this is just an inequality since being m * -invariant is only a necessary condition for a deformation to be allowed. Some of the m * -invariant deformations may be obstructed by other elements of the monodromy group; comparison with the constant period map case shows that the obstruction comes from the non-semisimple part of the monodromy group. The non-semisimple part of the monodromy measures the effective one-loop beta-function of the QFT (cfr. discussion around eqn.(2.40)). Thus the dimension formula (4.51) is consistent with the physical expectations on the conformal manifold.

The set Ξ(k) of allowed dimensions
We write Ξ(k) ⊂ (Q ≥1 ) for the set of all rational numbers which appear as dimension ∆ of a generator of the chiral ring R in a CSG of rank ≤ k. Clearly Ξ(k) is monotonic in k: Ξ(k − 1) ⊂ Ξ(k). If M is a rank k 2 CSG, its symmetric power M [k 1 ] -if not too singular -should also be a CSG. This rather sloppy argument would suggest that the set Ξ(k) also satisfies the following requirement: for all k 1 , where s · Ξ(k) stands for the set of rationals obtained by multiplying all rationals in Ξ(k) by the integer s.
To determine Ξ(k) it suffices to give the difference of the sets for two successive ranks k; we already know that Ξ(1) = {1, 2, 3, 4, 6, 3/2, 4/3, 6/5}. The Fact stated at then end of §.4.3.2 implies A priori this is just an inclusion, that is, the conditions we got insofar are just necessary conditions. However, experience with the first few k's suggests that the two sets Ξ(k) and Ξ(k) are pretty close, and likely equal. The discrepancy (if there is any) is expected to vanish as k increases. Note that (4.78) that is, all rational numbers ≥ 1 appear in the list for some (large enough) rank k. For instance if ∆ is a large integer, the Coulomb dimension ∆ will first appear in rank k min [108] k min = 1 2 φ(∆) > 1 2 ∆ e γ log log ∆ + A log log ∆ , where A = 2.50637, (4.79) while (for comparison) the minimal rank for a Lagrangian SCFT is Eqn.(4.77) is shown by recursion in k. For k = 1 we know that it is true with ⊆ replaced by =. We consider the rays M * ⊂ M generated by the vev of a single chiral field of dimension ∆; along M * a R-symmetry Z ℓ is preserved, ℓ being the order of 1/∆ in Q/Z, so that 1/∆ = s/ℓ with 1 ≤ s ≤ ℓ and (s, ℓ) = 1. We have ℓ = 1, 2 iff ∆ = 1, 2. Now let m * be the corresponding element of the monodromy. m ℓ * = 1. m * acts on the C-period a * non-zero on M * by a primitive ℓ-th root of unity, hence the cyclotomic polynomial The monotonicity of Ξ(k) is obvious. Eqn.(4.76) for Ξ(k) is equivalent to the inequality φ(ℓk 2 ) ≤ φ(ℓ) k 2 which holds by Euler's formula For bookkeeping, it is convenient to list the (candidate) "new-dimensions" at rank k After the completion of this paper, the paper [32] appeared on the arXiv in which Ξ(2) is also determined. The agreement with (4.83) is perfect.
The new-dimension sets N(k) up to k = 13 are listed in table 8.

4)
The Carmichael conjecture (still open) states that ν(d) = 1 for all d; the conjecture is known to be true for d < 10 10 10 . 5) One has k odd ⇒ |N(k)| ≤ 8k. (4.90) The cardinality N (k) of the set Ξ(k). N (k) may be written in the form of a Stieltjes integral where V (x) is the Erdös-Bateman Number-Theoretic function [109,111] V (x) = n≤x ν(n), (4.92) whose values for x ∈ N are given by the sequence A070243 in OEIS [114]. Note that V (2k) is also the number N (k) int of integral elements of Ξ(k)  Various expressions for the error term may be found in ref. [109,111].
To construct the putative new-dimension sets N(k) we only need to solve the equation φ(x) = 2k for all k ≥ 2. There is an explicit algorithm to solve recursively this equation for a given k once we know the solutions for all k ′ < k [110].

Analytic expressions of N (k) and N (k) int
We define N (x) int for real x as To obtain an analytic formula for N (x) int one starts from the Dirichlet generating function N (s) (chap. 1 of [112]) for the totient multiplicities ν(m). N (s) has an Euler product .
. . . , ∆ k }, a prototypical case being the Picard-Lefschetz theory of a SCFT engineered by F -theory on a singularity. More generally, from local considerations on a ray of regular rank k reg we get k reg out of the k dimensions. The number k-tuples of allowed dimensions in presence of rays of regularity k reg ≥ 2 is much less than |Ξ(k)| k due to correlations between the dimensions of the various chiral operators of a given SCFT.
At a regular ray we have  where the factors are all distinct. Two embeddings m * ֒→ Sp(2k, Z) are equivalent if they are conjugate in Sp(2k, Z). A weaker notion of equivalence is conjugacy in Sp(2k, R). It follows from the considerations in §. 4.2 that a Sp(2k, R) conjugacy class is characterized by a subset of k out of the 2k roots of the characteristic polynomial (4.103) (namely the spectrum of U(m * )), {ζ 1 , · · · , ζ k }, having the property that ζ i ζ j = 1 for all 1 ≤ i, j ≤ k (cfr. eqn.(4.31)). Once given the spectrum {ζ i , · · · , ζ k } of U(m * ), we have to select which one of the k roots is the eigenvalue associated to the non-zero (multivalued) period a * on M * . Renumbering the roots so that this is the first one, we have e 2πiα * = ζ 1 , e −2πiβ j = ζ j , j ≥ 2, (4.104) the dimension k-tuple {∆ i } then being given by eqn. (4.51). It may happen that some of the dimensions in {∆ i } so found do not belong to Ξ(k); such a k-tuple should be discarded.
A priori, there are 2 k ways of splitting the spectrum of m * into two non-overlapping sets satisfying eqn. (4.31). Taking into account the k choices of the root we call ζ 1 , this yield k · 2 k possibilities for {α * , β i } for a given characteristic polynomial of the form (4.103). However, some of these possibilities come with a restriction: it is not true in general that we may find arithmetical embeddings m * ֒→ Sp(2k, Z) which realize as spectrum of U(m * ) all the subsets of roots consistent with eqn. (4.31). The simplest example is provided by the regular embeddings In this case the sets {e 2πi/12 , e 14πi/12 } and {e 10πi/12 , e 22πi/12 } are realized as Spectrum U(m * ), while {e 2πi/12 , e 10πi/12 } and {e 14πi/12 , e 22πi/12 } are not realized. Thus the allowed dimension sets depends on subtle Number-Theoretical aspects of the classification of all inequivalent embedding Z n ֒→ Sp(2k, Z); this topic will be addressed in section 5. There we shall justify the above claim on the embeddings Z 12 ֒→ Sp(4, Z). In section 5 we shall also see that the two spectra {e 2πi/12 , e 10πi/12 } and {e 14πi/12 , e 22πi/12 } may be realized by an embedding in Z 12 ֒→ S(Ω) Z where Ω is a polarization with charge multiplies (1, e 2 ) with e 2 ≥ 2; this result agrees with Example 5.
Summarizing: a conjugacy class of regular embeddings Z n ֒→ Sp(2k, Z) is a candidate for the monodromy m * at a normal (regular) ray M * ⊂ M along which the discrete subgroup Z n ⊂ U(1) R is unbroken. The datum of the conjugacy class, together with a choice of ζ 1 , produces a candidate dimension k-tuple {∆ 1 , · · · , ∆ k } by eqn. (4.51). It is not guaranteed that {∆ 1 , · · · , ∆ k } ∈ Ξ(k) k , and k-tuples which do not belong to Ξ(k) k should be discarded. At this point we must also impose consistency between the various normal rays M * . This reduce the list of allowed k-tuple even further. While we have no proof that all the surviving k-tuples are actually realized by some CSG, the experience suggests that this algorithm produces few "spurious" k-tuples, if any. As an illustration, we now run the algorithm in detail for k = 2. Ranks k = 3 and k = 4 may be found in the tables of section 6.

{∆ 1 , ∆ 2 } for rank-2 SCFTs
The possible regular characteristic polynomial are In  table 5 we write only the embeddings of the corresponding regular cyclic groups in Sp(4, Z) which lead to dimensions {∆ 1 , ∆ 2 } ∈ Ξ(2) 2 . For instance, Φ 3 Φ 4 does not have any embedding with this property. We have written separately the dimensions pairs associated to embeddings in groups S(Ω) Z with Ω a non-principal polarization. The fine points on the conjugacy classes of elliptic elements in the Siegel modular group Sp(2k, Z) will be discussed in the next section; table 5 summarizes the results of that analysis in the special case k = 2.
Consistency between the two rays u 2 = 0 and u 1 = 0 leads us to consider three situations: RR pairs of dimensions which appear twice in table 5, once in the form {∆ 1 , ∆ 2 } and once in the form {∆ 2 , ∆ 1 } corresponding to the case of both rays being strongly regular; RN a pair of dimensions {∆ 1 , ∆ 2 } where the second one ∆ 2 ∈ Ξ(1) corresponding to one strongly regular and one weakly regular of irregular ray; NN {∆ 1 , ∆ 2 } ∈ Ξ(1) 2 two weakly regular/irregular rays.
All rank-2 CSG with a conformal manifold of positive dimension must be of type NN. This holds, in particular, for the weakly coupled Lagrangian models which have  [20,21] The authors of refs. [20,21] have given a (possibly partial) classification of the dimension pairs {∆ 1 , ∆ 2 } which may appear in a rank 2 SCFT using quite different ideas. It is interesting to compare their list with the present arguments. To perform the comparison, we need to keep in mind the two caveat in §.2.1.1.
Let us recall the framework of [20,21]. They start from the fact that all families of rank 2 principally polarized Abelian varieties are families of Jacobians of genus 2 hyperelliptic curves which they write in two ways y 2 = v −r/s x 5 + · · · and y 2 = v −r/s x 6 + · · · , (4.107) where · · · stand for certain polynomials in x, u, v depending on the particular CSG which are listed in [20,21]. u, v are the "global" (in their sense) coordinates in the conical Coulomb branch, with v the operator of larger dimension. 0 ≤ r/s ≤ 1 is a rational number written in minimal terms, i.e. (r, s) = 1. Their SW differential has the form The two rays M v = {u = 0} and M u = {v = 0} preserve a discrete R-symmetry which may be read for each CSG from the explicit polynomials in the large parenthesis. From the same expressions we may read if these rays are regular irreducible (RI), regular reducible (RR), or irregular (Ir). We may distinguish their geometries in two classes. The first one is when r/s ∈ Z, that is, the global pre-factor in the rhs of eqn.(4.108) is a univalued function of v. The geometries M v M u 10/7 8/7 x 5 + ux + v RI Z 10 RI Z 8 8/5 6/5 Lagrangian SCFT Table 6: Geometries in refs. [20,21] with univalued symplectic structure. First two columns give the Coulomb dimensions, third the family of hyperelliptic curves, third and fourth the regularity/irregularity of the rays along the axes together with the corresponding unbroken R-symmetry.
with this property listed in refs. [20,21] are recalled in table 6 (we do not bother to discuss the Lagrangian models since the agreement with our results is obvious in this case). The first two columns are the dimensions of v and u as listed in refs. [20,21]. We see that in all cases these dimensions belong to the intersection of the sets of dimensions associated with the cones M v , M u , yielding perfect agreement with our approach. Note that only dimensions consistent with a principal polarization appear, since this is an assumption in [20,21]. The second class of geometries is when r/s ∈ Z, that is, a multi-valued prefactor. As discussed in [20] these geometries lead to a susy central charge Z which is well-defined up to a (locally constant) unobservable phase. Here the first remark of §.2.1.1 applies: to get a univalued SW differential λ we need to go to a finite cover where a suitable fractional power of v becomes uni-valued. Then we consider as global coordinates on the cover the functions (v (1−r/s) , u) and compare their dimensions with the ones in our table. This leads to the dimension list in table 7; we see that all dimension pairs agree with our table on the nose.
Of course, if the physically correct Coulomb branch is the geometrically natural covering which has a well-defined holomorphic symplectic structure Ω or its quotient considered by Argyres et al. it is a question of physics not of geometry. There is one aspect that suggests that quotient of [20,21] is the physical Coulomb branch: the dimension pair {10/9, 4/3} enters in their list (twice) only through quotient CSG. The dimensions of the two quotient geometries are, respectively, {20/9, 10/9} and {20/9, 4/3}. Now, the both covering dimensions are < 2, and there is evidence that a consistent SCFT with all ∆ i < 2 should be an  [20,21] with multivalued symplectic structure. First two columns contain the cover Coulomb dimensions; third one the hyperelliptic curves.
Argyres-Douglas model of type ADE; since {10/9, 4/3} does not correspond to such a model, we are inclined to think that physics requires to take a discrete quotient of the geometrically natural geometry, as the authors of [20,21] do.

Elliptic conjugacy classes in Siegel modular groups
Listing the dimension k-tuples {∆ 1 , · · · , ∆ k } has been reduced to understand the conjugacy classes of finite order elements inside the Siegel modular group Sp(2k, Z) or, in case of more general polarizations (non-trivial charge multipliers e i , see eqn.(2.2)) in the commensurable arithmetic group S(Ω) Z . In this section we give an explicit description of such classes.
Readers not interested in Number Theoretic subtleties may skip the section.

Preliminaries
We write Ω for the 2k×2k symplectic matrix in normal form and −, − for the corresponding skew-symmetric bilinear pairing.

Elements of Sp(2k, Z) with spectral radius 1
m ∈ Sp(2k, Z) has (spectral) radius 1 iff its characteristic polynomial is a product of cyclotomic ones that is, if all its eigenvalues are roots of unit. An element m of spectral radius 1 is semi-simple (over C) iff its minimal polynomial is square-free, i.e.
2) Suppose that no ratio d i /d j (i = j) is a prime power. Then in (5.3) we may choose R ∈ Sp(2k, Z). More generally, if ℓ is a prime such that ℓ r = d i /d j for all i, j and r = 0, we may choose R ∈ Sp(2k, Z ℓ ).
Proof. For each d ∈ I we define the integral 2k × 2k matrices Since mΩ = Ωm −t , and we have Π d Ω = ΩΠ t d . Up to a rational multiple, the Π d form a complete set of orthogonal idempotents over Q compatible with the skew-symmetric pairing Ω. Then they split over Q the representation in the block diagonal form of item 1). The splitting is over Z iff the ̺ d are ±1. We have where R(P, Q) stands for the resultant of the two polynomials P and Q. Under the assumption that d/e, e/d are not prime powers, ̺ d = ±1 [113]. In facts, ̺ d is divisible only by the primes p such that there is e ∈ I with d/e = p r , 0 = r ∈ Z [113].
In other words, if no ratio d i /d j is a non-trivial prime power, all embeddings m ֒→ Sp(2k, Z) are block-diagonal up to equivalence. If some d i /d j is a prime power, in addition to the block-diagonal ones, we may have other inequivalent embeddings. We shall return to this aspect after studying the case that the minimal polynomial is irreducible over Q.

Regular elliptic elements of the Siegel modular group
We recall that a finite-order element m ∈ Sp(2k, Z) is regular iff the eigenvalues {ζ 1 , · · · , ζ k } of U(m) ≡ Cτ + D satisfy ζ i ζ j = 1. The spectrum S ≡ {ζ i } of U(m) will be called the spectral invariant of the regular elliptic element m ∈ Sp(2k, Z). It is a subset of k roots,

The spectral invariant as a sign function
We focus on a m ∈ Sp(2k, Z) whose minimal polynomial is Q-irreducible, We fix a primitive d-root, ζ, and write K ≡ Q[ζ] for the corresponding cyclotomic field and k = Q[ζ + ζ −1 ] for its maximal totally real subfield, Gal(K/k) = {±1}. Let ψ α 1 ∈ K (α = 1, . . . , s) be a basis of the ζ-eigenspace of the matrix m. Let σ ∈ Gal(K/Q) ∼ = (Z/dZ) × ; then ψ α σ ≡ σ(ψ α 1 ) form a basis of the σ(ζ)-eigenspace. We write ψ α 1 , ψ β −1 = t αβ ∈ K(s). Without loss of generality, we may assume t αβ to be diagonal t αβ = t α δ αβ with t α ∈ K. t α is odd (i.e. purely imaginary)t α = −t α . The symplectic structure is given by a 2-form Thus m is the direct sum of s (possibly inequivalent) embeddings Z d → Sp(φ(d), R). For each summand we define the odd sign (function) The spectral invariant of the α-th summand is It is obvious that S α satisfies condition (5.8). We shall denote by the same symbol, S α , both the spectral invariant and the corresponding sing function. A semi-simple element m satisfying (5.9) is regular iff the spectral invariant is the same for all its direct summands, i.e. S α = S β .
In particular if sign is realized −sign is also realized.
In §.4.5 that the list of possible Coulomb branch dimensions {∆ 1 , · · · , ∆ k } is determined from the Sp(2k, R)-conjugacy classes of regular elements of Sp(2k, Z) (or the corresponding arithmetic group for non principal polarizations) through their sign function invariant S. Then our main problem at this point is to understand the set {S} of signs which do are realized for a given polarization. This is the next task.

Cyclic subgroups of integral matrix groups
In this section we review the theory of the embedding of cyclic groups into groups of matrices having integral coefficients in a language convenient for our purposes (also providing explicit expressions for the matrices). See also [54,117]. Our basic goal is to describe the set {S} of signs which do appear and more generally the regular elliptic elements of the Siegel modular group.

Embeddings Z n ֒→ GL(2k, Z) vs. fractional ideals
We focus on a single block, that is, we consider a matrix m ∈ GL(2k, Z) with minimal polynomial Φ n (z). Then 2k ≡ s φ(n) for some s ∈ N.
Notations. We fix once and for all a primitive n-root of unity ζ ∈ C, and write K ≡ Q[ζ] for the n-th cyclotomic field, O ≡ Z[ζ] for its rings of integers, k ≡ Q[ζ +ζ −1 ] for its maximal real subfield, and o ≡ Z[ζ + ζ −1 ] for the ring of algebraic integers in k. Gal(K/k) ∼ = Z 2 , the non-trivial element ι being complex conjugation, ι(x) =x. Gal(k/Q) ∼ = (Z/nZ) × /{±1}. We writen for the conductor of the field K: n = n if n = 2 mod 4 n/2 otherwise. (5.14) We write C K (C k ) for the group of ideal classes in K (resp. in k). N will denote the relative norm K → k extended to the groups of fractional ideals I K N − → I k in the usual way [118,119].
The embedding Z n ֒→ GL(2k, Z) makes Z 2k into a finitely-generated torsion-less Omodule M, multiplication by ζ being given by m. Conversely, any finitely-generated torsionless O-module M defines an embedding Z n ֒→ GL(2k, Z) where 2k is the rank of M seen as a (free) Z-module. The ring of cyclotomic integers O is a Dedekind domain. The following statement holds for all such domains: Proposition 5.2.1 (see [118]). A finitely-generated torsion-less module M over the Dedekind domain O has the form a 1 ⊕ a 2 ⊕ · · · ⊕ a s , where a i are fractional ideals in K. Two modules a 1 ⊕ a 2 ⊕ · · · ⊕ a s and b 1 ⊕ b 2 ⊕ · · · ⊕ b t are isomorphic if and only if s = t and the ideal class In order to describe the explicit embeddings we sketch the proof in the special case s = 1.
Proof. Let a ⊂ K be a fractional ideal. In particular a is a torsion-free finitely generated Z-module, hence a lattice isogeneous to O, and thus of rank 2k. Choosing generators, we may write a = 2k a=1 Z ω a , with ω a ∈ K. Now ζ ω a ∈ a, and hence there is an integral 2k × 2k matrix m such that ζω a = m ab ω b .
The minimal polynomial of m is the n-th cyclotomic polynomial, Φ n (m) = 0. Thus the matrix m yields an explicit embedding of Z n into GL(2k, Z). Had we chosen a different set of generators for a, ω ′ a , we would had gotten an integral matrix m ′ which differs from m by conjugacy in GL(2k, Z). Indeed, ω ′ a = A ab ω b , for some A ∈ GL(2k, Z). Thus the map a → (conjugacy class of m) is independent of all choices. By construction, the vector ω ≡ (ω 1 , · · · , ω 2k ) ∈ K 2k is the eigenvector of m associated to the eigenvalue ζ. The eigenvector associated to the eigenvalue σ(ζ), σ ∈ Gal(K/Q), is the σ(ω).
Conversely, if m ∈ GL(2k, Z) with minimal polynomial Φ n , consider an eigenvector ω ≡ (ω 1 , · · · , ω 2k ) ∈ K 2k associated to the eigenvalue ζ, and set a = 2k a=1 Zω a . Clearly, if ω is such an eigenvector so is µ ω for all µ ∈ K × . Hence a and (µ)a (µ ∈ K × ) describe the same conjugacy class of integral matrices m, that is, the conjugacy class of m depends only on the class of the fractional ideal a in C K = I K /(K × ).
The action of ζ on the module s i=1 a i is unitary for the natural Hermitian form

The dual embedding
Given an embedding of Z n ֒→ GL(2k, Z), generated by the integral matrix m, we have a second embedding, the dual one, where the generator is represented by the integral matrix where Φ ′ n (z) is the derivative of Φ n (z) andā i is the complex conjugate ideal of a i .

Note that
In appendix A.1 we show a some properties of the map a → a * which greatly simplify the computations. In particular: Ifn is not the power of an odd prime, we may alternatively choose ̺ to be real by multiplying it by a purely imaginary unit, e.g. (ζ − ζ −1 ) forn = 2 r or i forn = 2 r .

Complex, real, quaternionic
At the level of underlying C-linear representations, An anti -linear morphism R : V M → V M ∨ is said to be a real (resp. quaternionic) structure iff it squares to +1 (resp. −1) [121]. A real (resp. quaternionic) structure embeds the matrix m in the orthogonal (resp. symplectic) group. To get an embedding Z n ֒→ Sp(2k, Z) we need a quaternionic structure defined over Z. First of all, this requires M and M ∨ to be isomorphic as O-modules. Writing M = (1) ⊕(s−1) ⊕ a, we must have which implies that Na · O = aā = η). [122], the fractional ideal Na is principal in k, that is, Na = (η) for some η ∈ k × .
From eqn.(5.21) we see that the construction of the embedding is essentially reduced to the case s = 1. From now on we specialize to this case, so that M ≡ a, M ∨ = a * . Fix a Z-basis 49 {ω a } of a and let φ a be the dual basis of a * , i.e. ω a , φ b = δ a b . If a satisfies condition (5.22), a * = ̺/ā = ̺η −1 a, and we write Then {λ v ω a } is also a Z-basis of the dual fractional ideal a * and there is a a matrix J ab ∈ GL(2k, Z) (depending on the unit v) such that i.e. the integral unimodular matrix J ab is skew-symmetric (resp. symmetric) if the unit v is such that λ v is purely imaginary (resp. real). In the first case J ab is a principal integral symplectic structure, hence similar over Z to the standard one Ω, i.e. J = h t Ωh for some h ∈ GL(2k, Z). In the second case J is a unimodular symmetric quadratic form. Thus for a fixed fractional ideal a, we find a symplectic structure (i.e. an embedding in Sp(2k, Z)) per each choice of the unit v such that λ v is purely imaginary. We shall count the inequivalent ones in the next subsection.
Since η ∈ k × is always real, and ̺ was chosen to be purely imaginary (cfr. Lemma 5.2.2) J ab is skew-symmetric iff v is real, and symmetric iff it is purely imaginary. In particular, the two obvious choices v = ±1, always produce embeddings Z n ֒→ Sp(2k, Z) (2k ≡ φ(n)).

Conjugacy classes of embeddings Z n ֒→ Sp(φ(n), Z)
Regular embeddings Z n ֒→ Sp(φ(n), Z) exist for all n ≥ 3. Indeed, the condition Na principal is trivially satisfied if a itself is principal. Thus the trivial ideal class (1) yields regular embeddings Z n ֒→ Sp(φ(n), Z) for all n ≥ 3. We proceed as follows: we fix an embedding (m, J) associated to the ideal (1) and call it the reference embedding. All inequivalent embeddings are obtained by acting on the reference one (m, J) with a certain Abelian group H defined in the next Proposition. A subgroup of H is easy to describe: in eqn.(5.23) we may choose a different real unit v ∈ O and still get an invariant quaternionic structure; this is the same as multiplying η ∈ k × by a unit of o. In this way we get new (inequivalent) embeddings (m, J ′ ) for a given fractional ideal a: they correspond to embeddings which are conjugate over GL(φ(n), Z) but not over the subgroup Sp(φ(n), Z). On the other hand, under a → µa with µ ∈ K × we have η → η Nµ, hence the image of η in the group k × /NK × is independent of the choice of the representative ideal a in the ideal class. To describe also the embeddings belonging to different GL(φ(n), Z) conjugacy classes, it is convenient to consider the group of fractional ideal classes in K whose relative norm is principal in k. Then we have group K = (a, η) ∈ L × k × : Na = (η) (5.27) and the group homomorphism The above discussion shows the Proposition 5.2.2 (see [117]). Let n ≥ 3. The (Abelian) group acts freely and transitively on the set of the Sp(φ(n), Z)-conjugacy classes of embeddings Z n ֒→ Sp(φ(n), Z). In particular, the number of Sp(φ(n), Z)-conjugacy classes is where U (resp. u) is the group of unities of O (resp. o).
To compute the second factor in (5.30) we consider the group of units in the relevant fields.
Group of units. We write µ for the group generated by the roots of unity in K µ = ± ζ k . while its square root 2 sin(π/n) is not in k. Hence, for n divisible by two distinct primes, In particular ε 1 ≡ N(1 − ζ) is a fundamental unit of k.
For n ≥ 3, let k = φ(n)/2. From Dirichlet unit theorem [118] we know that where ε a , a = 1, . . . , k − 1 are the real positive fundamental units. For Q = 1 we have In conclusion, where Q = 1, 2 is the Hasse unit index, h − the relative class number of the cyclotomic field K, and B 1,χ the first Bernoulli number of the odd Dirichlet character χ.
However, to fully solve our problem we need to know also when two distinct conjugacy classes are conjugate in the larger group Sp(φ(n), R) (or, equivalently, in Sp(φ(n), Q)).

Embeddings Z n ֒→ S(Ω) Z forΩ non-principal
The symplectic matrix J defined in eqn. (5.24), for λ v as in (5.23) (with v a unit of o), corresponds to a principal polarization, i.e. J is an integral skew-symmetric matrix with det J = 1. Let 0 = κ ∈ o and consider the matrix J κ defined by If κ is a unit, J κ is a principal-polarization. For κ just integer in k, J κ is an integral skew-symmetric matrix with determinat det J κ = N k/Q κ 2 . An element η ∈ k × is said to be totally positive iff σ(η) > 0 for all σ ∈ Gal(k/Q); the set of all totally positive elements k × + ⊂ k × form a subgroup while (for n ≥ 3) [118] k × Comparing with eqn.(5.29), we see that The group of principal fractional ideals (η) with η ∈ k × + is a subgroup of the group of all principal fractional ideals. The quotient I k /(k × + ) is called the narrow-ideal class, C nar k . Likewise we have the subgroup of totally positive units u + ⊂ u; from the ray class exact sequence [118,125] 1 50) and the number of Sp(φ(n), R)-conjugacy classes of embeddings Z n ֒→ Sp(φ(n), Z) (i.e. the number of possible sign assignments in the integral symplectic structure is Hence ker(C K and so that the number of Sp(φ(n), R) inequivalent embeddings is 2 φ(n)/2+(b−a) ≤ 2 φ(n)/2 .
From eqn.(5.52) we see that if b < a not all signatures of the symplectic structure may be realized. Indeed, from eqn. (5.49) we see that all signatures are realized iff the kernel of the natural map C nar k → C k is contained in the image of N. We mention a few known facts on a: a) (Weber) if n = 2 r , u + = u 2 , that is, a = 0; b) (Kummer, Shimura [126,127]) ifn is a prime, a = 0 if and only if the class number of K is odd; c) of course a > 0 ifn is divisible by two distinct primes.
Thus, for instance, if h K is odd andn composite = 2 r , not all signs of the symplectic form are realizable.
. This happens e.g. for n = 39, 56, 78. For these three instances the conductor is divisible by just two distinct primes, and hence (by a result of Sinnott [128]) u coincides with the group of cyclotomic units.

The sign function
From (5.25) we see that the sign function is A few examples are in order. We have seen above that if ( * * ) h K is odd and the conductorn is either a power of 2 or an odd prime (5.55) then all 2 φ(n)/2 signs are realized with Ω principal. Let us consider the first few n's which do not satisfy these condition ( * * ). The first one is 9.
Example 15. n = 9 In this case h K = Q = 1 so there are 2 φ(n)/2 = 8 distinct Sp(6, Z)conjugacy classes of order 9 elements. Since 9 is a prime power, u is the group of the the real cyclotomic units, that is, The sign table for the three elements σ a ∈ Gal(k/Q) are  (5.61) and then the signs of the reference embedding are Writing ξ = ζ + ζ −1 , we have (according to Mathematica) Writing ξ = ζ + ζ −1 , we have (according to Mathematica) so that u 1 u 2 u 3 u 4 u 5 ∈ u + and u/u + ∼ = Z φ(21)/2−1 2 ≡ Z 5 2 , and only 32 out of the possible 64 signs are actually realized. The allowed spectral invariants may be read from the above tables.

Explicit matrices
We now write explicitly the integral matrices yielding a reference embedding Z n ֒→ Sp(φ(n), Z) on which we act with the groups H or H R to get the inequivalent embeddings over Sp(φ(n), Z) and Sp(φ(n), R), respectively.
Let a be a fractional ideal of K such that Na = (η), η ∈ k × . We write k = φ(n)/2 and choose generators of the free Z-module a, a = 2k a=1 Zω a . Define the dual vector (φ a ) ∈ K 2k by the condition Tr K/Q ω aφ b = δ a b . (5.74) By definition, the dual ideal is a * = 2k a=1 Zφ a . Since a * = λa with λ purely imaginary (cfr. eqn.(5.23)), there exists Λ ∈ GL(2k, Z) such that where Λ ab is the inverse of Λ ab ; the second equation being a consequence of the first in view of (5.74)). Then 76) and the integral matrix Λ is antisymmetric with determinant 1, hence similar over the integers to the standard symplectic matrix Ω. Each ideal class [a] ∈ C − K = ker(C K N − − → C k ) yields an embedding Z n → GL(2k, Z) which is quaternionic with respect to 2 k /Q inequivalent (over Z) symplectic structures. To get the reference embedding, let us consider the trivial class in C − K ; as a representative ideal we take O ≡ (1) itself.
As a Z-basis of O we take ω x = ζ x−1 with x = 1, . . . , 2k. It is convenient to re-label the elements of this basis. Let n = p r 1 1 p r 2 2 · · · p rs s be the prime decomposition of n. Choose primitive p r i i -th roots of 1, ζ i . Then ζ = i ζ i is a primitive n-th root. By the Chinese remainder theorem, there exist integers e i , i = 1, . . . , s, such that e i = δ ij mod p  With these conventions, the action of Z n ∼ = i Z p r i i explicitly factorizes in the product of the action of the factor groups Z p r i We have to discuss separately the matrices m i associated to an odd prime and the one associated to 2 (if present). For p i odd, each m i factorizes in the matrix m (p) yielding the reference embedding Z p → GL(p − 1, Z) times the p r i −1 i -circulant where m (p) (resp. C p,r ) is the (p − 1) × (p − 1) matrix (resp. p r−1 × p r−1 ) and we used [129] Tr K/Q ζ t ≡ ℓ∈(Z/nZ) × ζ ℓt = φ(n) φ(n/(n, t)) µ(n/(n, t)), Using the reference ̺ described in appendix A.1, for s odd the symplectic matrix Λ of our reference embedding, is simply the tensor product of the reference symplectic matrices Λ i for each prime p i |n, Λ = i Λ i ; for p i odd Jordan block of eigenvalue −1, To set this matrix in the standard form Ω, it suffices to replace the above Z-basis with the Z-basis Therefore, for s odd the reference embedding Z n ∼ = s i=1 Z p r i i in Sp(φ(n), Z) is simply the tensor product of the embeddings of the factor groups Z p r i i → Sp(φ(p r i i , Z). For s even the above tensor product produces an orthogonal rather than a symplectic embedding since i Λ i is symmetric; to get an antisymmetric pairing, we multiply it by the reference imaginary unit of Lemma A.1.2. We get Then the change in the matrix Λ produced by multiplication by v is simply Replacing (1) with a non-principal fractional ideal. In a Dedekind domain O, an ideal a which is not principal may be generated by two elements, that is, has the form Let ω a a Z-basis of O and ̟ a a Z=basis of a. There are integral matrices X, Y such that The matrix M yielding the action of ζ ′ in the basis ̟ a is M = X −1 mX = Y −1 mY . The condition that Na is principal then implies that the induced Λ is defined over Z.

Reducible minimal polynomial
For completeness, we give some additional details on the case that the minimal polynomial of the elliptic element m ∈ Sp(2k, Z) is reducible over Q M(z) = Φ d (z) Φ n (z), n > d. (5.98) If n/d is not a prime power, all embeddings are conjugate to a block-diagonal one m d ⊕ m n , see Lemma 5.1.1. Suppose that n/d = p r with p prime while (d, p) = 1. We still have the block-diagonal embeddings, and all embeddings are conjugate to block-diagonal ones over Q.
Thus there is an element R ∈ Sp(2k, Q) which splits the Z[m]-module V and the symplectic structure R −1 mR acts as as multiplication by ζ n × ζ d , R t Ω R = J n ⊕ J d . for certain coefficients a s .

Tables of dimensions for small k
In this section we present some sample tables of both dimensions and dimension k-tuples for small values of the rank k.

Dimension k-tuples: USE OF THE TABLES
Tables of all ALLOWED dimension k-tuples become quite long pretty soon as we increase k. For conciseness we list only the strongly regular dimension k-tuples from which one can infer all allowed k-tuples. The tables of strongly regular k-tuples contain the basic informations needed to check whether a proposed dimension k-tuple {∆ 1 , · · · , ∆ k } is consistent or not with the arguments of the present paper. By definition, a strongly regular dimension k-tuples is a set of dimensions as computed using eqn.(4.51) along a normal ray M * ⊂ M with strongly regular monodromy m * (i.e. such that the characteristic polynomial of m * is square-free). In turn, the strongly regular monodromies may be distinguished in two kinds: the ones consistent with a principal polarization, m * ∈ Sp(2k, Z), and those associated to a suitable non principal polarization, m * ∈ S(Ω) Z (det Ω = 1). The complete list of all allowed dimension k-tuples is then recovered from the tables of the strongly regular ones by the algorithm described in section 4.5.1 which we review in the next subsection. In tables 9, 10, and 11 we present the list of the strongly regular dimension k-tuples for k = 3 and k = 4. For k = 3 we list both the principal (table 9) and non-principal 3-tuples (table 10), while for k = 4 we limit ourselves to the principal ones (table 11).

The algorithm to check admissibility of a given dimension k-tuple
Suppose we are given a would-be dimension k-tuple, {∆ 1 , · · · , ∆ k }, written in non-increasing order ∆ i ≥ ∆ i+1 , and we wish to determine whether it is consistent with the geometric conditions discussed in the present paper. In order to answer the question, we focus on the k normal rays in the Coulomb branch M i = u j = 0 for i = j ⊂ M, i = 1, 2, . . . , k. (6.4) The monodromy along M i , m i , may be either regular or irregular. From a regular monodromy m i we may read all k dimensions using the Universal Formula (4.51). Experience with explicit examples (e.g. the ones having constant period map or those engineered in F -theory) suggests that the ray M 1 associated with the chiral operator u 1 of largest dimension ∆ 1 = max i ∆ i always has a regular monodromy. 51 However m 1 may be just weakly regular; in this case eqn.(4.51) still applies but the corresponding k-tuple is not listed in the tables, and we need to follow the procedure described below. Recall that we have defined the regular rank k reg, 1 ≤ k of the monodromy matrix m 1 to be one-half the degree of the square-free part of its characteristic polynomial.
To run the algorithm, one begins writing the rational number ∆ 1 in minimal form, ∆ 1 ≡ n 1 /r 1 with (n 1 , r 1 ) = 1; we may assume n 1 > 2 by the argument in §.4.3.2. Let ℓ 1 be the largest integer such that ∆ ℓ 1 = ∆ 1 , that is, the multiplicity of the largest dimension. ∆ 1 is a new-dimension in some rank k 1 = 1 2 φ(n 1 ) and k 1 ℓ 1 ≤ k. If k 1 = k, the monodromy m 1 is automatically strongly regular, and hence the full k-tuple should appear in the tables of strongly regular k-tuples under the characteristic polynomial (C.P.) Φ n 1 . More generally, if k 1 ℓ 1 = k, the dimension k-tuple is the union of ℓ 1 strongly regular dimension k 1 -tuples for Φ n 1 . If ℓ 1 k 1 < k, the k-tuple is the union of ℓ 1 strongly regular k 1 -tuples and a residual (k − ℓ 1 k 1 )-tuple {∆ i } i∈A 1 (A 1 ⊂ {1, . . . , k}). Under the assumption that m 1 is (weakly) regular we have If our candidate k-tuple satisfies all these requirements at the ray M 1 , we next consider consistency conditions at the rays M 2 , M 3 , etc. along the lines of §.4.3.7. The arguments are parallel to the ones for M 1 except that now we do not expect the monodromy to be fully regular (not even in the weak sense) so that only a sub k reg, i -tuple of dimensions is fixed at each ray. This still yields non trivial consistency conditions as in the examples of §.4.3.7.
The algorithm is longer to explain than to run. We illustrate the method in a typical example.
Example 20. In rank 4 the following (non strongly regular) 4-tuple exists 52 14, 10, 8, 4 . (6.7) Let us apply the procedure to it. The largest dimension, ∆ 1 = 14 has multiplicity 1 and is a rank 3 new-dimension (see table 8); then 3 out of the 4 dimensions (6.7) should form a strongly regular 3-tuple to be found in

Constructions of the lists
The procedure to determine the lists is the one explained in sections 4, 5 which we sum up here. We start by computing ρ in the case of a cyclic group with an indecomposable characteristic polynomial Φ(z). We start the algorithm with ρ temp := 1 Φ(ξ) ′ . (6.10) If ρ temp is purely imaginary, then ρ = ρ temp , otherwise we find the unit u such that uρ temp is purely imaginary. 53 Once ρ is computed, we get the initial signs for each element σ i of the Galois group associated to Φ(z). From the positive signs we compute the dimension tuple.
All the other signs can be computed exploiting the fundamental units of the cyclotomic field (using Mathematica or PARI [130] when the former software fails): to each generator of the unit group we associate a sign tuple (the signs of the Galois elements). From the signs of ρ, it is easy to compute all possible signs by repeatedly multiplying the sings amongst one another. This is the algorithm to get all dimensions tuples. The embedding may be obtained by a direct sum of lower order cyclic elements. In this case, we get the product of cyclotomic polynomials Φ d 1 ,d 2 ,··· ,ds (z) := Φ d 1 (z) · · · Φ ds (z). (6.11) The procedure is similar to the above: we first compute all the signs separately for each factor -using the above algorithm -and then we put them all together to compute the full list of dimensions. Particular attention must be payed to those products in which the ratio of the conductors of the cyclotomic factors is a (power of a) prime number, e.g. Φ 12 Φ 4 in the rank 3 case. In this situation, the cyclic group representation is no longer irreducible, and thus the theory of Dedekind domains of rank 1 cannot be applied: ρ is no longer a number but rather a matrix. Since this branch of number theory is not well developed, we preferred the explicit construction of the symplectic matrices Λ's. We only consider the action of the group H = u/NU on the initial embedding, whose signs are still defined by those of ρ (the corresponding matrix shall be called m and is given in subsection 5.3). The action of H only modifies the symplectic matrix: the signs of the characteristic polynomial of the new symplectic matrix, evaluated at the cyclotomic roots, give the sign changes to be applied to the original signs of ρ. Hence, the problem is to find the group H. It is a very had task to find this group: fortunately we know how it acts on the symplectic matrices: where p v is a polynomial with integer coefficients of maximal rank φ(n) − 1. Thus, we can write an algorithm that looks for as many p v 's as possible: every time we find one we check whether Λ v is principal (i.e. of unit determinant) and symplectic; once these two conditions are matched, we add the sign tuple to our results. In the end, we compute all the dimensions starting from the signs of ρ and we multiply the signs with those explicitly found by our algorithm. If we find all possible signs, then the final result is definitive. In general, if we do not find all signs, we can only state our results with high confidence. In table 9 we list the fully regular 3-tuples for k = 3. The first column yields the characteristic polynomial of the embedding and the second column the corresponding dimension 3-tuples. Table 11 contains the fully regular 4-tuples for k = 4.