Tinkertoys for the Twisted D-Series

We study 4D N=2 superconformal field theories that arise from the compactification of 6D N=(2,0) theories of type D_N on a Riemann surface, in the presence of punctures twisted by a Z_2 outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by M_{g,n}, but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D_4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D_5 and D_6 theories. We find realizations of Spin(8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing beta-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.


Introduction
The study of four-dimensional N = 2 superconformal field theories (SCFTs) has benefited considerably in recent years from the construction of a class of such theories (sometimes called class S) as compactifications of the mysterious 6D (2,0) SCFTs on Riemann surfaces with a partial twist [1,2,3,4,5,6,7,8,9,10]. The realization of many Lagrangian theories whose Seiberg-Witten curves were previously unknown, the discovery of a multitude of interacting SCFTs that generalize the Minahan-Nemeschansky E N theories [11,12], and the understanding of S-duality [13] are just a few of the remarkable features of this class of theories.
The key ingredient, greatly expanding the class of 4D theories one can obtain, is the possibility of adding codimension-two defects of the (2,0) theories localized at points on the Riemann surface C. Depending on our choice of these punctures on C, we get different 4D N = 2 SCFTs. A yet-wider class of theories can be obtained by including outerautomorphism twists [9] on C, such that, when traversing an incontractible cycle on C (either going around a handle of C, or circling a puncture on C) the ADE Lie algebra comes back to itself up to an outer-automorphism. In particular, this introduces a new class of codimension-two defects, which we refer to as "twisted punctures", and whose local properties were studied in [8].
In [3], we started our program of classifying the 4D N=2 SCFTs that arise from the 6D (2,0) theories by focusing on the A N −1 series. In that paper, we constructed the possible "fixtures" (three-punctured spheres) and the cylinders that connect them, which are the basic building blocks for any pair-of-pants decomposition of a Riemann surface. In [4] we carried out a similar program for the D N theories, and in [14] we studied the SCFTs that arise from incorporating outer-automorphism twists in the A 2N −1 theories. In this paper, we want to continue our classification program by adding outer-automorphism twists to the theories of type D N . Preliminary studies of the twisted D N series were made by Tachikawa in [9,10].
The D N Dynkin diagram is invariant under a Z 2 outer automorphism group. Correspondingly, the possible twists are classified by giving an element γ ∈ H 1 (C − {p i }, Z 2 ). The forgetful map, which "forgets" the puncture, p, gives an inclusion , then we say that the puncture at p is twisted (otherwise, untwisted). (For the D 4 theory, the Z 2 enhances to a non-abelian S 3 group. The study of the 4D N = 2 SCFTs that arise from such enhancement is work in progress.) For a given puncture, we explain how to compute all the local properties that contribute to determining the 4D N = 2 SCFT. Among these, are the contribution to the graded Coulomb branch dimensions, the global symmetry group, flavour-current central charges, the conformal-anomaly central charges (a, c), and the "pole structure" and "constraints", which determine the contribution to the Seiberg-Witten curve. From this information, it is possible to determine gauge groups, hypermultiplet matter representations, and other properties.
As an application of our results, we are able to find realizations of Spin(8) gauge theory with matter in the 6(8 v ), or with matter in the 5(8 v ) + 1(8 s ). These two cases, of vanishing β-function for Spin (8), were the ones that were not captured by the untwisted sector of the D N series. Similarly, for Spin (7) gauge theory, we find the theory with matter in the 5 (7), and in the 1(8) + 4 (7); the other combinations with vanishing β-function were already found in the untwisted sector of the D N series. We also study various realizations of Sp(N ) gauge theory, including Sp(3) with matter in the 11 2 (6) + 1 2 (14 ) and in the 3(6) + 1 (14 ), where the 14 is the 3-index traceless antisymmetric tensor representation.

The Z 2 -twisted D N Theory
The Coulomb branch geometry of the 4D N = 2 compactification [1,2] of the 6D N = (2, 0) theories of type D N is governed by the Hitchin equations on C with gauge algebra so(2N ). In particular, the Seiberg-Witten curve Σ is a branched cover of C described by the spectral curve [9], where Φ is the so(2N )-valued Higgs field, while the k-differentials φ k (k = 2, 4, 6, . . . , 2N −2) and the Pfaffian N -differentialφ are associated with the Casimirs of the D N Lie algebra. In the rest of the paper, N will always stand for the rank of D N .
Introducing punctures on C corresponds to imposing local boundary conditions on the Hitchin fields. We consider untwisted and twisted punctures under the action of the Z 2 outerautomorphism group of the so(2N ) Lie algebra. Untwisted punctures are labeled by sl (2) embeddings in so(2N ), or, equivalently, by nilpotent orbits in so(2N ), or by D-partitions 1 of 2N . Instead of a compact curve, C, consider a semi-infinite cigar, with the puncture at the tip. Reducing along the circle action, we get 5D SYM on a half-space, with a Nahm-type boundary condition of the sort studied by Gaiotto and Witten in [15]. For that reason, we call the D-partition that labels the untwisted puncture the Nahm pole.
To describe the local Hitchin boundary condition for an untwisted puncture with Nahmpole D-partition p, one must recall the Spaltenstein map 2 , which takes p into a new Dpartition d(p), called the Hitchin pole of the puncture 3 . Then, the local boundary condition corresponding to p is where X is an element of the nilpotent orbit 4 associated to d(p), and so(2N ) above denotes a generic regular function in z valued in so(2N ).
On the other hand, we have a sector of twisted punctures, with monodromy given by the action of the nontrivial element o of the Z 2 outer automorphism group of D N . The action of o splits so(2N ) as where so(2N − 1) and o −1 are the eigenspaces with eigenvalues +1 and -1, respectively. The action of o on the k-differentials is also quite simple: Following [8], the twisted punctures of the D N series are labeled by embeddings of sl (2) in sp(N − 1) (the Langlands dual of so(2N − 1)), or, equivalently, by nilpotent orbits in sp(N − 1), or by C-partitions 5 of 2N − 2.
To describe the local boundary condition for a twisted puncture, we need to recall the relevant Spaltenstein map 6 . This is a map d that takes a C-partition p of 2N − 2 into a B-partition d(p) of 2N − 1. A B-partition of 2N − 1 labels an sl(2) embedding in so(2N − 1), or equivalently a nilpotent orbit in so(2N − 1). So, in our nomenclature, the Nahm pole p of a twisted puncture is a C-partition of 2N − 2, and its Hitchin pole 7 is a B-partition d(p) of 2N − 1. The local boundary condition for the Higgs field is then: Here X is an element of the so(2N − 1) nilpotent orbit d(p), while o −1 and so(2N − 1) in the equation above denote generic regular functions in z valued in these linear spaces, respectively. 4 Using a nilpotent element X in this equation amounts to writing the local boundary condition in the absence of mass deformations. The mass-deformed boundary condition involves semisimple (diagonalizable) elements of so(2N ), whose eigenvalues take values in the Cartan subalgebra of the flavour Lie algebra for the puncture. For the untwisted A series, a recipe for mass-deformed local boundary conditions was given in [5]. A general prescription is given in Sec. 2.4 of [8]. 5 A C-partition of 2N is a partition of 2N where each odd part appears with even multiplicity. A Bpartition of 2N − 1 is a partition of 2N − 1 where each even part appears with even multiplicity. 6 This Spaltenstein map consists in adding a part "1" to a C-partition p, taking the transpose, and then doing a B-collapse. The result is always a B-partition. The "B-collapse" is discussed in [8] and in [16]. 7 Again, when the Nahm pole p is non-special, the complete Hitchin pole information is not just d(p), but a pair (d(p), C), with C the Sommers-Achar group [8].

Global Symmetry Group and Central Charges
The local properties of a puncture that we list in our tables are the pole structure (with constraints), the flavour group (with flavour-current central charges for each simple factor) and the contributions (δn h , δn v ) to, respectively, the effective number of hypermultiplets and vector multiplets (or, equivalently, to the conformal-anomaly central charges (a, c)). We will discuss how to compute pole structures and constraints in §2.1.2,2.1.3. Here we will briefly focus on the other properties.

Pole Structures
The pole structure of a puncture is the set of leading pole orders {p 2 , p 4 , p 6 , . . . , p 2N −2 ;p} in the expansion of the k-differentials φ k (z) (k = 2, 4, 6, . . . , 2N − 2) and the Pfaffianφ(z) around the position of the puncture on C. Knowing the pole structures of the various punctures allows us to write down the Seiberg-Witten curve (2.1) of a theory. The pole orders are all integers, except forp in a twisted puncture, which must be a half-integer because of the monodromy (2.2).
We already saw in [4] how to read off the pole structure of an untwisted puncture from its Hitchin-pole D-partition p. Basically, regard p as a partition in the untwisted A-series, use the procedure to write down the pole structure [3], and discard the pole orders that would correspond to φ k with odd k. Finally, divide the pole order p 2N of φ 2N by two, to obtain the pole orderp of the Pfaffianφ. p 2N will always be even, so thatp will come out to be an integer, as expected for an untwisted puncture.
To compute the pole structure of a twisted puncture, we use its Hitchin B-partition p. Simply, add 1 to the first (i.e., the largest) part in p, and use the same procedure to compute the pole structure as for an untwisted D-series puncture. Notice that upon adding 1 to the largest part, the B-partition becomes a partition of 2N , and one can show that the pole order p 2N of φ 2N is always odd, so that the pole orderp of the Pfaffian is a half-integer, as it should be.

Constraints
In the untwisted D-series, punctures featured "constraints", which are either: 1) relations among leading coefficients in the k-differentials ("c-constraints"); or 2) expressions defining new parameters a (k) of scaling dimension k as, roughly, the square roots of a leading coefficient c (2k) of dimension 2k ("a-constraints"). Both kinds of constraints affect the counting of graded Coulomb branch dimensions of the theory, as well as the Seiberg-Witten curve. As expected, we find a-constraints and c-constraints also in the twisted sector. The pole structure and the constraints provide a "fingerprint" [17] that allows us to identify the puncture uniquely. Let us briefly review our nomenclature. For a puncture at z = 0, we consider the coefficients c (2k) l andc l of the leading singularities in the expansion in z of the 2k-differentials (2k = 2, 4, . . . , 2N − 2) and the Pfaffianφ, respectively, . . denotes less singular terms. (The pole orders l above are, of course, the same as those in the pole structure, so we have l = p 2k or l =p, respectively; in this subsection we just write l to keep expressions simple.) An a-constraint of scaling dimension 2k is an expression linear in c (2k) l that defines (up to sign) a new parameter a (k) where . . . stands for a polynomial in leading coefficients (of dimension less than 2k) as well as new coefficients a (j ) l (which would themselves be defined by other a-constraints). This polynomial is homogeneous in dimension and pole order, i.e., in every term in the polynomial, the sum of the scaling dimensions of every factor must be 2k, and the sum of pole orders must be l. The existence of an a-constraint implies that, in counting graded Coulomb branch dimensions, a parameter of scaling dimension 2k is to be replaced by one of dimension k.
A c-constraint of dimension 2k is an expression linear in c (2k) l , which relates it to other leading coefficients, and perhaps also to new parameters a j l defined by a-constraints, where, again, the ellipsis denotes a homogeneous polynomial in leading coefficients and new parameters. For even N , if the puncture is very-even, a "very-even" c-constraint, which is linear in the leading coefficients of both φ N and the Pfaffian, may appear, Unlike an a-constraint, a c-constraint does not define any new parameters; it simply tells us that c (2k) l (or, say, c (N ) for a very-even c-constraint) is not independent, and so it should not be considered when counting Coulomb branch dimensions.
Finally, at every scaling dimension 2k, we find at most one constraint, which can be either an a-constraint or a c-constraint.
Below, we present algorithms to compute the scaling dimensions 2k at which a-constraints and c-constraints appear for a given puncture. This information is enough to compute the local contribution to the graded Coulomb branch dimensions.
Untwisted punctures Let p be the Nahm pole D-partition of an untwisted puncture. Also, let q = {q 1 , q 2 , . . . } be the transpose partition, and s = {s 1 , s 2 , . . . } the sequence of partial sums of q (s i = q 1 + q 2 + · · · + q i ). Below, s 1 denotes the first element of s, and p last , the last element of the D-partition p. (By the conditions that define a D-partition, s 1 is always an even number.) Then, an a-constraint of dimension 2k exists if the following conditions are met: 1. 2k belongs to s, say, s j = 2k.
2. j is even.
3. If s j is a multiple of s 1 , say, s j = rs 1 , one has r ≥ 2 p last 2 + 1.

s j is not the last element of s.
On the other hand, a c-constraint of scaling dimension 2k exists if the following conditions are met: 1. 2k belongs to s, say, s j = 2k.
2. If j is even, one has that: a) s j is a multiple of s 1 , say, s j = rs 1 ; b) p last 2 + 1 ≤ r ≤ 2 p last 2 ; c) s j is not the last element of s.
3. If j is odd, one has that: a) s j is neither the first nor the last element of s; b) both s j−1 and s j+1 are even; c) s j = s j−1 +s j+1 2 ; d) if s j is divisible by s 1 , say, s j = rs 1 , one has r ≥ p last 2 + 1.
Finally, if p is very even, an additional, "very-even", c-constraint exists at 2k = N if N belongs to s and N = s 1 p last 2 . As already mentioned, this very-even c-constraint is linear in both leading coefficients c Twisted punctures Suppose we have a twisted puncture labeled by the Nahm-pole C-partition p. Let q be the transpose partition, and s the sequence of partial sums of q. It is convenient to define another sequence s , obtained by adding 2 to every element in s. (As a check, the last element of s must be 2N .) Let s = {s 1 , s 2 , . . . }.
Then, an a-constraint of scaling dimension 2k exists if the following conditions are met: 1. 2k belongs to s , say, s j = 2k.
3. s j is not the last element of s .
On the other hand, a c-constraint of scaling dimension 2k exists if the following conditions are met: 1. 2k belongs to s , say, s j = 2k.
2. j is even.
3. s j is not the last element of s 4. Both s j−1 and s j+1 are even, and s j = Constraint structure The constraints of twisted punctures are very simple. c-constraints are always "cross-terms" between a-constraints, or between an a-constraint and the Pfaffian (where φ 2N =φ 2 is seen as another "a-constraint"). As a schematic example, c (k+m) below is a cross-term for the "squares" at dimensions 2k and 2m: In an actual example, k+m would always turn out to be even). a-constraints also generically contain cross-terms, in addition to the quadratic term in the new parameter. Many examples can be found in the Tables. The constraints of untwisted punctures are slightly more complicated, but they resemble very much the constraints of twisted punctures in the A 2N −1 series [14], so we refrain from repeating the details. To be brief, there is a sequence of c-constraints (illustrated below in an example), all related to each other, and which are associated to the first terms in the set of partial sums s. c-constraints outside this sequence are simply cross-terms between a-constraints and/or the Pfaffian, as in (2.4). For a very-even puncture, the very-even cconstraint, if it exists, becomes part of the sequence just mentioned. As usual, a-constraints can include cross-terms in addition to the quadratic term that defines the new parameter.
Let us discuss the constraints of a D 6 very-even puncture, [6 2 ]. In this case, q = [2 6 ] and s = [2,4,6,8,10,12]. Also, p last = 6 and s 1 = 2. So, there are c-constraints at 2k = rs 1 with 4 ≤ r ≤ 6, that is, at 2k = 8, 10. There is also a very-even c-constraint (at 2k = 6). All c-constraints in this case constitute the sequence mentioned in the previous paragraph. There are no a-constraints. We can also compute the pole structure to be {1, 2, 3, 4, 5; 6}. Let us see the structure of these c-constraints by writing: . Flipping the sign ofc 3 switches between the constraints for the red and the blue versions of this puncture.

Collisions
When two punctures collide, a new puncture appears. This process can be described at the level of the Higgs field, using the local boundary conditions discussed in §2, or at the level of the k-differentials, using the pole structures and the constraints of §2.1.2 and §2.1.3. Of course, both mechanisms are quite related, because the k-differentials are, essentially, the trace invariants of the Higgs field. These procedures are analogous to those for the twisted A 2N −1 series described in [14].
Let us start by discussing collisions using the Higgs field. Consider two untwisted punctures at z = 0 and z = x on a plane. The respective local boundary conditions are: where X 1 and X 2 are representatives of the respective Hitchin-pole orbits for the punctures. Then, in the collision limit, x → 0, a new untwisted puncture appears at z = 0, Here, X 1 + X 2 is an element of the mass-deformed Hitchin-pole orbit for the new puncture, and the mass deformations correspond to the VEVs of the decoupled gauge group. Taking the mass deformations to vanish, X 1 + X 2 becomes the Hitchin-pole nilpotent orbit for the new puncture. The fact that the new residue is X 1 + X 2 also follows from the residue theorem applied to the three-punctured sphere that appears in the degeneration limit; another derivation ensues from an explicit ansatz for the Higgs field on the plane with two punctures [14], where the limit x → 0 can be taken. Now consider an untwisted and a twisted puncture, at z = 0 and z = x, respectively. The respective local boundary conditions are: Then, the local boundary condition for the new twisted puncture is: Finally, consider two twisted punctures at z = 0 and z = x, Then, the local boundary condition for the new untwisted puncture is: where o −1 denotes a generic element in such space. The procedure to collide punctures using k-differentials is explained in [14] for the case of the twisted A 2N −1 series. The discussion is entirely analogous, so we leave the details to that paper. Here we will just give an example of how to use it.

Gauge Couplings
Consider an N = 2 supersymmetric gauge theory, with simple gauge group, G, and matter content chosen so that the β-function vanishes. This gives rise to a family of SCFTs, parametrized by A rich class of (though not all) such theories can be realized as compactifications of the (2, 0) theory on a sphere with four untwisted punctures. If the four puncture are distinct, then the S-duality group, Γ(2) ⊂ P SL(2, Z), is generated by The fundamental domain for Γ(2) is isomorphic to M 0,4 CP 1 . In particular, the coordinate on the complex plane, f , is given by 9 9 Our θ-function conventions are Since Γ(2) is index-6 in P SL(2, Z), the generators of the latter group act on M 0,4 as These generate an S 3 action on M 0,4 , as depicted in the figure The points, {0, 1, ∞}, of the compactification divisor, are fixed points with stabilizer group Z 2 . The points {−1, 1/2, 2} are also fixed points with stabilizer group Z 2 . Finally, the points Of course, while the j-invariant is invariant under the full P SL(2, Z), the physics generically is not If two of the punctures are identical, then τ → −1/τ leaves the physics unchanged. The S-duality group is Γ 0 (2) ⊂ P SL(2, Z), generated by T 2 : τ → τ + 2 and S : τ → −1/τ , whose fundamental domain is the Z 2 quotient of M 0,4 by f → 1/f . The physics at f = 0 and at f = ∞ are both that of a weakly-coupled G gauge theory. The other boundary point, f = 1, and the interior point, f = −1 are fixed-points of the Z 2 action.
If three of the punctures (or all four) are identical, then the S-duality group is the full P SL(2, Z), the physics at all three boundary points is that of a weakly-coupled G-gauge theory and the fundamental domain is just the shaded region in the figure.
How this picture gets modified, in the presence of twisted punctures, will be one of our main themes in this paper.

Very-even Punctures
In the A 2N −1 series, the outer automorphism twists acted trivially on the set of nilpotent orbits. So the identities of the untwisted punctures were unaffected by the introduction of twisted punctures. By contrast, in the D N series (for N even), the outer automorphism twists act by exchanging the "red" and "blue" very-even punctures. Dragging an untwisted very-even puncture around a twisted puncture turns it from red to blue, or vice-versa.
To illustrate the phenomenon, let us look at an example in the twisted D 4 theory.
Here, it is useful to recall [4] that the very-even puncture 10 has only one constraint, which is a very-even c-constraint, 10 As in [3,4,14], a Nahm-pole partition p is represented by a Young diagram such that the column heights are equal to the parts of p. (So is the puncture with Nahm pole D-partition [2 4 ].) In this paper we do not use Young diagrams to represent Hitchin-pole partitions.
where the top (bottom) sign corresponds to a red (blue) puncture.
The Higgs field (with Coulomb branch parameters u 2 , u 4 ,ũ, u 6 ) yields the differentials The powers of z ij ≡ z i − z j have been introduced to make the above expressions Möbiusinvariant 11 , and hence well-defined on the moduli space. However, the (unavoidable) squareroots mean that moduli space is, itself, a double-cover (in fact, a 4-fold cover, but the SW geometry factors through a Z 2 quotient) of the moduli space of the 4-punctured sphere.
Whether a very-even puncture is red or blue depends on the relative sign of the residues of the cubic poles of φ 4 (z) andφ(z) at the location of the puncture. But the square-roots are such that if we drag the very-even puncture (say, the one located at z 3 ) around one of the twisted punctures (say, the one located at z 1 ), the relative sign changes, indicating that the puncture has changed from red to blue, or vice versa.
Since the formulae are a little bit formidable-looking in their fully Möbius-invariant form, it helps to fix the Möbius invariance by setting The expressions for φ 4 (z),φ(z) (which are all we need for the present discussion) simplify to To minimize the number of ensuing branch cuts, we have chosen not to preserve the obvious z 3 ↔ z 4 symmetry. We can restore it by redefining the Coulomb branch parameter u =ũ z 13 z 24 z 12 z 34 The resulting theory lives naturally on the 4-fold branched cover of M 0,4 .
Dragging the point z 3 = w 2 around the origin changes the sign of w in the above expressions. This changes the relative sign of the residues of φ 4 andφ at z = w 2 , whilst preserving the relative sign of the residues at z = 1.
Of course, the Seiberg-Witten geometry is invariant under the operation of simultaneously flipping all of the colours of all of the very-even punctures. This gives a Z 2 which acts freely on the gauge theory moduli space. We will often find it useful to work on the quotient, fixing the colour of one of the very-even punctures.
Having seen the phenomenon is global example, let us recover the same result, working locally on the plane, with the Higgs field itself (rather than the gauge-invariant kdifferentials). Consider a very-even Higgs-field residue B ∈ so(2N ), which belongs to a, say, red nilpotent orbit. We can write in the other (blue) nilpotent orbit. This map defines an isomorphism between the elements of the red and the blue nilpotent orbits. Now suppose that the twisted puncture (with residue A ∈ so(2N − 1) is at z = 0 and the very-even puncture (with residue B ∈ so(2N )) is at z = x. Then, the Higgs field for this system is: where D is a generic element in o −1 , and the . . . denote regular terms. The factor of x 1/2 is necessary to make Φ well-defined as a one-form. Then, x parametrizes the distance between the very-even puncture and the twisted puncture, and if x circles the origin, x 1/2 → −x 1/2 , it enforces B| o −1 → −B| o −1 , so our red puncture becomes blue, or vice versa.

Atypical Punctures
As an application of the formulas in §2.1, let us find the series of punctures with contribution n 2 = 2. We will call these "atypical punctures", as they give rise to theories where the number of simple factors in the gauge group is not equal to the dimension of the moduli space of the punctured Riemann surface, C. We have seen this phenomenon already in the twisted A 2N −1 series [14]. From our rules for a-constraints, it is easy to see that there are no untwisted atypical punctures, and that for a twisted puncture to be atypical, its Nahm pole C-partition must consist of exactly two parts. Hence, the atypical punctures are if N is even.
These arise, respectively, as the coincident limit of (for N even) Normally, the OPE of two (regular) punctures, p and p , yields a third (regular) puncture, p , coupled to a gauge theory, (X, H), where • The gauge group, H, is a subgroup of the global symmetry group of p .
• In the coincident limit, the gauge coupling of H goes to zero.
Here, when p is atypical, the would-be gauge theory is empty: (X, H) = (∅, ∅). Instead, the theory with an insertion of p has one more simple factor in the gauge group than the "expected" 3g − 3 + n.
For a surface, C, with n punctures, m of which are atypical, the number of simple factors in the gauge group is 3g − 3 + n + m. "Resolving" each atypical puncture by the pair of punctures, above, yields a surface with n + m punctures and the moduli space of the gauge theory is a branched cover of M g,n+m . In contrast to the usual case, where each component of the boundary of the moduli space corresponds to one simple factor in the gauge group becoming weakly-coupled, the boundaries of M g,n+m , where an atypical puncture arises in the OPE, do not typically correspond to any gauge coupling becoming weak (that is, under the branched covering, they are the image of loci in the interior of the gauge theory moduli space).

Gauge Theory Fixtures
In particular, for n = 3, m = 1 (or 2), we have a "gauge theory fixture." Resolving the atypical puncture yields a gauge theory moduli space which is branched cover of M 0,4 . We may well ask, "Where, in the gauge theory moduli space, have we landed, in the coincident limit which yields the atypical puncture?" The answer is that we are at the interior point, "f (τ ) = −1", though the mechanics of how this happens varies between the cases.
Let us resolve We have parametrized M 0,4 by x, but the gauge theory moduli space is a branched cover, parametrized by w, with The gauge coupling so that f = 0 and f = ∞ both map to x = 1, while f = 1 maps to x = ∞. Our gauge-theory fixture is whatever lies over the point x = 0. From (2.6), this is the interior point, f (τ ) = −1, of the gauge theory moduli space.
As an example, let us consider the D 4 gauge theory fixture whose resolution is Actually, since we have two very-even punctures, the full moduli space is a 4-sheeted cover of M 0,4 . The SW geometry is invariant under simultaneously flipping the colours of both punctures, so we can consistently work on the quotient by that Z 2 , and take the colour of the puncture to be red. SU (4) gauge theory, with matter in the 1(6) + 4(4) was studied in [3]. Near f (τ ) = 0, the weakly-coupled description is the Lagrangian field theory. Near f (τ ) = 1, the weakly-coupled description is an SU (2) gauging of the SU (8) 8 × SU (2) 6 SCFT, R 0,4 . Near f (τ ) = ∞, the weakly-coupled description is SU (3), with two hypermultiplets in the fundamental, coupled to the (E 7 ) 8 SCFT.
In the present case, the f → 1 theory arises as x → ∞ SU(2) , SU (2) Having fixed the behaviour of f over this two-sheeted cover of M 0,4 , by reproducing the correct asymptotics as x → 1 and x → ∞, we can now take x → 0 ∅ , ∅ ) ( gauge theory fixture empty and recover that the gauge theory fixture is the aforementioned SU (4) gauge theory at f (τ ) = −1.

Gauge Theory Fixtures with Two Atypical Punctures
When we resolve the gauge theory fixtures with two atypical punctures, we obtain a branched covering of M 0,5 .
The geometry of M 0,5 , and the relevant branched covering thereof, were discussed in detail in section 5.1.2 of [14]. Here, we will simply borrow the relevant results.
The (compactified) M 0,5 is a rational surface. The boundary divisor consists of ten (−1)curves (CP 1 s with normal bundle O(−1)). We label these curves as D ij , corresponding to the locus where the punctures p i and p j collide. The D ij , in turn, intersect in 15 points.
The moduli space of the (2, 0) compactification is a branched covering,M → M 0,5 , which is branched over the boundary divisor.
The D 4 gauge theory fixture is an Sp(2) × SU (2) gauge theory, with matter in the 6(4, 1) + 4(1, 2), with gauge couplings (f Sp(2) , f SU (2) ) = (−1, −1). Resolving the atypical punctures, we have a 5-punctured sphere, Since the resolution has two very-even punctures,M is an 8-sheeted branched cover of M 0,5 . However since the gauge couplings (and the rest of the physics) are invariant under simultaneously flipping the colours of both very-even punctures, we can pass to the quotient, X =M/Z 2 , and it is the geometry of 4-sheeted branched cover, X → M 0,5 , that was studied in detail in [14]. Meromorphic functions on M 0,5 are rational functions of the cross-ratios s 1 = z 13 z 25 z 15 z 23 , s 2 = z 14 z 25 z 15 z 24 X is a branched 4-fold cover of M 0,5 , whose ring of meromorphic functions is generated by rational functions of w 1 , w 2 w 2 1 = s 1 , w 2 2 = s 2 The gauge couplings are meromorphic functions on X, given by There is a natural action of the dihedral group, D 4 , on X. The Z 2 ×Z 2 subgroup is generated by the deck transformations, which act on the gauge couplings as Both α and β change the relative colour of the two very-even punctures. The additional generator of D 4 , γ :, w 1 ↔ w 2 acts as S-duality for the Sp (2), At the boundary, various sheets come together, and the behaviour of the gauge couplings is • Over D 15 and D 25 , both couplings go to f = 1, but the ratio • Over D 35 , both couplings are weak (f = 0 or f = ∞), but the ratio f Sp (2) f SU (2) is arbitrary.
• Over D 12 , one coupling is weak (f = 0 or ∞), while the other is arbitrary.
• Over D 34 , one coupling is f = 1, while the other is arbitrary.
Over the intersections of these divisors, we see the various S-duality frames of the gauge theory.

. Atypical Degenerations and Ramification
Once we introduce outer-automorphism twists, the moduli space of the gauge theory no longer coincides with M g,n , the moduli space of punctured curves. As we saw, in §2.5.1, even the dimensions don't agree, until we "resolve" each atypical puncture, replacing M g,n by M g,n+m (for m atypical punctures). Even then, the moduli space of the gauge theory is a branched covering of M g,n+m , branched over various components of the boundary. Over a generic point on "most" of the components of the boundary, the covering is unramified, and the gauge couplings behave "normally": one (and only one) gauge coupling becomes weak at that irreducible component of the boundary. Here, we would like to catalogue the exceptions: those components of the boundary where • the covering is ramified • an "unexpected" (either 0 or 2, in the cases at hand) number of gauge couplings become weak • both Let us denote, by D p 1 ,p 2 ,...p l , the component of the boundary of M g,n+m where the punctures p 1 , p 2 , . . . p l collide, bubbling off an (l + 1)-punctured sphere. All of our exceptional cases will involve either D p 1 ,p 2 or D p 1 ,p 2 ,p 3 .
D T,V The first source of ramification, as we saw in §2.4, is that the outer automorphism changes the colour of a very even puncture from red to blue and vice versa. In general, this changes the physics of the gauge theory. So, for a theory with v very-even punctures, we get a 2 v sheeted cover of the moduli space of curves, ramified (with ramification index 2) over D T,V where "T " denotes any twisted-sector puncture and "V " represents any very-even. As already noted, simultaneously changing the colour of all of the very-even punctures leads to isomorphic physics so we can (and usually will) pass to the Z 2 quotient. Generically, the gauge couplings behave "normally," with one gauge coupling becoming weak at D T,V .
When N is even, there is one such collision where, in addition to ramification, no gauge coupling becomes weak. Instead, the two punctures fuse (in non-singular fashion) into an atypical puncture.

Computing the Index in the Hall-Littlewood Limit
Each puncture has a "manifest" global symmetry associated to it. The global symmetry group of the SCFT associated to a fixture contains the product of the "manifest" global symmetry groups, associated to each of the punctures, as a subgroup. But, in general, it is larger. Here, we will outline how to use the superconformal index [18,19,20,21] to determine the global symmetry group of the fixture and (in the case of a mixed fixture) the number of free hypermultiplets that it contains.
The prescription to compute the superconformal index of an interacting SCFT defined by a D N -series fixture was given in [22]. For a D N Z 2 -twisted sector fixture with punctures (Λ 1 ,Λ 2 , Λ 3 ), whereΛ denotes a twisted puncture and Λ an untwisted puncture, the index is given by 12 The various elements of this formula are summarized below. Detailed explanations can be found in [22]: • A(τ ) is the overall (fugacity-independent) normalization, given by (1 − τ 4j ).
• P λ are the Hall-Littlewood polynomials of type SO(2N ) and Sp(N ), given by with (w) denoting the length of the Weyl group element w.
• The prescription for writing the K-factors can be found in [22]. Their precise form will not be important here.
• The fugacities a I dual to the Cartan subalgebra of the flavor symmetry group of the puncture Λ I (Λ I ) are assigned by setting the character of the fundamental representation of SO(2N ) (Sp(N − 1)) equal to the sum of SU (2) characters corresponding to the decomposition determined by the puncture, with SU (2) fugacity equal to τ . The multiplicity of each SU (2) representation is then replaced by the character of the fundamental representation of the flavor symmetry determined by that multiplicity. From this equation, one can simply read off the fugacities. 13 13 If the puncture is not "very even", different choices of fugacities are related by a Weyl transformation, under which the Hall-Littlewood polynomials are invariant. For "very even" punctures there are two inequivalent choices, which are permuted by the Z 2 outer-automorphism, corresponding to the red and blue coloring. For examples, see [22].
For example, the D 4 twisted puncture corresponds to the SU (2) embedding under which the 6 of Sp(3) decomposes as 2 + 4(1). So setting we can take fugacities To determine the global symmetry, as well as any decoupled sector, of an interacting SCFT fixture from its superconformal index, we need only compute (2.8) to order τ 2 : as explained in [23], the contribution at order τ is due to free hypermultiplets while the contribution at order τ 2 is due to moment map operators of flavor symmetries.
Computing the index to order τ 2 while keeping only the term λ = 0 in the sum over representations gives the contribution encoding the manifest global symmetry. The global symmetry of the SCFT is enhanced if there are additional terms contributing at order τ 2 coming from the sum over λ > 0.
As an example, consider the fixture .
Letting (a 1 , a 2 ), (b 1 , b 2 ) be Sp(2) fugacities and c an SU (2) fugacity, from (2.8) we find The order τ term signals the contribution of a free hypermultiplet in the 1 2 (1, 1, 2) of Sp(2) × Sp(2) × SU (2), the index of which is given by where P E denotes the plethystic exponential [22]. Removing the contribution of the free hypermultiplet, the index of the interacting SCFT is given by a 1 , a 2 , b 1 , b 2 , c)τ 2 + . . . and hence this SCFT has an enhanced Sp(5) global symmetry. We can also use the second order expansion of (2.8) as a check on our identifications for the gauge theory fixtures. For example, the fixture is an SU (2) × SU (2) We have used this technique to check the global symmetries and the number of free hypermultiplets in our tables of fixtures for the Z 2 -twisted D 4 theory.

The Sp(4) 6 × SU (2) 8 SCFT
Here we use the superconformal index to argue that the D 4 interacting fixture gives rise to the Sp(4) 6 × SU (2) 8 SCFT. For this fixture, we cannot use any S-dualities to study its properties as none of the flavor symmetries carried by the punctures can be gauged. The Sp(4) 6 × SU (2) 8 SCFT first appeared in [14] as the twisted-sector fixture in the A 3 theory. It also appears, accompanied by six free hypermultiplets, as (2.9) in our list of twisted-sector mixed fixtures in the D 4 theory. In those cases, we are able to use various S-dualities to study it. Letting a and b be SU (2) fugacities and c 2 1 , c 2 2 U (1) fugacities, the expansion of the index of this fixture is given by indicating that the manifest SU (2) 2 24 ×U (1) 2 global symmetry is enhanced to Sp(4)×SU (2). This, along with the other numerical invariants of this fixture agree with our previous results for the Sp(4) 6 × SU (2) 8 SCFT.
Since A 3 ∼ = D 3 , we can use (2.8) to compute the index of the twisted A 3 fixture by appropriately identifying fugacities and replacing P λ SO(6) (P λ Sp(2) ) → P µ SU (4) (P µ SO(5) ) where µ (µ ) is the highest weight of the SU (4) (SO(5)) representation corresponding to λ (λ ). Letting a be an SU (2) fugacity and (b 1 , b 2 ), (c 1 , c 2 ) SO(5) fugacities, the expansion of the index of the twisted A 3 fixture is in agreement with (2.10). We have checked further that the unrefined indices (obtained by setting all flavor fugacities to "1") of these two fixtures agree to tenth order in τ . The unrefined index of each fixture is given by We can also compare with the mixed fixture (2.9). After removing the contribution to the index of a free hypermultiplet in the 6 of Sp (3), the index of this fixture is given by Again, the numerical invariants of this fixture imply the SCFT is the Sp(4) 6 × SU (2) 8 theory. We have computed the unrefined index of this fixture to fourth order in τ ; removing the contribution of the free hypermultiplet, we find agreement with the fixtures above.

Regular Punctures
The untwisted sector of regular punctures was discussed in [4]. The Z 2 -twisted regular punctures are shown in the

Irregular Punctures
A fairly lengthy list of irregular untwisted punctures, arising from the OPE of untwisted punctures, was discussed in [4]. Additional ones arise from considering the OPE of two As was the case in [4], there are three inequivalent embeddings of Sp(2) → Spin(8), exchanged by triality, under which one of the 8-dimensional representations decomposes as 5 + 3(1) while the other two decompose as 2 (4). To indicate which we mean, we assign a green/red/blue colour to . The same remark applies to the three index-1 embeddings of SU (2) × SU (2) in the SU (2) 3 of which are exchanged by triality.

Mixed Fixtures
Three new SCFTs make their appearance in the list of "mixed" fixtures (accompanied by some number of free hypermultiplets).

Gauge Theory Fixtures
For each gauge theory fixture, we list the gauge group, G, and the representation content of the hypermultiplets, (R F 1 , R F 2 , R F 3 ; R G ). Here, R G is the representation of the gauge group and R F i is the representation of the semisimple part of the flavour symmetry of the i th puncture (where we work counterclockwise from the upper-left, and omit F i if it is abelian or empty).

19
, Sp (2) ) (   [24,25,26,27,28]. In particular, the (2,0) theory of type D N is the theory on 2N coincident M5-branes at an orientifold singularity and, in that realization of these theories [9], the key building block is the fixture consisting of a twisted-sector minimal puncture, a twisted-sector full puncture and an untwisted-sector full puncture, which is a free-field fixture transforming as a bifundamental half-hypermultiplet of Sp(N − 1) × SO(2N ).
Taking two of these fixtures and connecting them with a cylinder yields the aformentioned SO(2N ) gauge theory. Connecting them, instead, with a cylinder yields the Sp(N − 1) gauge theory.
Here, we read off the S-dual strong-coupling descriptions. In the SO(2N ) case, ) + 1(1) + 2( ) 1 ( For this theory, by contrast, all the degeneration limits are (isomorphic) weakly-coupled Lagrangian field theories. The flavour symmetry group for this family of field theories is F = SU (2) 2N 2 −N −1 × Spin(8) 2N . As is the case for SU (2), N f = 4, the S-duality, which acts as an S 3 symmetry on M 0,4 , acts as outer automorphisms of the Spin(8) flavour symmetry. Moreover, the Seiberg-Witten curve takes the absurdly simple form where the quadratic differential

Spin(8) Gauge Theory
Spin(8) gauge theory, with matter in the n v (8 v ) + n s (8 s ) + n c (8 c ), is superconformal for n v + n s + n c = 6. Up to permutations, related to triality, the list of possible values for n v , n s , n c is quite short and we discussed most of them in [4]. There were, however, two cases which were not realizable with only untwisted sector punctures.
One is n v = 6, which is a special case of the construction in §4.1. The other case is n v = 5, n s = 1 (which, as we shall presently see, lies in the same moduli space as n v = 5, n c = 1).
Consider the 4-punctured sphere This is a weakly-coupled Spin (8)

Spin(7) Gauge Theory
Similar to the case of Spin(8) gauge theory, realizations of most cases of conformally-invariant Spin(7) gauge theory were already discussed in [4]. Here we show realizations of the missing two cases.

Sp(3) Gauge Theory
In this section, we will consider various cases of Sp(3) gauge theory, with vanished β-function. We have already discussed the theory with 8(6) and the theory with 1(14) + 4(6) (special cases of the discussion of §4.1).
The gauge coupling is f (τ ) = 2w 1 + w In particular, the S-duality group is the Γ 0 (2), generated by Here, T acts as the deck transformation, w → −w, and ST 2 S acts trivially on the w-plane.
The theory at f (τ ) = 0 is the Lagrangian field theory; at f (τ ) = 1, ∞ (which project to x = 1) we have the Sp(2) gauging of the (E 8 ) 12 SCFT. The gauge theory fixture, at x = ∞, is the theory at the Z 2 -invariant interior point of the moduli space, f (τ ) = 2.

Higher Genus
In almost all of the discussion in this paper, we have taken C to be genus-zero. We should close with at least one example of higher-genus, so that we can see the effect of twists around handles of C.
Consider a genus-one curve, with one minimal puncture, in the D 4 theory.
Under the action of the modular group, H 1 (T 2 − p, Z 2 ) breaks up into two orbits: the zero orbit (the "untwisted theory") and the nonzero orbit ("the twisted theory"). The untwisted theory is a Spin(8) gauging of the (E 8 ) 12 SCFT. There are three inequivalent index-2 embeddings of Spin(8) in E 8 . They can be characterized by how the 248 decomposes (up to outer automorphisms of Spin (8) The untwisted theory corresponds to (4.6a). The twisted theory, depending on the Sduality frame chosen, corresponds either to a Spin(8) gauging of the (E 8 ) 12 SCFT using the embedding (4.6b), or to an Sp(3) gauging of the Sp(6) 8 SCFT.
For the untwisted theory, the gauge theory moduli space is the fundamental domain for P SL(2, Z) in the UHP, and τ is the modular parameter of the torus. For the twisted theory, the moduli space of the gauge theory is the moduli space of pairs (C, γ), where γ is a nonzero element of H 1 (C, Z 2 ). This is the fundamental domain of Γ 0 (2), as discussed in §2.3.