Tinkertoys for the twisted D-series

We study 4D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 superconformal field theories that arise from the compactification of 6D N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (2, 0) theories of type DN on a Riemann surface, in the presence of punctures twisted by a ℤ2 outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by ℳg,n, but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D5 and D6 theories. We find realizations of Spin(8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing β-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.

1 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 in [9,10,15].
The D N Dynkin diagram is invariant under a Z 2 outer automorphism group. Correspondingly, there is a Z 2 bundle over the punctured surface; if the monodromy is nontrivial around the puncture, then we say that the puncture at p is twisted (otherwise, untwisted). 1 (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 2 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],
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 outer-automorphism 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 Dpartitions 3 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 [16]. 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 Nahm-pole D-partition p, one must recall the Spaltenstein map, 4 which takes p into a new D-partition d(p), called the Hitchin pole of the puncture. 5 Then, the local boundary condition corresponding to p is where X is an element of the nilpotent orbit 6
To describe the local boundary condition for a twisted puncture, we need to recall the relevant Spaltenstein map. 8 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 9 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.

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" [18] 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, where . . . 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

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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 10 to compute the scaling dimensions 2k at which aconstraints 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.
4. 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. 10 As in our previous papers, the a-constraints and c-constraints of a special puncture can be read off directly from the k-differentials for the Hitchin field with the appropriate boundary condition at the puncture. It is harder to compute the constraints of a non-special puncture in this way. In practice, we use the fact that every non-special puncture belongs to a special piece which contains a (unique) special puncture; their constraints are the same, except that some a-constraints are relaxed as dictated by the Sommers-Achar group; see §3.4 and §3.5 of [8], and §2.1 of [4]. For the twisted D-series, one can also compute the c-constraints, and partial information about the a-constraints, both for special and non-special punctures, by studying the linear SO −Sp quiver corresponding to the puncture; see appendix B of [9]. The constraints are non-trivially consistent with the dimension of the Hitchin nilpotent orbit, the value of δnv, and the collisions of punctures using k-differentials. The general recipes for the constraints in terms of partitions that we show in this paper were found by studying a large number of examples, and we have checked their consistency up to high N . However, we do not have a rigorous proof of them.

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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 (N ) l andc l . (The pole orders of φ N andφ are the same if the conditions just mentioned hold, so such a linear constraint is possible.) A generic veryeven puncture may or may not have this very-even c-constraint. In particular, a very-even puncture could have a c-constraint of dimension N which is not very even (in the sense that it is not linear in both c (N ) l andc l ).
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. 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: (2.4) (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 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 JHEP04(2015)173 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). These c-constraints all belong to the sequence mentioned in the previous paragraph. Also, there are no a-constraints, and the pole structure is {1, 2, 3, 4, 5; 6}. The structure of the c-constraints is the clearest in terms of auxiliary quantities t 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 section 2, or at the level of the k-differentials, using the pole structures and the constraints of section 2.1.2 and section 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: where X| so(2N −1) is the restriction of X ∈ so(2N ) to the subalgebra so(2N − 1).

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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.
Colliding the new puncture [2N − 13, 2 6 , 1] with the minimal twisted puncture is much easier, because none is constrained. So all we need to do is add up pole orders, and identify gauge-group Casimirs. The sum of the pole structures is {2, 4, 6, 7, 9, 11, 13, 13, . . . , 13; 13 2 }. Hence, we have again a gauge group with Casimirs 2, 4, 6, and a new puncture with pole structure {1, 3, 5, 7, 9, 11, 13, 13, . . . , 13; 13 2 }, with no constraints. These properties correspond to the puncture [2N − 14, 1 12 ], which has flavour symmetry Sp (6), and the gauge group is an Sp(3) subgroup of Sp (6). Actually, the Sp(3) flavour group of the [2N −13, 2 6 , 1] puncture that we found in the previous paragraph can be interpreted as the commutant in Sp (6) (6) are being gauged by two consecutive cylinders. (We studied examples of this "atypical" gauging in the twisted A 2N −1 series [14]; see also [9] for an earlier discussion in the context of SO − Sp linear quivers.) Let us derive the same result by doing the collisions in a different order: first, we collide a [2N − 7, 7] puncture (at z = 0) with the minimal twisted puncture (at z = x). We use the k-differentials 11 This time, solving the c-constraints is less simple. The constraints are not solvable unless one introduces parameters r 2 , r 4 , r 6 of dimension 2,4,6 such that: (See section 4.1.3 of [14] for a similar example in more detail.) Then, the constraints imply: and in the limit x → 0, we get a pole structure {1, 3, 4, 5, 6, 7, 7, 7, . . . , 7 2 }, with constraints In this subsection, we use generic names for Coulomb branch parameters such as u 2k , v 2k , r k , etc. They are understood to be different variables in different collisions.

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that is, we have a-constraints at 2k = 4, 8, 12 and c-constraints at 2k = 6, 10. These properties uniquely identify the twisted puncture [2N − 8, 6]. Notice that there are no gauge-group Casimirs, so our interpretation is that the cylinder is "empty". This is an example of an "atypical degeneration", as we will recall in section 2.5.4.

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) ⊂ PSL(2, Z), is generated by

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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 12 Since Γ(2) is index-6 in PSL(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 under the action of PSL(2, Z)) is Of course, while the j-invariant is invariant under the full PSL(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) ⊂ PSL(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 PSL(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 13 has only one constraint, which is a very-even c-constraint, where the top (bottom) sign corresponds to a red (blue) puncture. 13 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.

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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 14 , and hence well-defined on the moduli space. However, the (unavoidable) square-roots 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 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.
14 To minimize the number of ensuing branch cuts, we have chosen not to preserve the obvious z3 ↔ z4 symmetry. We can restore it by redefining the Coulomb branch parameter The resulting theory lives naturally on the 4-fold branched cover of M0,4.
Having seen the phenomenon in a global example, let us recover the same result, working locally on the plane, with the Higgs field itself (rather than the gauge-invariant k-differentials). Consider a very-even Higgs-field residue B ∈ so(2N ), which belongs to a, say, red nilpotent orbit. We can write puts the residue B 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, , so our red puncture becomes blue, or vice versa.

Atypical punctures
As an application of the formulas in section 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 a) (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.

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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 respectively. 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 JHEP04 (2015)173 so that f = 0 and f = ∞ both map to x = 1, while f = 1 maps to x = ∞. Our gaugetheory 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 weaklycoupled 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

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Over x = 1, we have two distinct degenerations, which are exchanged by dragging the puncture around the origin and returning it to its original position: the Lagrangian field theory (f = 0) 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 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 .

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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 5punctured 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 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 .

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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 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. Over In the first case, f Sp(2) = 0 or ∞ and f SU(2) = 1; in the latter, f Sp(2) = 1 and f SU(2) = 0 or ∞.
Over D 12 ∩ D 35 and D 12 ∩ D 45 , we have In both cases, the underlined gauge group on the right-hand cylinder is identified with the gauge group on the left-hand cylinder. The notation, which we introduced in [14], indicated that when the cylinder on the right pinches off, both factors in the gauge group become weakly-coupled (f → 0 or ∞). When the cylinder on the left pinches off, only one of the gauge group factors becomes weakly-coupled.

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Over D 34 ∩ D 15 and D 34 ∩ D 25 , f Sp(2) = f SU(2) = 1. So we have These differ only very subtly, as to "which" SU(2) gauge coupling is controlled by the cylinder on the left. In the first case, it is the SU(2) which couples to the (E 7 ) 8 (i.e., the one which becomes weakly-coupled at f Sp(2) = 1); in the second case, it is the SU (2)

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Finally, over D 13 ∩ D 24 and D 14 ∩ D 23 , we recover our gauge theory fixture, and read off that its gauge theory couplings are f Sp (2)

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 section 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 section 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 veryeven 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 . [N 2 ] . 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.

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, we again obtain an atypical puncture as the OPE. No gauge coupling become weak, but the moduli space is ramified (with ramification index 2).
The moduli space is unramified over this component of the boundary. Nonetheless, two gauge couplings become weak.
Here, again, an SU(2) × SU(2) gauge group becomes weak, but now the moduli space is also ramified (with ramification index 2) In all of the remaining cases, the moduli space is ramified (with ramification index 2) and two gauge couplings become weak.
and, for N even, the gauge group which becomes weak is

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 [19][20][21][22] 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 [23]. 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 15 . (2.8)
• 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 [23]. Their precise form will not be important here.

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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 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 [24], 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 .

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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
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.
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].

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 Z 2 -twisted punctures. Moreover, twisted-sector irregular twisted punctures arise from the OPE of an untwisted puncture and a Z 2 -twisted puncture. These two sets of new irregular punctures are listed in the tables below. 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). To understand the ambiguity in the matter content listed for gauge theory fixture (2), consider resolving it to the 5-punctured sphere Spin (7) , Spin (7)) ( which is attached to the rest of the surface by a weakly-coupled Spin(7). In the limit corresponding to the gauge theory fixture, both SU(2)s are strongly-coupled (τ = i). But there are three distinct limits where both SU(2)s are weakly coupled: , Spin(7)) ( SU(2)̲ × SU (2) , SU(2)̲ × SU(2) ) ( 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, we have an SU(2) gauging of the SU(2) 8 ×Sp 2(N −1) 2N SCFT. In the Sp 2(N −1) case, we have an SU(2) gauging of the SU(2) 8 × Spin(4N ) 4(N −1) SCFT.

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For completeness, let us note that the other Sp(N ) gauge theory which is superconformal for arbitrary N > 1, namely the one with one hypermultiplet in the traceless antisymmetric tensor and four hypermultiplets in the fundamental representation, was already realized 17 (with the addition of a single free hypermultiplet) in the untwisted sector of the A 2N −1 theory [3]: 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 η is (8), Spin (7) and Sp(3) gauge theory

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 section 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).

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Consider the 4-punctured sphere This is a weakly-coupled Spin(8) gauge theory with matter in either the 5(8 v ) + 1(8 s ) or the 5(8 v ) + 1(8 c ). The two realizations are exchanged by dragging the puncture around one of the twisted-sector punctures and returning it to its original location.
The strong coupling limits are SU(2) gauge theories (where we gauge an SU(2) subgroup of Sp(6) 8 ) and where the SU(2) 5 is gauged.

Spin(7) gauge theory
Similar to the case of Spin(8) gauge theory, realizations of most cases of conformallyinvariant Spin(7) gauge theory were already discussed in [4]. Here we show realizations of the missing two cases. 5 (7). With the addition of three free hypermultiplets, we have a realization of the theory with 5 hypermultiplets in the vector representation as The S-dual theory is an SU(2) gauging of the Sp(5) 7 × SU(2) 8 SCFT, plus 3 free hypermultiplets. (7). The Spin (7) gauge theory, with one spinor and four vectors, can be realized in a couple of different ways. With the addition of three free hypermultiplets, we have

Sp(3) gauge theory
In this section, we will consider various cases of Sp(3) gauge theory, with vanishing βfunction. We have already discussed the theory with 8(6) and the theory with 1(14) + 4(6) (special cases of the discussion of section 4.1).
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.

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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 S-duality 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 PSL(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 section 2.3.