1-form Symmetries of 4d N=2 Class S Theories

We determine the 1-form symmetry group for any 4d N = 2 class S theory constructed by compactifying a 6d N=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d, modulo screening and flavor charges. Complete specification of a 4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.


Introduction
A massive vacuum of a 4d theory T is called confining if it preserves a non-trivial subgroup of the 1-form symmetry group of T [1]. Motivated by confinement in 4d N = 1 theories obtained by deforming 4d N = 2 theories that we will study in [2], we develop in this paper, as a precursor, the tools to determine the 1-form symmetry of 4d N = 2 theories. More specifically, we consider 4d N = 2 theories of Class S that can be obtained by compactifying 6d N = (2, 0) SCFTs on a Riemann surface [3]. We allow the Riemann surface to contain closed twist lines and arbitrary regular punctures which can be either untwisted or twisted.
It is well-known that 6d N = (2, 0) SCFTs are classified by a Lie algebra g of ADE-type, and that they are relative QFTs [4][5][6], which for the purposes of this paper can be understood as follows. The (2, 0) theory contains dimension-2 surface operators which are not mutually local, i.e. there is an ambiguity in defining a correlation function containing two such surface operators [7]. If there is no such ambiguity, then we call the theory an absolute QFT instead.
Fusion (OPE) of these surface operators lends the set of surface operators the structure of an abelian group. Moreover, the surface operators can be screened by dynamical strings in the theory. We denote the group of surface operators modulo screening by Z.
Upon compactification to 4d, one can wrap these surface operators along various 1-cycles on the Riemann surface to generate an abelian group L of line operators modulo screening in 4d. The non-locality of 6d surface operators descends to non-locality of these 4d line operators.
In other words, we obtain a relative 4d theory upon such a compactification. To obtain an absolute 4d theory T, one needs to choose a maximal subgroup Λ T ⊂ L of mutually local 4d line operators 1 . The group Λ T can be identified with the set of charges for the 1-form symmetry group of T. In other words, the 1-form symmetry group of T is identified as the Pontryagin dual Λ T of Λ T [1].
Special cases of the problem explored in this paper have been discussed previously in the literature. For example, in the case where the Riemann surface C g has no punctures and no closed twist lines, the group L was already determined in [9] (see also the recent paper [8]) to be H 1 (C g , Z). For the case of g = A 1 and arbitrary C g with arbitrary number of regular punctures, this problem was discussed in [10,11]. Another situation where this problem has been discussed arises whenever there exists a degeneration limit of C g in which the 4d theory can be identified as a weakly coupled 4d gauge theory. In such a situation, one finds a canonical splitting L L e × L m , where L e is associated to Wilson line operators and L m is associated to 't Hooft line operators. In such a situation, the constraint of mutual locality can also be understood as the constraint of Dirac quantization, and choosing a way of satisfying Dirac quantization condition (i.e. a choice of Λ T ⊂ L e × L m ) can be interpreted as choosing a global form of the gauge group and possible discrete theta parameters [12] (see also [1]). More recently work related to the higher form symmetry of 4d SCFTs and holography was studied in [13]. In the context of non-Lagrangian 4d N = 2 SCFTs of Argyres-Douglas type the 1-form symmetries were computed using the Type IIB realization using canonical singularities in [14,15], using the general observations in [16][17][18], which are applicable more generally to geometric engineering of SCFTs in string theory. Many recent papers have tackled the problem of determining higher-form symmetries in lower-dimensional QFTs starting from 1 The choice of Λ T is only part of the full set of choices one needs to make in order to define an absolute 4d N = 2 theory of Class S. For example, one can obtain a group L0 of dimension-0 and a group L2 of dimension-2 operators in the 4d theory by compactifying the 6d surface operators along the whole Riemann surface and along a point on the Riemann surface respectively. Then the non-locality of the 6d surface operators descends to a non-locality between elements of L0 and L2, and to choose an absolute 4d N = 2 theory, one also needs to choose subgroups Λ0 and Λ2 of L0 and L2, such that there is no non-locality between elements of Λ0 and Λ2. See [8] for a recent discussion.
supersymmetric QFTs in six dimensions [8,16,19,20] (see also [21] for a related discussion), and we expect many more interesting developments in this direction.
Our key proposal that lets us generalize the result of [9] is that a 6d surface operator wrapping a cycle surrounding a regular puncture does not contribute to the set L of 4d line operators (modulo screening and flavor charges). In the case of untwisted regular punctures, any 6d surface operator can be wrapped around the puncture, and hence according to the above proposal, untwisted regular punctures are invisible to the determination of L and 1form symmetry Λ T of a 4d N = 2 theory T obtained after choosing a polarization Λ T ⊂ L. On the other hand, twisted regular punctures do have a non-trivial influence on the calculation of L. This is because such a puncture lives at the end of a twist line which acts non-trivially on the 6d surface operators, and hence only the 6d surface operators left invariant by this action can be inserted along a loop surrounding the twisted regular puncture. Thus, according to the above proposal, a twisted regular puncture is only invisible to the 6d surface operators invariant under the action of the corresponding twist line. As we discuss in various examples throughout the paper, a justification for the above proposal is that L obtained using it matches the L obtained using the gauge theory analysis of [12] (see also [1]) whenever there exists a limit of the compactification in which a weakly coupled 4d N = 2 gauge theory arises [3,[22][23][24][25][26][27][28][29][30][31][32][33][34][35].
Let us now discuss a subtlety that arises due to the fact that one needs to take the area of C g to zero in passing from the 6d theory to the 4d N = 2 theory. One might worry that the set L discussed might not be the true set of line operators modulo screening in the 4d theory. However, this worry is alleviated by the fact that in order to define the 4d N = 2 theory one often needs to perform a non-trivial topological twist 2 on C g , due to which one expects protected quantities to be independent of the area of C g . The set L is such a protected quantity as each element in the set can be represented by a BPS line operator in the 4d N = 2 theory, and the screenings can also be understood in terms of BPS particles. On the other hand, in situations where one does not need to perform a non-trivial topological twist, one cases are computed from the relative homology three-cycles of the non-compact Calabi-Yau, or equivalently, the second homology of the link (i.e. the boundary five-fold). From the local Higgs bundle realization of the ALE-fibration of the Calabi-Yau three-fold, we determine these homology groups and confirm the defect group for the case of no punctures and for regular untwisted and twisted punctures: The defect group L has a simple description, purely in terms of the data on the boundary of the non-compact Calabi-Yau threefold, namely the boundary B F = S 3 /Γ ADE → C fibration, where C is the Gaiotto curve, and the base of the ALE-fibration. Then the defect group is simply given in terms of the 2-cycles of B F , which extend trivially to the Calabi-Yau.
In fact, as we discuss in section 6.2, this approach can be viewed to provide a justification for our key proposal that a 6d surface operator wrapping a cycle surrounding a regular puncture does not contribute to L. Moreover, this approach might shed light on the irregular punctures as, e.g. generalized AD theories have a realization in terms of Type IIB on canonical singularities, from which in turn the 1-form symmetry can be computed [14,15].
We find that L can be roughly constructed from the various kinds of data on the Riemann surface used for compactification. We collect this rough decomposition of L in Table 1 to be used as a reference. It is important to note that the table only captures the group-structure of L, while one of the key ingredients is the pairing on L capturing the mutual non-locality of 4d line operators. This pairing is required to choose a polarization Λ and determine the corresponding 1-form symmetry Λ. The explicit form of the pairing can be found in the main text.
The paper is organized as follows. In section 2 we review some properties of dimension-2 surface operators and outer-automorphism discrete 0-form symmetries in 6d N = (2, 0) SCFTs. In section 3 we discuss 1-form symmetry in absolute 4d N = 2 theories obtained by compactifying 6d (2, 0) theories on a genus g Riemann surface in the presence of arbitrary twists by outer-automorphism discrete 0-form symmetries, but without involving any punctures. In section 4 we extend our analysis of previous section to includ arbitrary untwisted and twisted regular punctures. In section 5 we sketch how our analysis can be extended to include irregular punctures, giving explicit results for a specific class of irregular punctures of A n−1 (2, 0) theories. Finally, in section 6 we argue from a Type IIB realization of class S theories for the 1-form symmetries. Our notation is summarized in appendix A. Table 2: For the ADE Lie algebras g we denote by G the simply-connected Lie group, and list the center Z(G), the Pontryagin dual group to the center Z(G), and the bihomomorphism ·, · . E 8 has a trivial center group, which has been denoted by 0 since we use an additive notation for the group multiplication law throughout this paper. We denote a generator of Z(G) for g = A n−1 , E 6 , E 7 as f ; a generator of Z(G) for g = D 2n+1 as s; and generators of Z(G) Z 2 × Z 2 for g = D 2n as s, c. We also define v := s + c for g = D 2n . These surface operators are not all mutually local. Consider a correlation function containing two surface operators α, β ∈ Z(G). As α is moved around β, the correlation function is transformed by a phase factor 4 exp 2πi α, β

Surface Operators and Outer
The bihomomorphism can be specified by providing its values on the generators of Z(G) [8].
These are also listed in table 2.
The (2, 0) theory admits a discrete 0-form symmetry which can be identified with the group of outer-automorphisms O g of g, which are Then the elements of S 3 can be written as 1, a, a 2 , b, ab, a 2 b. An important conjugation relation we will use throughout the paper is bab = a 2 .

Compactifications without Punctures
In this section we consider compactifications of 6d (2, 0) theories on a Riemann surface C g of genus g without any punctures. If there are no other ingredients involved in the compactification, such a compactification is called as an untwisted compactification. On the other hand, we can also consider twisted compactifications which means the following. The outerautomorphism 0-form symmetry in 6d (2, 0) theory discussed in the last section is generated by topological operators of codimension-1 in the 6d theory. Inserting such a topological operators along a cycle of the Riemann surface gives rise to a "codimension-0 object" in the 4d theory, which means that the resulting 4d theory itself is different from the 4d theory arising when no such topological operators are inserted. We often refer to the locus of the topological operator on C g as a twist line, and when this locus is a 1-cycle on C g we say that the twist line is closed. In the presence of punctures this picture is enhanced by the alternative of open twist lines. Open twist lines emanate and end at punctures and we discuss their effect in section 4.
Twisted and untwisted compactifications can equivalently be distinguished in the Higgs bundle description of the compactification. Here the insertion of topological operators along twist lines gives rise to an action on the Higgs field by an outer automorphism o across these.
The insertions alter the gauge group of the effective 4d N = 2 theory and have a geometric interpretation in the IIB dual description as we explain in more detail in section 6. In this geometric picture we are further able to justify the key assumption that regular untwisted punctures are irrelevant in determining the defect group, which we also argue for in the section 4.

Untwisted Case
Let us compactify a (2, 0) theory on a Riemann surface C g of genus g without any punctures or twists. This gives rise to a relative 4d N = 2 theory with a set of line defects descending from the elements of Z(G) wrapped along various cycles of C g . That is, the set L of 4d line defects (modulo screening) can be identified with These line defects are not all mutually local. The violation of mutual locality between two where a, b is the intersection pairing on H 1 (C g , Z). This gives rise to a pairing on H 1 (C g , Z) which is the natural combination of the intersection pairing and the bihomomorphism (2.2) ·, · : We can specify an absolute 4d N = 2 theory by choosing a maximal set of line operators , (3.4) which are all mutually local, i.e. the phase (3.2) is trivial for any two elements in Λ. Such a set Λ is also referred to as a 'maximal isotropic subgroup' or as a 'polarization' in what follows. The 1-form symmetry of the absolute 4d N = 2 theory can then be identified with the Pontryagin dual Λ of Λ.
Once we choose a set of A and B cycles on C g , we can decompose where Z g A is the contribution of A-cycles, and Z g B is the contribution of B-cycles. Moreover, Z g A and Z g B are maximal isotropic sublattices, and hence provide canonical choices of Λ once a choice of A and B cycles has been made.
Example: When (2, 0) theory of type g is compactified on a torus, we obtain 4d N = 4 SYM with gauge algebra g. Choosing an A-cycle and a B-cycle, we write (3.6) We assume without loss of generality that the A-cycle is much shorter than the B-cycle. Then, and all discrete theta parameters turned off. In these cases, we have 1-form symmetry which matches with the 1-form symmetry obtained using the Lagrangian description of 4d N = 4 SYM: when the gauge group is G, this is the electric 1-form symmetry; and then the gauge group is G/Z(G), this is the magnetic 1-form symmetry.
Other choices of global forms of the gauge group and discrete theta angles are obtained by choosing other polarizations. For concreteness, consider the case of g = su (4). In this case, The P SU (4) theory with a discrete theta parameter n ∈ {0, 1, 2, 3} turned on is obtained by choosing Λ to be the sublattice generated by the element (n, 1) ∈ Z 4 × Z 4 Z A × Z B (where we have represented Z 4 as the additive group Z/4Z). Any such choice leads to the 1-form symmetry Λ Z 4 . (3.8) If we choose the polarization Λ generated by elements (0, 2) and (2, 0) in Z 4 × Z 4 , then we obtain the SO(6) SU (4)/Z 2 theory with the discrete theta parameter turned off. In this case the 1-form symmetry is From the point of view of the Lagrangian description, the two Z 2 factors are electric and magnetic 1-form symmetries respectively. The remaining su(4) theory has SO(6) gauge group and a discrete theta parameter turned on. This is obtained by choosing Λ to be generated by the element (1, 2) ∈ Z 4 × Z 4 Z A × Z B , and the 1-form symmetry group of the theory is (3.10) Example: Consider compactifying A 1 (2, 0) theory on C g with g ≥ 2.
In an S-duality frame, in which A-cycles are much shorter than B-cycles, we obtain the following Lagrangian 4d where we have a total of 2g − 1 nodes. Each node describes a gauge algebra and an edge between two nodes denotes a half-bifundamental 5 between the two nodes. An edge connecting an su(2) node to a node labeled 1 2 F implies that the corresponding su(2) gauge algebra carries an extra half-hyper charged in fundamental rep. If we choose Λ = (Z/2Z) g A , we obtain the 4d theory with all the gauge groups being simply connected. In this case, we have 1-form which can be easily matched with the above Lagrangian description with all the gauge groups chosen to be the simply connected ones. A Z 2 factor arises from each of the g number of so(n) nodes (where n = 3, 4 and the corresponding gauge group is Spin(n)). This Z 2 is the subgroup of center of Spin(n) that acts trivially on the fundamental representation of so(n) as defined in the above footnote.

Including Closed Z 2 Twist-lines
We can also consider twisted compactifications of 6d N = (2, 0) on C g (without punctures).
This involves wrapping the topological defects generating the outer-automorphism discrete 0-form symmetries along cycles on C g . In this subsection we either consider those g for which the outer-automorphism group is Z 2 , or the case g = D 4 with twist lines valued only in the Z 2 subgroup of the S 3 outer-automorphism group generated by the element b (see section 2). We can wrap the Z 2 twist lines along some L ∈ H 1 (C g , Z 2 ). Let us first discuss the case of For g = A 2n−1 , we can only wrap the element nf ∈ Z Z 2n and hence (3.14) Similarly, for g = D n , only v can be wrapped and hence An absolute 4d N = 2 theory is then specified by choosing with Λ being maximally isotropic. The 1-form symmetry of the 4d N = 2 theory can then be identified with Λ.
For a general C g with arbitrary g, the twist lines are specified by picking an element L ∈ H 1 (C g , Z 2 ). By Poincare duality, we can work with the dual element L ∈ H 1 (C g , Z 2 ).  Now, combining the results discussed previously, we easily identify the set L of 4d line operators (modulo screening). We find that the polarization is chosen as where the pairing is obvious from our previous discussion. The 1-form symmetry group of such an absolute 4d N = 2 theory is identified as Λ.
Example: Consider compactifying D n+1 (2, 0) theory on a torus, and wrap a Z 2 twist line along the B-cycle. We can write the set of line defects as First, assume that the A-cycle is much shorter than the B-cycle. This corresponds to first compactifying D n+1 (2, 0) theory on a circle with outer-automorphism twist, leading to 5d N = 2 SYM with sp(n) gauge algebra and discrete theta angle θ = 0 [36]. We further compactify this 5d theory on another circle obtaining 4d N = 4 SYM with sp(n) gauge algebra. Choosing Λ = (Z/2Z) A corresponds to picking the simply connected Sp(n) gauge group for the 4d theory, and the 1-form symmetry Λ Z 2 can be identified as the electric 1-form symmetry from the point of view of this Lagrangian 4d theory. Choosing Λ = (Z/2Z) B leads to gauge group Sp(n)/Z 2 with the discrete theta parameter turned off, and the 1-form symmetry Λ Z 2 can be identified as the magnetic 1-form symmetry from the point of view of this Lagrangian 4d theory. Now, assume that the B-cycle is much shorter than the A-cycle. This corresponds to first compactifying D n+1 (2, 0) theory on a circle without outer-automorphism twist, leading to 5d N = 2 SYM with so(2n + 2) gauge algebra. We further compactify this 5d theory on another circle with a Z 2 outer-automorphism twist, leading to 4d N = 4 SYM with so(2n + 1) gauge algebra. Choosing Λ = (Z/2Z) A corresponds to picking the SO(2n + 1) gauge group for the 4d theory with all discrete theta parameters turned off, and the 1-form symmetry Λ Z 2 can be identified as the magnetic 1-form symmetry from the point of view of this Lagrangian 4d theory. Choosing Λ = (Z/2Z) B leads to the simply connected gauge group Spin(2n + 1), and the 1-form symmetry Λ Z 2 can be identified as the electric 1-form symmetry from the point of view of this Lagrangian 4d theory.
Non-example: Consider compactifying A 2n (2, 0) theory on a torus, and wrap a Z 2 twist line along the B-cycle. Our proposal would predict the set L of 4d line defects (modulo screening) which is the trivial group. That is, all the 4d line defects are proposed to be screened.
Correspondingly, the 1-form symmetry of the resulting 4d theory is predicted to be trivial.
These predictions are incorrect as we now show.
The limit for which the A-cycle is much shorter than the B-cycle corresponds to first compactifying the A 2n (2, 0) theory on a circle of radius R 6 with an outer-automorphism twist, thus leading to 5d N = 2 SYM with sp(n) gauge algebra with gauge coupling g 2 Y M = R 6 and discrete theta angle θ = π [36]. Due to the presence of non-trivial discrete theta angle the BPS instanton particle in this 5d theory transforms in the fundamental representation of sp(n).
Thus, the group of line operators (modulo screening) in this 5d theory is trivial. Moreover, every possible 't Hooft dimension-2 surface operator in the 5d theory which is local with the above mentioned instanton BPS particle is screened. Thus, the group of surface operators (modulo screening) in this 5d theory is also trivial.
Compactifying the above 5d theory further on a circle of finite non-zero radius R 5 , one expects the 4d theory obtained to have no line defects (modulo screening), since there are no line or surface defects (modulo screening) in the 5d theory as we saw above. This is so far consistent with our above predictions.
However, as we send R 5 , R 6 → 0 while keeping R 6 /R 5 preserved, we obtain the 4d N = 4 theory having g = sp(n) with gauge coupling g 2 Y M = R 6 /R 5 and theta angle θ = π. This 4d theory clearly has a Z 2 × Z 2 group of 4d line operators (modulo screening). Thus, our above predictions do not provide the correct answer in the limit when the torus is shrunk to zero size.
From the point of view of the above 5d theory, this limit decouples the BPS instanton particle responsible for screening the fundamental Wilson line, since the mass m of the BPS Figure 4: Resolving various S 3 lines into a-lines and b-lines.
instanton particle scales as m ∼ 1/R 6 → ∞. This means that the fundamental Wilson line is not screened after taking this limit. Moreover, the 't Hooft operator which was not mutually local with the BPS instanton particle becomes available, and we recover the correct result that the set of 4d line operators (modulo screening) is Z 2 × Z 2 . There are 3 distinct choices of polarization corresponding to choosing the 4d gauge group Sp(n) and Sp(n)/Z 2 with a discrete Z 2 valued theta parameter. In each of the three cases, the true 1-form symmetry is Z 2 , which is interpreted as an emergent 1-form symmetry from the point of view of the above 6d → 4d compactification.
The fact that our predicted result for L does not capture the true L is not surprising as explained in the introduction. As discussed there, the predicted L is guaranteed to match the true L only when a non-trivial topological twist is performed on C g . When no non-trivial topological twist is needed, the predicted L is only expected to be a subgroup of the true L.
In the presence of a non-trivial topological twist, the set of BPS particles would be protected as we take the limit of zero area. When there is no topological twist, the set may not be protected, as we saw in the example above where a 4d BPS particle (descending from the 5d BPS instanton particle) was decoupled in the limit of zero area.

Including Closed S 3 Twist-lines
An arbitrary S 3 twist on C g can be manufactured by combining a and b twist lines, which are two elements of orders three and two respectively inside S 3 (see section 2). An arbitrary S 3 twist is described as a trivalent network of topological lines valued in S 3 obeying group composition law. One can separate the a-dependent part out of each edge in this network.
That is, an edge carrying ab can be separated into b and a, and an edge carrying a 2 b can be The network of b lines can be smoothened, and hence is an element L ∈ H 1 (C g , Z 2 ) along which we wrap b lines. In the previous subsection, we have seen that we can always choose If we choose L = 0, then we can represent the network of a lines as an can be identified with the trivial element of Z due to the action of a twist line. Thus, in this case, an absolute 4d N = 2 theory is chosen by wrapping B 1 by performing gauge transformation inside S 3 . Thus, we can always ensure that Figure 6: A Riemann surface of genus g with a closed Z 2 twist line b wrapped along the B 1 cycle and a closed Z 3 twist line a wrapped along the B 2 cycle. The cycle A (which is homologically equivalent to A 1 + A 2 ) has been divided into two sub-segments, denoted respectively by green and blue. The color is changed as A crosses a twist line, indicating that an element of Z wrapped along one sub-segment is in general different from the element of Z wrapped along the other sub-segment, due to the action of outer-automorphism associated to the twist line. Moreover, we need to consider a rather small, special subset of regular punctures separately.
The punctures in this subset are referred to as atypical punctures. In the presence of atypical regular punctures, the number of simple factors in the gauge algebra arising in a degeneration limit of the Riemann surface is not equal to the dimension of the moduli space of the Riemann surface [24][25][26] (see also [22]). We call a regular puncture which is not atypical as a typical puncture. An atypical regular puncture can be resolved into some number of typical regular punctures. Throughout this section until subsection 4.4, a regular puncture always refers to a typical regular puncture.
In this section, we consider compactifications of 6d N = (2, 0) theories on a Riemann surface C g with an arbitrary number of (untwisted and twisted) regular punctures, and an arbitrary number of closed twist lines (which do not have end-points).

Untwisted Regular Punctures
Let L be the set of 4d line operators (modulo screening) when a (2, 0) theory is compactified on a Riemann surface C g without any punctures, but possibly in the presence of closed twist • su(n) + 2nF by compactifying A n−1 (2, 0) theory.
For this case L is trivial, which is what is expected from the 4d gauge theory description as it can be checked that the line operators (modulo screening and flavor charges) form a trivial set in all of the above gauge theories. Consequently, the 1-form symmetry is also trivial for all of these theories, and the gauge group must be the simply connected one.

Torus with 1 regular untwisted puncture and twisted line:
We can obtain the following 4d N = 2 gauge theories by compactifying (2, 0) theories on a torus with 1 regular untwisted puncture and a twisted line wrapped along a non-trivial cycle 7 : 6 The notation gi + niRi denotes a 4d N = 2 gauge theory with gauge algebra g along with ni full hypers in irrep Ri. If ni is half-integral, it means that there is an additional half-hyper in Ri along with ni number of full hypers in Ri. F denotes fundamental irrep for su(n) and sp(n), and vector irrep for so(n). S denotes spinor irreps for so(n) and C denotes co-spinor irrep for so(2n). Λ 2 denotes 2-index antisymmetric irrep for su(n) and sp(n). See also appendix A. 7 S 2 denotes the 2-index symmetric representation of su(n). See also appendix A.
In the former case, we have . Now we add in the matter. The hypermultiplets in S 2 and Λ 2 screen 2W , and thus the set of Wilson lines (modulo screening and flavor charges) can be identified with Z 2 , generated by W . On the other hand, the 't Hooft lines must be mutually local with 2W , and hence the set of 't Hooft lines (modulo screening and flavor charges) can be identified with Z 2 , generated by nH. Thus, we verify the prediction (4.2). Choosing the polarization Λ to be the Z 2 generated by W leads to gauge group SU (2n). Choosing Λ to be the Z 2 generated by nH or W + nH leads to gauge group SU (2n)/Z 2 with discrete theta parameter turned off or on respectively. In all these cases, the 1-form symmetry is In the latter case, L is trivial. Correspondingly, the set of line operators in the gauge theory (modulo screening and flavor charges) is trivial. The set of Wilson lines is trivial because 2W is a generator of Z 2n+1 , and the set of 't Hooft lines is trivial because they need to be mutually local with W (as W is screened). There is no 1-form symmetry, and the gauge group must be the simply connected SU (2n + 1).
Torus with k regular untwisted punctures: 4d N = 2 su(n) k necklace quiver can be obtained by compactifying A n−1 (2, 0) theory on a torus with k regular untwisted punctures.
In this case, which can be verified from the 4d gauge theory description. For example, choosing all gauge groups to be SU (n) corresponds to choosing one of the two Z n factors as the polarization.
The 1-form symmetry is then predicted to be which can be identified as the diagonal subgroup of the Z k n center of the gauge group SU (n) k .
C g with n regular untwisted punctures: Consider compactifying A 1 (2, 0) theory on C g in the presence of n regular untwisted punctures [3]. According to our proposal, we predict There are a number of degeneration limits which lead to a variety of S-dual weakly-coupled 4d conformal gauge theories. The predicted answer for L and the pairing on it can be verified from the point of view of any of these 4d gauge theories. For example, one such degeneration limit (which exists for n ≥ 2) leads to the following 4d gauge theory where an edge between two su(2) gauge algebras denotes a full hyper in bifundamental, while an between an su(2) and an so(n) gauge algebra denotes a half-hyper in bifundamental (see earlier discussion for our slightly non-standard definition of fundamental of so(3) and so (4)).
The edge between a node labeled nF and a node labeled su(2) denotes that the corresponding su(2) gauge algebras carries n extra hypers in fundamental representation, where n is allowed to be a half-integer to account for half-hypers in fundamental. Choosing a particular Λ Z g 2 ⊂ L corresponds to choosing all the gauge groups to be simply connected. The 1-form symmetry is predicted to be Λ Z g 2 for this choice, which can be verified easily from the 4d gauge theory description. A Z 2 factor arises as the subgroup of the center of each Spin(n) (where n = 3, 4) gauge group that leaves the vector rep of Spin(n) invariant.
Example and Comparison with 6d (1, 0) on T 2 : The last class of example has an alternative realization in terms of a 6d (1, 0) on T 2 [37,38]: For g = 1 and n = 2 the A 1 theory on C 1,2 has defect group L = Z 2 × Z 2 . We can alternatively think of this as the compactification of the 6d (1,0) theory that is the SU (2) − SU (2) conformal matter theory of rank 2, i.e. 2 M5-branes probing C 2 /Z 2 . The 6d theory has a tensor branch geometry, which has two non-compact curves, with SU (2) singularities, sandwiching a (−2)-curve, with SU (2) gauge group. The defect group given by Z 2 , and the dimensional reduction of this on T 2 , results in L A = L B = Z 2 . More generally, 2 M5-branes probing a Z k singularity results in a 'hybrid' class S theory, where an A 1 -trinion is glued to an A k−1 one (see (2.6) in [38]). The tensor branch-geometry changes simply to SU (k) groups both on the non-compact curves as

Z 2 -twisted Regular Punctures
In this subsection we either consider those g for with the outer-automorphism group is Z 2 , or the case g = D 4 with twist lines valued only in the Z 2 subgroup of the S 3 outer-automorphism group generated by the element b (see section 2).
Consider a (2, 0) theory compactified on C g with Z 2 twist lines, in the presence of both We choose f wrapped along B o i as the generator g B i of L B i , and the element obtained by wrapping f along the green sub-segment of A o i,i+1 to be the generator the non-trivial pairings 9 mod 1 are along with the previously discussed pairing on Z g A × Z g B . For g = A 2n , we have We choose f wrapped along B o i as the generator g B i of L B i , and the element obtained by wrapping f along the green sub-segment of A o i,i+1 to be the generator g A i,i+1 of L A i,i+1 . Then, the non-trivial pairings are (4.14) 9 The pairing is a product of an intersection number and the value of the bihomomorphism (2.2). Intersections are taken to be positive if complementing the direction of the first and second argument by a vector pointing outward from the page results in a right handed basis.
For g = D n , we have (4.15) We choose s wrapped along B o i as the generator g B i of L B i , and the element obtained by wrapping s along the green sub-segment of A o i,i+1 to be the generator g A i,i+1 of L A i,i+1 . Then, the non-trivial pairings are For g = E 6 , we have With these pairings, an absolute 4d N = 2 theory is chosen by a maximal isotropic subgroup The 4d theory carries a 1-form symmetry Λ. In the rest of this subsection, we substantiate our proposal by discussing a few Lagrangian examples.
In such a compactification, our proposal predicts that L is trivial, which can be verified by computing the set of line operators (modulo screening and flavor charges) in all of the above gauge theories. Hence, the 1-form symmetry is trivial and the gauge group in all these examples must be the simply connected one.
In both of these cases we have with the generators g A 1,2 and g B 1 having the pairing which can also be easily verified. For any consistent choice of gauge group and discrete theta parameters in the above gauge theories, the 1-form symmetry of the gauge theory is Z 2 .
Let us consider another example, which is of the 4d N = 2 quiver gauge theory where an edge between two nodes denotes a bifundamental hyper between the corresponding gauge algebras. This theory can be produced by compactifying A 2n−1 (2, 0) theory on a sphere with 4 regular twisted punctures and k + 3 regular untwisted punctures [24]. In this case, our proposal predicts that with the generators g A 1,2 and g B 1 having the pairing can be constructed by compactifying D 2n+1 (2, 0) theory on a torus with 6 regular twisted punctures [22]. Our proposal would predict that for this gauge theory we have and The non-trivial pairings on L are defined in terms of generators Let us reproduce this result by explicitly studying the line operators of the 4d gauge theory.
Before accounting for the matter content, the Wilson lines for all the gauge algebra factors form the group where (Z 4 ) i is associated to gauge algebra so(4n + 2) i , and (Z 2 ) i is associated to gauge algebra sp(2n) i . We choose generators W so i for (Z 4 ) i and W sp i for (Z 2 ) i . The matter content implies that the set of Wilson lines (modulo screening) can be generated by W so The first two generators are of order two, and the last generator is of order four. Thus, the contribution of Wilson lines to the set of line operators (modulo screening and flavor charges) is with the generators identified above. On the other hand, before accounting for the matter content, the 't Hooft lines for all the gauge algebra factors form the group where (Z 4 ) i is associated to gauge algebra so(4n + 2) i , and (Z 2 ) i is associated to gauge algebra sp(2n) i . We choose generators H so i for (Z 4 ) i and H sp i for (Z 2 ) i . The matter content requires us to choose the subset L H of Z H which is mutually local with the matter content. We can choose the generators for L H to be 2H so 1 , 2H so 2 , i (H so i + H sp i ). The first two generators are of order two, and the last generator is of order four. Thus, the contribution of 't Hooft lines to the set of line operators (modulo screening and flavor charges) is with the generators identified above. We thus see that clearly and the generators can be identified as

S 3 -twisted Regular Punctures
Now we consider incorporating more general regular twisted punctures in the D 4 (2, 0) theory.
We can have the following various irreducible configurations of twisted regular punctures as shown in figure 11: • Two punctures joined by an oriented Z 3 twist line implementing the transformation a ∈ S 3 as one crosses it in a particular direction (which is left to right in the fourth configuration of figure 11). We refer to this configuration as a "meson".
• Three punctures acting as sources of three a twist lines. The three twist lines meet at a point and annihilate each other. We refer to this configuration as a "baryon".
• Three punctures acting as sinks of three a twist lines. The three twist lines originate from a common point. We refer to this configuration as an "anti-baryon". • Two punctures emitting a 2 b and b Z 2 twist lines which combine to form an a twist line which ends at a puncture. We refer to this configuration as a "mixed" configuration.
• A puncture emitting an a twist line which then splits into a 2 b and b Z 2 twist lines. Each Z 2 twist line ends on a puncture. We refer to this configuration as an "anti-mixed" configuration.
There are plenty of redundancies when we try to combine the above configurations:  • l mesons.
• l mesons plus a Z 2 closed twist line.
• k open b lines plus l mesons.
• p baryons plus l mesons.
• One baryon plus l mesons plus a Z 2 closed twist line.
with the non-trivial pairings being  is broken into green and blue sub-segments lying between two difference kinds of twist lines the cycle crosses.   and c on the blue sub-segment of A b i,i+1 ; g A,b i,i+1 by inserting c on the green sub-segment of A b i,i+1 and v on the blue sub-segment of A b i,i+1 . Then the non-trivial pairings are and v A,a i,i+1 as the generators for L A,a i,i+1 . In total, we have with the non-trivial pairings being  figure 24 to be c wrapping the cycle B 1 . Then, we obtain that the total set of 4d line operators is (4.47)  which is obtained by wrapping c along the green sub-segment and s along the blue sub-segment of A b,a k,1 . In total, we have The non-trivial pairings are those on Z g A × Z g B , those given in (4.45), and those listed below arising by wrapping s along the green sub-segment of A a i,i+1 (which implies that v is wrapped along the red sub-segment and c is wrapped along the blue sub-segment) is called s A,a i,i+1 , and the element of L A,a i,i+1 arising by wrapping v along the green sub-segment of A a i,i+1 is called v A,a i,i+1 . We choose s A,a i,i+1 and v A,a i,i+1 as the generators for L A,a i,i+1 . In total, we have with the non-trivial pairings being . In total, we have is obtained by wrapping c along the green sub-segment of A b,a k,1 . Moreover, we can write c B,a . We can quickly deduce that There are no new non-trivial pairings.
being the only new non-trivial pairing not discussed previously. L is trivial for k = 0.   . This is the same configuration that we obtained above, and hence we expect that L should be trivial. Indeed, this is the case for the two 4d N = 2 gauge theories so(8) + 3F + 3S and so(7) + 2F + 3S.

Atypical Regular Punctures
Atypical regular punctures can be straightforwardly included in our analysis by resolving each atypical regular puncture into typical regular punctures. See the beginning of Section 4 for the definition of atypical regular punctures and further references which discuss them in detail. After this resolution, we observe that we have a sphere with what we referred to as a "mixed" configuration in section 4.3. Our analysis there suggests that we should have a trivial L, which matches the result obtained using the sp(2) + 6F gauge theory description.
As another example, we can obtain the 4d N = 2 gauge theory which matches the result obtained using the above gauge theory description, as the reader can readily verify.
As another example, we can obtain the 4d N = 2 gauge theory with gauge algebra sp(2) ⊕ sp(2) and hypermultiplet content 1 2 (Λ 2 , F) + (Λ 2 , 1) + 7 2 (1, F) by compactifying D 4 (2, 0) theory on a sphere with three twisted regular punctures acting as sources of a twist lines, thus forming a "baryon-like" configuration [26]. See figure 37. One out of these three punctures is typical, while two of them are atypical. Each atypical puncture can be resolved into two which matches the result obtained using the above gauge theory description, as the reader can readily verify.

Towards Irregular Punctures
The analysis of this paper has focused on compactifications of 6d (2, 0) theories involving only regular (either untwisted or twisted) punctures. In this section, we discuss how our analysis can be generalized to incorporate irregular punctures, which are the punctures where the Hitchin field has poles of higher-order than simple poles.
A class of irregular punctures for A n−1 (2, 0) theory were discussed in [39]. The Hitchin field at an irregular puncture of type P k in this class can be written as where 0 ≤ k ≤ n − 2 and the mass parameters satisfy k+1 i=1 m i = 0. Here ω is an n-th root of unity and Λ denotes the dynamically generated scale. For irregular puncture of type P n−1 , we can instead write ϕ = 1 z 2 diag(Λ, Λ, · · · , Λ, −(n − 1)Λ)dz + 1 z diag(m 1 , m 2 , · · · , m n )dz + · · · (5.2) Figure 38: A line carrying α ∈ Z can be deformed across an irregular puncture of type P k for 1 ≤ k ≤ n − 1. where n i=1 m i = 0. We would now like to understand how these irregular punctures impact the determination of 1-form symmetry. In particular, we would like to understand whether an element α of Z Z n can be moved across an irregular puncture of type P k (where k can take values in {0, 1, · · · , n − 1}). To answer this question, we consider compactifying A n−1 (2, 0) theory on a sphere with two irregular punctures both of same type P k . This leads to the 4d N = 2 asymptotically free gauge theory su(n) + 2kF [39]. We know from the gauge theory viewpoint that L is trivial if k > 0, from which we can bootstrap that an element α of Z Z n wrapped along the cycle W displayed in figure 39 can be contracted to a trivial loop. In other words, we learn that any element α of Z Z n can be moved across an irregular puncture of type P k if 1 ≤ k ≤ n − 1, see figure 38. Thus, as far as considerations about 1-form symmetry are concerned, an untwisted irregular puncture of type P k for k > 0 behaves exactly like an untwisted regular puncture, i.e. such an untwisted irregular puncture can be neglected when determining the 1-form symmetry.
On the other hand, for k = 0, we obtain the 4d N = 2 pure su(n) gauge theory. This gauge theory has L = Z n × Z n (5.3) Notice that our above proposal for 't Hooft line operators implies that a 6d surface operator can end on the codimension-2 defect associated to an irregular puncture of type P 0 . This is our first example of a puncture having this property. One would imagine that more general irregular punctures discussed in [40][41][42] allow a subgroup of Z to end on them, depending on the type of puncture. We defer a more thorough analysis to a future work, but finish this section by substantiating our proposal for the properties of punctures of type P k by studying the following example.

Class S from Type IIB
Class S theories can also have a realization in terms of a dual, Type IIB compactification, using geometric field theory methods, developed for general N = 2 theories, predating class S [43]. Type IIB on a canonical singularity gives rise to N = 2 SCFTs, and more generally can provide a way to engineer gauge theories. The Calabi-Yau X geometries that realize class S theories, can be constructed as ALE-fibrations over a curve where the resolutions parametrized for the ALE-fiber are encoded in a Higgs field ϕ. The connection is made through the Higgs bundle, [44,45]. The Higgs field ϕ is a meromorphic 1-form valued in the respective ADE Lie algebra g. We consider the 6d (2,0) theory of type We assume that the Higgs bundle is diagonalizable, i.e. ϕ = diag(λ 1 , · · · , λ r ). The spectral data encodes a local Calabi-Yau, which defines an ALE-fibration over C. Each sheet is labeled by a fundamental weight of g. For simplicity let us focus on the A N −1 case. There are N sheets, associated to the L i , i = 1, · · · , N fundamental weights, with the simple roots realized as α i = L i − L i+1 . The Higgs field eigenvalues λ i encode the volumes of the rational curve in the ALE-fibration, where each simple root is associated to a rational curve P 1 i , whose volume is determined by The Poincaré dual of these three-cycles are used to expand the supergravity four-form and construct the gauge bosons of the effective 4d theory. The gauge algebra of the theory is therefore determined by the initial choice of ADE gauge group and twist line structure.
Example: Consider the 6d (2, 0) A 2n−1 theory compactified on the torus C g = T 2 with a closed b twist line along the B cycle. The spectral cover Σ is a 2n-sheeted cover of the torus T 2 . Each sheet can be thought of as associated to a fundamental weight L i , i = 1, · · · , 2n, and the outer automorphism acts as o : which induces an action on the simple roots α i = L i − L i+1 ↔ α 2n−i . The root α n is fixed.
There are n 3-cycles, one for each orbit of the outer automorphism on the P 1 fibers which determine the root system of the 4d gauge algebra. These 3-cycles intersect linearly with the 3-cycle corresponding to the fixed P 1 lying at the end of this chain. The root originating from this P 1 is shorter than than the remaining n − 1 roots and we find the roots system of type B n . Overall we find the gauge group to reduce from SU (2n) to Spin(2n + 1) when introducing the twist line, the center of Spin(2n + 1) is Z 2 .

Line operators from IIB
The line operators in this context are realized in terms of wrapped D3-branes, on non-compact three-cycles, modulo screening by particles, which are D3-branes wrapped on compact threecycles. To study these, consider the analog arguments as in [14,16,18]. In relative homology, where ∂X is the boundary link 5-fold of the Calabi-Yau three-fold, the line operators are thereby realized in terms of where the topology of B k is given by and the topology of B F is given by (6.11) The contribution of (6.11) part of ∂X Cg,n to H 2 (∂X Cg,n , Z) is obtained by choosing an element α ∈ H 1 (S 3 /Γ ADE ), which is then fibered over a loop L in C g,n . We have where L is defined in (6.8).
Now, notice that the above contributions of the kind (α, L) are precisely the contributions we have been considering throughout the paper. Let us label the group of line operators obtained using the earlier considerations in the paper as L 0 . Then we clearly have L 0 ⊆ L F . (6.14) Thus, the only way for our previous calculation L 0 and the Type IIB calculation L to match is if In the rest of this subsection, we justify this equality.
First thing we need to show is that the contribution of all boundary components B k to L is trivial. Indeed, the only 2-cycles in B k are the exceptional P 1 s in C 2 /Γ ADE , but none of these 2-cycles is trivial when embedded into X, and hence do not contribute to L. The above argument can be viewed as a justification of our key assumption used in the earlier parts of the paper: If L is a loop surrounding a regular (untwisted or twisted) puncture carrying an element α ∈ Z(G) left invariant by the twist line emanating from the puncture, then such a loop is trivial in L. As an alternative approach one might consider arguing that closing an untwisted regular puncture does not change the defect group. It would be interesting to develop this point of view. Here we note, that in the geometric descriptions one could argue as folllows: regular punctures characterize base points at which fibral P 1 's both decompactify and potentially braid upon. For line operators the decompactification of cycles is immaterial. We can therefore rescale Higgs field with a factor of the base coordinate z preserving the braiding structure. This completely removes regular punctures. In other words, regular punctures can be filled in from the perspective of line operators and do not contribute to the defect group. It would be interesting to develop the precise dictionary, and to expand it to include irregular punctures.
Generically the above procedure can be applied to any canonical singularity. E.g. even in the case of general irregular punctures, which realize Argyres Douglas theories, that do not necessarily admit a Lagrangian description. The theories of type AD[G, G ] have a realization in terms of Type IIB on a canonical singularity and for AD[G, G ] theories, the 1-form symmetries are non-trivial only for G = A N with N > 1 and G = D, E type, see [14,18]. These results should provide further insights into computing the one-form symmetry for irregular punctures more generally.
• Z(G): The Pontryagin dual of the center of a simply connected group G. For a 6d N = (2, 0) theory associated to an A,D,E Lie algebra g, Z(G) captures the group of dimension-2 surface operators modulo screenings, also known as the defect group of the 6d theory.
• C g : A Riemann surface of genus g which might carry punctures depending on the context.
• L: The set of line operators (modulo screenings and flavor charges) for a relative 4d N = 2 theory obtained by compactifying a 6d N = (2, 0) theory on a Riemann surface C g , possibly in the presence of twist lines and (untwisted and twisted) regular punctures.
• ·, · : Often referred to as pairing. It takes two elements of Z(G) or of L, and outputs an element of R/Z which captures the phase associated to mutual non-locality of the defect operators associated to the two elements.
• Λ: Often referred to as polarization or maximal isotropic subgroup. This is a maximal subgroup of L such that the pairing ·, · on L restricted to this subgroup Λ vanishes. A choice of such a Λ is correlated to the choice of an absolute 4d N = 2 theory.
• Λ: Pontryagin dual of polarization Λ. Captures the 1-form symmetry of the absolute 4d N = 2 theory associated to a polarization Λ.
• S: Denotes the irreducible spinor representation for g = so(n).
• nR: Denotes n full hypermultiplets transforming in a representation R.
• 2n+1 2 R: Denotes n full hypermultiplets and a half-hypermultiplet transforming in a pseudo-real representation R.