1-form Symmetries of 4d N=2 Class S Theories
11-form Symmetries of 4 d N = 2 Class S Theories Lakshya Bhardwaj, Max Hübner, Sakura Schäfer-Nameki
Mathematical Institute, University of Oxford,Andrew-Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK
We determine the 1-form symmetry group for any 4 d N = 2 class S theory constructed bycompactifying a 6 d N = (2 ,
0) SCFT on a Riemann surface with arbitrary regular untwistedand twisted punctures. The 6 d theory has a group of mutually non-local dimension-2 sur-face operators, modulo screening. Compactifying these surface operators leads to a groupof mutually non-local line operators in 4 d , modulo screening and flavor charges. Completespecification of a 4 d theory arising from such a compactification requires a choice of a maximalsubgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4 d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment onhow to generalize our results to compactifications involving irregular punctures. Finally, tocomplement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realizationof class S theories. a r X i v : . [ h e p - t h ] F e b ontents d (2 ,
73 Compactifications without Punctures 8 Z Twist-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Including Closed S Twist-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Z -twisted Regular Punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 S -twisted Regular Punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Atypical Regular Punctures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 A Summary of Notation 57
A massive vacuum of a 4 d theory T is called confining if it preserves a non-trivial subgroupof the 1-form symmetry group of T [1]. Motivated by confinement in 4 d N = 1 theoriesobtained by deforming 4 d 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 4 d N = 2 theories. Morespecifically, we consider 4 d N = 2 theories of Class S that can be obtained by compactifying6 d N = (2 ,
0) SCFTs on a Riemann surface [3]. We allow the Riemann surface to containclosed twist lines and arbitrary regular punctures which can be either untwisted or twisted.It is well-known that 6 d N = (2 ,
0) SCFTs are classified by a Lie algebra g of ADE-type,and that they are relative QFTs [4–6], which for the purposes of this paper can be understoodas follows. The (2 ,
0) theory contains dimension-2 surface operators which are not mutuallylocal, i.e. there is an ambiguity in defining a correlation function containing two such surface2perators [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 anabelian group. Moreover, the surface operators can be screened by dynamical strings in thetheory. We denote the group of surface operators modulo screening by b Z .Upon compactification to 4 d , one can wrap these surface operators along various 1-cycleson the Riemann surface to generate an abelian group L of line operators modulo screening in4 d . The non-locality of 6 d surface operators descends to non-locality of these 4 d line operators.In other words, we obtain a relative d theory upon such a compactification. To obtain an absolute d theory T , one needs to choose a maximal subgroup Λ T ⊂ L of mutually local4 d line operators . The group Λ T can be identified with the set of charges for the 1-formsymmetry group of T . In other words, the 1-form symmetry group of T is identified as thePontryagin dual b Λ T of Λ T [1].Special cases of the problem explored in this paper have been discussed previously inthe literature. For example, in the case where the Riemann surface C g has no puncturesand no closed twist lines, the group L was already determined in [9] (see also the recentpaper [8]) to be H ( C g , b Z ). For the case of g = A and arbitrary C g with arbitrary numberof regular punctures, this problem was discussed in [10, 11]. Another situation where thisproblem has been discussed arises whenever there exists a degeneration limit of C g in whichthe 4 d theory can be identified as a weakly coupled 4 d gauge theory. In such a situation,one finds a canonical splitting L ’ L e × L m , where L e is associated to Wilson line operatorsand L m is associated to ’t Hooft line operators. In such a situation, the constraint of mutuallocality can also be understood as the constraint of Dirac quantization, and choosing a wayof satisfying Dirac quantization condition (i.e. a choice of Λ T ⊂ L e × L m ) can be interpretedas choosing a global form of the gauge group and possible discrete theta parameters [12] (seealso [1]). More recently work related to the higher form symmetry of 4d SCFTs and holographywas studied in [13]. In the context of non-Lagrangian 4d N = 2 SCFTs of Argyres-Douglastype the 1-form symmetries were computed using the Type IIB realization using canonicalsingularities in [14, 15], using the general observations in [16–18], which are applicable moregenerally to geometric engineering of SCFTs in string theory. Many recent papers have tackledthe problem of determining higher-form symmetries in lower-dimensional QFTs starting from The choice of Λ T is only part of the full set of choices one needs to make in order to define an absolute 4 d N = 2 theory of Class S. For example, one can obtain a group L of dimension-0 and a group L of dimension-2operators in the 4 d theory by compactifying the 6 d surface operators along the whole Riemann surface andalong a point on the Riemann surface respectively. Then the non-locality of the 6 d surface operators descendsto a non-locality between elements of L and L , and to choose an absolute 4 d N = 2 theory, one also needsto choose subgroups Λ and Λ of L and L , such that there is no non-locality between elements of Λ andΛ . See [8] for a recent discussion. d surface operatorwrapping a cycle surrounding a regular puncture does not contribute to the set L of 4 d lineoperators (modulo screening and flavor charges). In the case of untwisted regular punctures,any 6 d surface operator can be wrapped around the puncture, and hence according to theabove proposal, untwisted regular punctures are invisible to the determination of L and 1-form symmetry b Λ T of a 4 d N = 2 theory T obtained after choosing a polarization Λ T ⊂ L . Onthe 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 onthe 6 d surface operators, and hence only the 6 d surface operators left invariant by this actioncan be inserted along a loop surrounding the twisted regular puncture. Thus, according tothe above proposal, a twisted regular puncture is only invisible to the 6 d surface operatorsinvariant under the action of the corresponding twist line. As we discuss in various examplesthroughout the paper, a justification for the above proposal is that L obtained using it matchesthe L obtained using the gauge theory analysis of [12] (see also [1]) whenever there exists alimit of the compactification in which a weakly coupled 4 d N = 2 gauge theory arises [3,22–35].Let us now discuss a subtlety that arises due to the fact that one needs to take the areaof C g to zero in passing from the 6 d theory to the 4 d N = 2 theory. One might worry thatthe set L discussed might not be the true set of line operators modulo screening in the 4 d theory. However, this worry is alleviated by the fact that in order to define the 4 d N = 2theory one often needs to perform a non-trivial topological twist on C g , due to which oneexpects protected quantities to be independent of the area of C g . The set L is such a protectedquantity as each element in the set can be represented by a BPS line operator in the 4 d N = 2theory, and the screenings can also be understood in terms of BPS particles. On the otherhand, in situations where one does not need to perform a non-trivial topological twist, oneexpects that in general L should only be a subset of line operators (modulo screening) in the4 d N = 2 theory. An example where L does not capture the correct set of line operators isdiscussed towards the end of section 3.2.Many class S theories have known realizations in terms of local Calabi-Yau compactifica-tions in Type IIB in terms of an ALE-fibration over the curve C g,n . The defect group in those Note that when we refer to untwisted/twisted in this paper, we usually refer to the absence/presence ofouter-automorphism twist lines, not to the topologial twist. Although in principle any class S theory should have a IIB compactification associated to it, the preciseconstruction in particular in the case of non-diagonalizable Higgs fields and irregular punctures is – to ourknoweldge – not developed. √L untwisted handle b Z ( G ) o -twisted handle Inv( b Z ( G ) , o )untwisted regular puncture 0(open Z twist line of type o , open Z twist line of type o ) b Z ( G ) / Inv( b Z ( G ) , o )(open b line, a -twisted handle) Z (open b line, open b line) 0(meson, meson) Z × Z (meson, b -twisted handle) Z (open b line, meson) Z (baryon, baryon) Z × Z (meson, baryon) Z × Z (baryon, b -twisted handle) Z (open b line, baryon) Z (open b line, mixed configuration) Z Table 1: Summary of class S data and their impact on the defect group L . The contribution isalways squared, so we only list half of the contribution to L for each kind of Class S datum. Forexample, the first entry describes that an untwisted handle of the Riemann surface contributes b Z ( G ) × b Z ( G ) to L . The first four entries are universal for any class S construction – includingcontributions from the genus, punctures and twist-lines. Here b Z ( G ) is the Pontryagin dualof the center Z ( G ) of the simply connected group G associated to the ADE-algebra g of the6 d (2 ,
0) theory. Inv( b Z ( G ) , o ) is the subgroup of b Z ( G ) left invariant by the action of outer-automorphism o on b Z ( G ). An o -twisted handle refers to a handle carrying a closed o -twist linewrapped along either A or B cycle of the handle. An untwisted puncture does not contributeanything to L . The entries after the double-line refer to the S twisted compactifications of D (2 ,
0) theory, where open twist lines form a variety of irreducible configurations (meson,baryon etc.) and this comprises a summary of our findings in section 4.3, and we refer thereader there for a detailed discussion. 5ases 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 lo-cal Higgs bundle realization of the ALE-fibration of the Calabi-Yau three-fold, we determinethese homology groups and confirm the defect group for the case of no punctures and forregular untwisted and twisted punctures: The defect group L has a simple description, purelyin terms of the data on the boundary of the non-compact Calabi-Yau threefold, namely theboundary B F = S / Γ ADE → C fibration, where C is the Gaiotto curve, and the base of theALE-fibration. Then the defect group is simply given in terms of the 2-cycles of B F , whichextend trivially to the Calabi-Yau.In fact, as we discuss in section 6.2, this approach can be viewed to provide a justifica-tion for our key proposal that a 6 d surface operator wrapping a cycle surrounding a regularpuncture does not contribute to L . Moreover, this approach might shed light on the irregularpunctures as, e.g. generalized AD theories have a realization in terms of Type IIB on canonicalsingularities, 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 Riemannsurface used for compactification. We collect this rough decomposition of L in Table 1 to beused as a reference. It is important to note that the table only captures the group-structureof L , while one of the key ingredients is the pairing on L capturing the mutual non-localityof 4 d line operators. This pairing is required to choose a polarization Λ and determine thecorresponding 1-form symmetry b Λ. The explicit form of the pairing can be found in the maintext.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 6 d N = (2 , d N = 2 theories obtained bycompactifying 6 d (2 ,
0) theories on a genus g Riemann surface in the presence of arbitrarytwists by outer-automorphism discrete 0-form symmetries, but without involving any punc-tures. In section 4 we extend our analysis of previous section to includ arbitrary untwistedand twisted regular punctures. In section 5 we sketch how our analysis can be extended toinclude irregular punctures, giving explicit results for a specific class of irregular puncturesof A n − (2 ,
0) theories. Finally, in section 6 we argue from a Type IIB realization of class Stheories for the 1-form symmetries. Our notation is summarized in appendix A.6 Z ( G ) b Z ( G ) h· , ·i A n − Z n Z n h f, f i = n D n Z × Z Z × Z h s, s i = 0 , h c, c i = 0 , h s, c i = D n +1 Z Z h s, s i = D n +2 Z × Z Z × Z h s, s i = , h c, c i = , h s, c i = 0 D n +3 Z Z h s, s i = E Z Z h f, f i = E Z Z h f, f i = E − Table 2: For the ADE Lie algebras g we denote by G the simply-connected Lie group, andlist the center Z ( G ), the Pontryagin dual group to the center b Z ( G ), and the bihomomorphism h· , ·i . E has a trivial center group, which has been denoted by 0 since we use an additivenotation for the group multiplication law throughout this paper. We denote a generator of b Z ( G ) for g = A n − , E , E as f ; a generator of b Z ( G ) for g = D n +1 as s ; and generators of b Z ( G ) ’ Z × Z for g = D n as s, c . We also define v := s + c for g = D n . d (2 , d N = (2 ,
0) SCFTs are relative QFTs classified by a simple Lie algebra g of A, D, E type.Such a theory contains surface defect operators of dimension 2. Modulo screening by dynamicalobjects, these operators can be classified by the Pontryagin dual b Z ( G ) of the center Z ( G ) of thesimply connected group G associated to g , which are summarized in table 2. The Pontryagindual b Z ( G ) := Hom( Z (cid:0) G ) , R / Z (cid:1) of a finite abelian center group is isomorphic to the centergroup itself.These surface operators are not all mutually local. Consider a correlation function con-taining two surface operators α, β ∈ b Z ( G ). As α is moved around β , the correlation functionis transformed by a phase factor exp (cid:0) πi h α, β i (cid:1) (2.1)with a bihomomorphism h· , ·i : b Z ( G ) × b Z ( G ) → R / Z . (2.2)The bihomomorphism can be specified by providing its values on the generators of b Z ( G ) [8].These are also listed in table 2. Notice (in the following equation) that we define the pairing with a negative sign as compared to thestandard choice, which can be found for example in [8]. ,
0) theory admits a discrete 0-form symmetry which can be identified with thegroup of outer-automorphisms O g of g , which are O g = Z (2.3)for g = A n ≥ , D n ≥ , E , and O D = S , (2.4)namely the group formed by permutations of three objects. O g is trivial for E and E . Theouter-automorphisms act on representations of g , and hence on b Z ( G ). For g = A n , D n +1 , E ,the non-trivial element of O g = Z acts by sending the generator of b Z ( G ) to its inverse. For g = D n and n ≥
3, the non-trivial element of O g = Z acts by exchanging the two chosengenerators s, c of b Z ( G ) ’ Z × Z . For g = D , we generate O D = S in terms of a Z and a Z subgroup of it. We choose generators a ∈ Z and b ∈ Z , which act as follows a : s → v, v → c, c → sb : s → c, c → s, v → v . (2.5)Then the elements of S can be written as 1 , a, a , b, ab, a b . An important conjugation relationwe will use throughout the paper is bab = a . 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 compact-ification, such a compactification is called as an untwisted compactification. On the otherhand, we can also consider twisted compactifications which means the following. The outer-automorphism 0-form symmetry in 6 d (2 ,
0) theory discussed in the last section is generatedby topological operators of codimension-1 in the 6 d theory. Inserting such a topological op-erators along a cycle of the Riemann surface gives rise to a “codimension-0 object” in the 4 d theory, which means that the resulting 4 d theory itself is different from the 4 d theory arisingwhen no such topological operators are inserted. We often refer to the locus of the topologicaloperator on C g as a twist line , and when this locus is a 1-cycle on C g we say that the twist lineis closed . In the presence of punctures this picture is enhanced by the alternative of open twistlines. 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 Higgsbundle description of the compactification. Here the insertion of topological operators alongtwist lines gives rise to an action on the Higgs field by an outer automorphism o across these.8he insertions alter the gauge group of the effective 4d N = 2 theory and have a geometricinterpretation in the IIB dual description as we explain in more detail in section 6. In thisgeometric picture we are further able to justify the key assumption that regular untwistedpunctures are irrelevant in determining the defect group, which we also argue for in thesection 4. Let us compactify a (2 ,
0) theory on a Riemann surface C g of genus g without any puncturesor twists. This gives rise to a relative 4 d N = 2 theory with a set of line defects descendingfrom the elements of b Z ( G ) wrapped along various cycles of C g . That is, the set L of 4 d linedefects (modulo screening) can be identified with H ( C g , b Z ) ’ H ( C g , Z ) ⊗ b Z . (3.1)These line defects are not all mutually local. The violation of mutual locality between twoelements a ⊗ α, b ⊗ β ∈ H ( C g , Z ) ⊗ b Z ’ H ( C g , b Z ) is captured by the phaseexp (cid:0) πi h α, β ih a, b i (cid:1) , (3.2)where h a, b i is the intersection pairing on H ( C g , Z ). This gives rise to a pairing on H ( C g , b Z )which is the natural combination of the intersection pairing and the bihomomorphism (2.2) h· , ·i : H ( C g , b Z ) × H ( C g , b Z ) → R / Z h a ⊗ α, b ⊗ β i = h a, b ih α, β i . (3.3)We can specify an absolute 4 d N = 2 theory by choosing a maximal set of line operatorsΛ ⊂ H ( C g , b Z ) , (3.4)which are all mutually local, i.e. the phase (3.2) is trivial for any two elements in Λ. Sucha set Λ is also referred to as a ‘maximal isotropic subgroup’ or as a ‘polarization’ in whatfollows. The 1-form symmetry of the absolute 4 d N = 2 theory can then be identified withthe Pontryagin dual b Λ of Λ.Once we choose a set of A and B cycles on C g , we can decompose H ( C g , b Z ) ’ b Z gA × b Z gB , (3.5)where b Z gA is the contribution of A-cycles, and b Z gB is the contribution of B-cycles. Moreover, b Z gA and b Z gB are maximal isotropic sublattices, and hence provide canonical choices of Λ oncea choice of A and B cycles has been made. 9 xample : When (2 ,
0) theory of type g is compactified on a torus, we obtain 4 d N = 4 SYMwith gauge algebra g . Choosing an A-cycle and a B-cycle, we write H ( T , b Z ) ’ b Z A × b Z B . (3.6)We assume without loss of generality that the A-cycle is much shorter than the B-cycle. Then, b Z A can be identified as the set of 4 d Wilson line operators, and b Z B can be identified as the setof 4 d ’t Hooft line operators. Choosing Λ = b Z A , we obtain 4 d N = 4 SYM with gauge group G . On the other hand, choosing Λ = b Z B , we obtain 4 d N = 4 SYM with gauge group G /Z ( G )and all discrete theta parameters turned off. In these cases, we have 1-form symmetry b Λ ’ Z ( G ) , (3.7)which matches with the 1-form symmetry obtained using the Lagrangian description of 4 d N = 4 SYM: when the gauge group is G , this is the electric 1-form symmetry; and then thegauge 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 bychoosing other polarizations. For concreteness, consider the case of g = su (4). In this case, b Z A ’ b Z B ’ Z . The P SU (4) theory with a discrete theta parameter n ∈ { , , , } turnedon is obtained by choosing Λ to be the sublattice generated by the element ( n, ∈ Z × Z ’ b Z A × b Z B (where we have represented Z as the additive group Z / Z ). Any such choice leadsto the 1-form symmetry b Λ ’ Z . (3.8)If we choose the polarization Λ generated by elements (0 ,
2) and (2 ,
0) in Z × Z , then weobtain the SO (6) ’ SU (4) / Z theory with the discrete theta parameter turned off. In thiscase the 1-form symmetry is b Λ ’ Z × Z . (3.9)From the point of view of the Lagrangian description, the two Z factors are electric andmagnetic 1-form symmetries respectively. The remaining su (4) theory has SO (6) gauge groupand a discrete theta parameter turned on. This is obtained by choosing Λ to be generated bythe element (1 , ∈ Z × Z ’ b Z A × b Z B , and the 1-form symmetry group of the theory is b Λ ’ Z . (3.10) Example : Consider compactifying A (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 4 d = 2 theory so (4) so (3) su (2) su (2) so (4) · · · so (4) su (2) so (3) F F (3.11)where we have a total of 2 g − between the two nodes. An edge connectingan su (2) node to a node labeled F implies that the corresponding su (2) gauge algebra carriesan extra half-hyper charged in fundamental rep. If we choose Λ = ( Z / Z ) gA , we obtain the4 d theory with all the gauge groups being simply connected. In this case, we have 1-formsymmetry b Λ ’ Z g , (3.12)which can be easily matched with the above Lagrangian description with all the gauge groupschosen to be the simply connected ones. A Z factor arises from each of the g number of so ( n )nodes (where n = 3 , n )). This Z is the subgroupof center of Spin( n ) that acts trivially on the fundamental representation of so ( n ) as definedin the above footnote. Z Twist-lines
We can also consider twisted compactifications of 6 d N = (2 ,
0) on C g (without punctures).This involves wrapping the topological defects generating the outer-automorphism discrete0-form symmetries along cycles on C g . In this subsection we either consider those g for whichthe outer-automorphism group is Z , or the case g = D with twist lines valued only in the Z subgroup of the S outer-automorphism group generated by the element b (see section 2).We can wrap the Z twist lines along some L ∈ H ( C g , Z ). Let us first discuss the case of g = 1. Without loss of generality we can choose L to be the B-cycle of the torus.Then, along the dual A-cycle, we can only wrap those elements of b Z which are left invariantby the action of Z outer-automorphism o – see figure 1. Let us denote this subgroup of b Z byInv( b Z, o ) := b Z | o = { z ∈ b Z : o · z = z } . (3.13) Here, for ease of notation, we are using the convention that the fundamental representation of so ( n ) isthe n -dimensional vector representation. So, the fundamental representation for so (3) is not the fundamentalrepresentation for su (2), but rather the adjoint representation. Similarly, the fundamental representation of so (4) is the (2 ,
2) rep of su (2) ⊕ su (2) ’ so (4). o · αα = ⇒ o · α = α Figure 1: A closed Z twist line o is inserted along the B-cycle of a torus. An element α ∈ b Z inserted along the A-cycle is acted upon by o as it crosses the closed twist line. Since theA-cycle closes back to itself we deduce that only the elements α left invariant by the action of o can be inserted along the A-cycle. = = αo o · αooα Figure 2: A closed Z twist line o is inserted along the B-cycle of a torus. An element α ∈ b Z inserted along the B-cycle can be moved around and converted to the element o · α insertedalong the B-cycle.For g = A n − , we can only wrap the element nf ∈ b Z ’ Z n and henceInv( b Z, o ) ’ Z . (3.14)Similarly, for g = D n , only v can be wrapped and henceInv( b Z, o ) ’ Z . (3.15)For g = A n and g = E , no element in b Z can be wrapped and hence Inv( b Z, o ) is trivial. For g = E , E the group of outer automorphisms is trivial.On the other hand, along the B-cycle we can wrap any element α ∈ b Z , but moving itacross the twist line implies that α can be identified with the element o · α ∈ b Z where o · α isobtained by applying the Z action on α . See figure 2. The set of 4 d line operators descendingfrom the B-cycle, which we denote asProj( b Z, o ) := b Z h g − o · g i (3.16)can be obtained by modding out b Z by the identifications imposed by o , that is by moddingout b Z by the subgroup h g − o · g i ⊆ b Z generated by the element g − o · g ∈ b Z where g is a12enerator of b Z . For g = D n , the action of o implies that s ∼ c , which implies v ∼
0, andconsequently Proj( b Z, o ) ’ Z , (3.17)whose non-trivial element can be identified either with s or with c . For g = A n − and g = E ,we have f ∼ − f , which implies that 2 mf ∼ m ∈ Z . Thus for g = A n − , we haveProj( b Z, o ) ’ Z , (3.18)whose non-trivial element can be represented by any element of the form (2 m +1) f ∈ b Z ’ Z n .For g = A n and g = E , we can write f = 2 mf for m = n + 1 and m = 2 respectively, andhence Proj( b Z, o ) is trivial.Now, notice that Inv( b Z, o ) and Proj( b Z, o ) have a non-trivial mutual pairing which descendsfrom the mutual pairing (3.3) between b Z A and b Z B . For example, for g = D n , the generatorfor Inv( b Z, o ) ’ Z is v ∈ b Z A , and the generator for Proj( b Z, o ) ’ Z can be taken to be s ∈ b Z B .Then the pairing between the generators is h v, s i = 12 . (3.19)Had we chosen the generator of Proj( b Z, o ) to be c ∈ b Z B instead, we would have obtained thesame pairing as above. For g = A n − , the generator for Inv( b Z, o ) ’ Z is nf ∈ b Z A , and thegenerator for Proj( b Z, o ) ’ Z can be taken to be some (2 m + 1) f ∈ b Z B . The pairing betweenthe generators is h nf, (2 m + 1) f i = 12 , (3.20)irrespective of the value of m .An absolute 4 d N = 2 theory is then specified by choosingΛ ⊂ L ’ Inv( b Z, o ) × Proj( b Z, o ) , (3.21)with Λ being maximally isotropic. The 1-form symmetry of the 4 d N = 2 theory can then beidentified with b Λ.For a general C g with arbitrary g , the twist lines are specified by picking an element L ∈ H ( C g , Z ). By Poincare duality, we can work with the dual element b L ∈ H ( C g , Z ).Choose a set of A and B cycles on C g . Then b L assigns values b L ( A i ) , b L ( B i ) ∈ { , } to allcycles A i , B i . We can perform Sp (2 g, Z ) transformations to transform to a new set of A andB cycles such that only b L ( A ) = 1, while b L ( A i ) = 0 for all i = 1 and b L ( B i ) = 0 for all i . Thatis, in this frame, which can always be chosen, the twist line L wraps only the cycle B . Seefigure 3. 13 g B o A g A Figure 3: A Riemann surface of genus g with a closed Z twist line o wrapped along the B cycle.Now, combining the results discussed previously, we easily identify the set L of 4 d lineoperators (modulo screening). We find that the polarization is chosen asΛ ⊂ L ’ Inv( b Z, o ) × Proj( b Z, o ) × b Z g − A × b Z g − B , (3.22)where the pairing is obvious from our previous discussion. The 1-form symmetry group ofsuch an absolute 4 d N = 2 theory is identified as b Λ. Example : Consider compactifying D n +1 (2 ,
0) theory on a torus, and wrap a Z twist linealong the B-cycle. We can write the set of line defects asInv( b Z, o ) × Proj( b Z, o ) ’ ( Z / Z ) A × ( Z / Z ) B . (3.23)First, assume that the A-cycle is much shorter than the B-cycle. This corresponds to firstcompactifying D n +1 (2 ,
0) theory on a circle with outer-automorphism twist, leading to 5 d N = 2 SYM with sp ( n ) gauge algebra and discrete theta angle θ = 0 [36]. We furthercompactify this 5 d theory on another circle obtaining 4 d N = 4 SYM with sp ( n ) gaugealgebra. Choosing Λ = ( Z / Z ) A corresponds to picking the simply connected Sp ( n ) gaugegroup for the 4 d theory, and the 1-form symmetry b Λ ’ Z can be identified as the electric1-form symmetry from the point of view of this Lagrangian 4 d theory. Choosing Λ = ( Z / Z ) B leads to gauge group Sp ( n ) / Z with the discrete theta parameter turned off, and the 1-formsymmetry b Λ ’ Z can be identified as the magnetic 1-form symmetry from the point of viewof this Lagrangian 4 d theory.Now, assume that the B-cycle is much shorter than the A-cycle. This corresponds to firstcompactifying D n +1 (2 ,
0) theory on a circle without outer-automorphism twist, leading to 5 d N = 2 SYM with so (2 n + 2) gauge algebra. We further compactify this 5 d theory on another14ircle with a Z outer-automorphism twist, leading to 4 d N = 4 SYM with so (2 n + 1) gaugealgebra. Choosing Λ = ( Z / Z ) A corresponds to picking the SO (2 n + 1) gauge group for the4 d theory with all discrete theta parameters turned off, and the 1-form symmetry b Λ ’ Z canbe identified as the magnetic 1-form symmetry from the point of view of this Lagrangian 4 d theory. Choosing Λ = ( Z / Z ) B leads to the simply connected gauge group Spin(2 n + 1), andthe 1-form symmetry b Λ ’ Z can be identified as the electric 1-form symmetry from the pointof view of this Lagrangian 4 d theory. Non-example : Consider compactifying A n (2 ,
0) theory on a torus, and wrap a Z twist linealong the B-cycle. Our proposal would predict the set L of 4 d line defects (modulo screening)to be L ’
Inv( b Z, o ) × Proj( b Z, o ) ’ , (3.24)which is the trivial group. That is, all the 4 d line defects are proposed to be screened.Correspondingly, the 1-form symmetry of the resulting 4 d 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 firstcompactifying the A n (2 ,
0) theory on a circle of radius R with an outer-automorphism twist,thus leading to 5 d N = 2 SYM with sp ( n ) gauge algebra with gauge coupling g Y M = R anddiscrete theta angle θ = π [36]. Due to the presence of non-trivial discrete theta angle theBPS instanton particle in this 5 d theory transforms in the fundamental representation of sp ( n ).Thus, the group of line operators (modulo screening) in this 5 d theory is trivial. Moreover,every possible ’t Hooft dimension-2 surface operator in the 5 d theory which is local with theabove mentioned instanton BPS particle is screened. Thus, the group of surface operators(modulo screening) in this 5 d theory is also trivial.Compactifying the above 5 d theory further on a circle of finite non-zero radius R , oneexpects the 4 d theory obtained to have no line defects (modulo screening), since there are noline or surface defects (modulo screening) in the 5 d theory as we saw above. This is so farconsistent with our above predictions.However, as we send R , R → R /R preserved, we obtain the 4 d N = 4theory having g = sp ( n ) with gauge coupling g Y M = R /R and theta angle θ = π . This 4 d theory clearly has a Z × Z group of 4 d line operators (modulo screening). Thus, our abovepredictions do not provide the correct answer in the limit when the torus is shrunk to zerosize.From the point of view of the above 5 d theory, this limit decouples the BPS instantonparticle responsible for screening the fundamental Wilson line, since the mass m of the BPS15 → ab b a −→ a b b a ;Figure 4: Resolving various S lines into a -lines and b -lines.instanton particle scales as m ∼ /R → ∞ . This means that the fundamental Wilson line isnot screened after taking this limit. Moreover, the ’t Hooft operator which was not mutuallylocal with the BPS instanton particle becomes available, and we recover the correct resultthat the set of 4 d line operators (modulo screening) is Z × Z . There are 3 distinct choicesof polarization corresponding to choosing the 4 d gauge group Sp ( n ) and Sp ( n ) / Z with adiscrete Z valued theta parameter. In each of the three cases, the true 1-form symmetry is Z , which is interpreted as an emergent d → d compactification.The fact that our predicted result for L does not capture the true L is not surprising asexplained in the introduction. As discussed there, the predicted L is guaranteed to matchthe true L only when a non-trivial topological twist is performed on C g . When no non-trivialtopological 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 protectedas we take the limit of zero area. When there is no topological twist, the set may not beprotected, as we saw in the example above where a 4 d BPS particle (descending from the 5 d BPS instanton particle) was decoupled in the limit of zero area. S Twist-lines
An arbitrary S twist on C g can be manufactured by combining a and b twist lines, whichare two elements of orders three and two respectively inside S (see section 2). An arbitrary S twist is described as a trivalent network of topological lines valued in S obeying groupcomposition 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 b can beseparated into b and a , while an edge carrying either of 1 , a, a , b is left alone without anydecomposition (See figure 4). Each trivalent vertex is similarly decomposed into a vertex for b lines and a vertex for a, a lines. To decompose the vertices, we have to sometimes cross an a or a line across a b line. Such a crossing transforms a to a and a to a . See figure 5. Afterdecomposing the vertices, the original network has been decomposed into a network of b lines,16 b a ba −→ b ba a aa Figure 5: An example of resolving a trivalent S vertex into an a -vertex and a b -vertex. Noticethat two b lines meet to form a trivial line (since b = 1), which has not been displayed. Thevertex formed by b lines can now be smoothened out.with a network of a, a lines placed on top of the network of b lines such that any time a linecarrying a or a crosses a b line, it is transformed to an a or a line respectively.The network of b lines can be smoothened, and hence is an element L ∈ H ( C g , Z ) alongwhich we wrap b lines. In the previous subsection, we have seen that we can always choose L = 0 or L = B . If we choose L = 0, then we can represent the network of a lines as anelement ‘ ∈ H ( C g , Z ) with the dual element being b ‘ ∈ H ( C g , Z ). We can perform Sp (2 g, Z )transformations on cycles A i , B i to obtain a frame such that b ‘ ( A ) ∈ { , } , b ‘ ( B i ) = 0 for all i , and b ‘ ( A i ) = 0 for all i = 1. If b ‘ ( A ) = 0, then we are back in the completely untwisted casediscussed earlier. If b ‘ ( A ) = 1, then we have an a line wrapping B . We can see that thereis no element of b Z that can wrap A . On the other hand, any element of b Z which wraps B can be identified with the trivial element of b Z due to the action of a twist line. Thus, in thiscase, an absolute 4 d N = 2 theory is chosen byΛ ⊂ L ’ b Z g − A × b Z g − B (3.25)with b Z ’ Z × Z . The 1-form symmetry is identified with b Λ.Now, let us choose L = B . Since there is only a single b line, it is not possible to closean a line that crosses b line. Using this fact, one can argue that the only possible networkof a, a lines can be represented as an element ‘ ∈ H ( C g , Z ) which does not wrap A . Letthe dual element be b ‘ ∈ H ( C g , Z ), which has the property that b ‘ ( B ) = 0. Now we canperform Sp (2 g − , Z ) transformations on cycles A i , B i for i = 1 to obtain a frame such that b ‘ ( A i ) ∈ { , } for i = 1 , b ‘ ( B i ) = 0 for all i , and b ‘ ( A i ) = 0 for all i = 1 ,
2. If b ‘ ( A ) = 1, thenin total we have the ab line wrapping B (where we are representing S as a multiplicativegroup). But since ab is in the same conjugacy class as b , we can replace ab wrapping B by b wrapping B by performing gauge transformation inside S . Thus, we can always ensure that17 g b aB B B g A Figure 6: A Riemann surface of genus g with a closed Z twist line b wrapped along the B cycleand a closed Z twist line a wrapped along the B cycle. The cycle A (which is homologicallyequivalent to A + A ) has been divided into two sub-segments, denoted respectively by greenand blue. The color is changed as A crosses a twist line, indicating that an element of b Z wrapped along one sub-segment is in general different from the element of b Z wrapped alongthe other sub-segment, due to the action of outer-automorphism associated to the twist line. b ‘ ( A ) = 0. Now if b ‘ ( A ) = 0, then we are back in the Z twisted case discussed before.The only new case therefore is when b ‘ ( A ) = 0 and b ‘ ( A ) = 1, which has been representedin figure 6. Consider the impact of twist lines on the elements of b Z wrapping B-cycles first.Notice that s wrapped along the red cycle can be identified with v ( B ) (i.e. v wrapped alongthe cycle B ) by moving it to the left, and with c ( B ) by moving it to the right. See figure6. Thus v ( B ) = c ( B ). Similarly, c wrapped along the red cycle can be identified with v ( B ) by moving it to the left, and with v ( B ) by moving it to the right. See figure 6. Thus v ( B ) = c ( B ) = v ( B ) and we deduce that the B-cycles give rise to a groupProj( b Z ; a, b ) ’ Z × Z (3.26)generated by s ( B ) , c ( B ). Now, consider the impact of twist lines on the elements of b Z wrapping A-cycles. We can have v ( A ), but nothing can wrap A alone. Consider the cycle A = A + A which is denoted as partly blue and partly green in figure 6. We can wrap s along the blue sub-segment of A , and c along the green sub-segment of A . This configurationis consistent with the twist lines along B and B . We label the 4 d line operator obtained viathis configuration as s ( A ). Thus, the A-cycles give rise to the groupInv( b Z ; a, b ) ’ Z × Z (3.27)generated by v ( A ) , s ( A ). It can be easily seen that the pairs of dual elements in Inv( b Z ; a, b ) × Proj( b Z ; a, b ) are { v ( A ) , s ( B ) } , { s ( A ) , c ( B ) } . Thus, an absolute 4 d N = 2 theory is chosenby Λ ⊂ Inv( b Z ; a, b ) × Proj( b Z ; a, b ) × b Z g − A × b Z g − B (3.28)18 Twisted PunctureUntwisted PunctureFigure 7: A twisted puncture lives at the end of a non-trivial twist line t , while an untwistedpuncture does not live at the end of a non-trivial twist line.with Inv( b Z ; a, b ) and Proj( b Z ; a, b ) given by (3.27) and (3.26) respectively. The 1-form symme-try of this absolute 4 d N = 2 theory is identified with b Λ. Regular punctures are a special set of punctures defined by the condition that the Hitchin fieldhas (at most) a simple pole at the location of the puncture. These punctures can be either untwisted or twisted . Twisted regular punctures arise at the ends of twist lines, and hencethe Hitchin field transforms by the action of the corresponding outer-automorphism as oneencircles a twisted regular puncture. On the other hand, untwisted punctures do not live atthe ends of non-trivial twist lines, and correpondingly the Hitchin field does not pick up theaction of any non-trivial outer automorphism as one encircles an untwisted regular puncture.See figure 7.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 atypicalregular punctures, the number of simple factors in the gauge algebra arising in a degenerationlimit of the Riemann surface is not equal to the dimension of the moduli space of the Riemannsurface [24–26] (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 regularpunctures. Throughout this section until subsection 4.4, a regular puncture always refers toa typical regular puncture.In this section, we consider compactifications of 6 d N = (2 ,
0) theories on a Riemannsurface C g with an arbitrary number of (untwisted and twisted) regular punctures, and anarbitrary number of closed twist lines (which do not have end-points). Let L be the set of 4 d line operators (modulo screening) when a (2 ,
0) theory is compactifiedon a Riemann surface C g without any punctures, but possibly in the presence of closed twist19ines. The set L (and Dirac pairing on it) was determined in the last few subsections. Now,insert n regular untwisted punctures on C g . We propose that the set of 4 d line operatorsmodulo flavor charges (and screening) can again be identified with L . Moreover, an absolute4 d N = 2 theory is obtained by choosing a maximal isotropic subgroupΛ ⊂ L (4.1)and the 1-form symmetry of such an absolute 4 d N = 2 theory can be identified with b Λ.In other words, regular untwisted punctures turn out to be irrelevant for the considerationsof this paper. In the rest of this section, we substantiate this proposal by studying someexamples.
Sphere with 4 regular untwisted punctures : As a few examples, we can obtain the fol-lowing 4 d N = 2 gauge theories by compactifying (2 ,
0) theories on a sphere with 4 regularuntwisted punctures :• su ( n ) + 2 n F by compactifying A n − (2 ,
0) theory.• so (8) + 2 F + 2 S + 2 C , so (8) + 4 S + 2 C , so (8) + 4 S + C + F by compactifying D (2 , so (9) + 3 S + F , so (10) + 4 S , so (10) + 2 S + 4 F by compactifying D (2 ,
0) theory [28].• so (11)+ S +5 F , so (11)+ S +3 F , so (12)+ S + C +4 F , so (12)+ S +6 F , so (12)+ S + C +2 F by compactifying D (2 ,
0) theory [28].• su (4) + 2 Λ + 4 F , sp (2) + 6 F by compactifying A (2 ,
0) theory [27].For this case L is trivial, which is what is expected from the 4 d gauge theory description asit can be checked that the line operators (modulo screening and flavor charges) form a trivialset in all of the above gauge theories. Consequently, the 1-form symmetry is also trivial forall 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 follow-ing 4 d N = 2 gauge theories by compactifying (2 ,
0) theories on a torus with 1 regular un-twisted puncture and a twisted line wrapped along a non-trivial cycle : The notation g i + P n i R i denotes a 4 d N = 2 gauge theory with gauge algebra g along with n i full hypersin irrep R i . If n i is half-integral, it means that there is an additional half-hyper in R i along with b n i c numberof full hypers in R i . F denotes fundamental irrep for su ( n ) and sp ( n ), and vector irrep for so ( n ). S denotesspinor irreps for so ( n ) and C denotes co-spinor irrep for so (2 n ). Λ denotes 2-index antisymmetric irrep for su ( n ) and sp ( n ). See also appendix A. S denotes the 2-index symmetric representation of su ( n ). See also appendix A. su (2 n ) + S + Λ by compactifying A n − (2 ,
0) theory.• su (2 n + 1) + S + Λ by compactifying A n (2 ,
0) theory.In the former case, we have
L ’ Z × Z , (4.2)which can be matched with the 4 d gauge theory expectation. For a pure su (2 n ) gauge theory,the set of Wilson lines (modulo screening) is Z n with generator W being the Wilson linein fundamental rep of su (2 n ). The set of ’t Hooft lines (modulo screening) is also Z n withgenerator H . The Dirac pairing between W and H is h W, H i = n . Now we add in thematter. The hypermultiplets in S and Λ screen 2 W , and thus the set of Wilson lines (moduloscreening and flavor charges) can be identified with Z , generated by W . On the other hand,the ’t Hooft lines must be mutually local with 2 W , and hence the set of ’t Hooft lines (moduloscreening and flavor charges) can be identified with Z , generated by nH . Thus, we verify theprediction (4.2). Choosing the polarization Λ to be the Z generated by W leads to gaugegroup SU (2 n ). Choosing Λ to be the Z generated by nH or W + nH leads to gauge group SU (2 n ) / Z with discrete theta parameter turned off or on respectively. In all these cases, the1-form symmetry is b Λ ’ Z . (4.3)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 2 W is a generator of Z n +1 , and the set of ’t Hooft lines is trivial because they need to be mutuallylocal with W (as W is screened). There is no 1-form symmetry, and the gauge group must bethe simply connected SU (2 n + 1). Torus with k regular untwisted punctures : 4 d N = 2 su ( n ) k necklace quiver can beobtained by compactifying A n − (2 ,
0) theory on a torus with k regular untwisted punctures.In this case, L ’ b Z A × b Z B ’ Z An × Z Bn , (4.4)which can be verified from the 4 d gauge theory description. For example, choosing all gaugegroups 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 b Λ ’ Z n , (4.5)which can be identified as the diagonal subgroup of the Z kn center of the gauge group SU ( n ) k .21 g with n regular untwisted punctures : Consider compactifying A (2 ,
0) theory on C g in the presence of n regular untwisted punctures [3]. According to our proposal, we predict L ’ Z g × Z g . (4.6)There are a number of degeneration limits which lead to a variety of S-dual weakly-coupled4 d conformal gauge theories. The predicted answer for L and the pairing on it can be verifiedfrom the point of view of any of these 4 d gauge theories. For example, one such degenerationlimit (which exists for n ≥
2) leads to the following 4 d gauge theory su (2) su (2) · · · su (2) so (4)2 F n − su (2) · · · so (4) su (2) so (3)2 g − F (4.7)where an edge between two su (2) gauge algebras denotes a full hyper in bifundamental, whilean between an su (2) and an so ( n ) gauge algebra denotes a half-hyper in bifundamental (seeearlier discussion for our slightly non-standard definition of fundamental of so (3) and so (4)).The edge between a node labeled n F and a node labeled su (2) denotes that the corresponding su (2) gauge algebras carries n extra hypers in fundamental representation, where n is allowedto be a half-integer to account for half-hypers in fundamental. Choosing a particular Λ ’ Z g ⊂ L corresponds to choosing all the gauge groups to be simply connected. The 1-formsymmetry is predicted to be b Λ ’ Z g for this choice, which can be verified easily from the 4 d gauge theory description. A Z 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 d (1 , on T : The last class of example has an alter-native realization in terms of a 6 d (1 ,
0) on T [37, 38]: For g = 1 and n = 2 the A theoryon C , has defect group L = Z × Z . We can alternatively think of this as the compact-ification of the 6 d (1,0) theory that is the SU (2) − SU (2) conformal matter theory of rank2, i.e. 2 M5-branes probing C / Z . The 6d theory has a tensor branch geometry, whichhas two non-compact curves, with SU (2) singularities, sandwiching a ( − SU (2)gauge group. The defect group given by Z , and the dimensional reduction of this on T ,results in L A = L B = Z . More generally, 2 M5-branes probing a Z k singularity results in a‘hybrid’ class S theory, where an A -trinion is glued to an A k − one (see (2.6) in [38]). Thetensor branch-geometry changes simply to SU ( k ) groups both on the non-compact curves as22 = Figure 8: A configuration involving a closed Z twist line and an open Z twist line is topo-logically equivalent to a configuration involving on an open Z twist line.well as on the ( − Z -twisted Regular Punctures In this subsection we either consider those g for with the outer-automorphism group is Z , orthe case g = D with twist lines valued only in the Z subgroup of the S outer-automorphismgroup generated by the element b (see section 2).Consider a (2 ,
0) theory compactified on C g with Z twist lines, in the presence of bothuntwisted and twisted regular punctures. Twisted regular punctures are the regular puncturesthat appear at the ends of open Z twist lines. First thing to note is that if we have an opentwist line (i.e. a twist line with two end points), then we can always remove all the closed Z twist lines, since combining a closed twist line with an open twist line results in a single opentwist line, shown in figure 8.Recall that in the last subsection we proposed that any element of b Z that can be insertedon a loop surrounding an untwisted regular puncture does not contribute to the set of 4 d linedefects (modulo flavor charges). Similarly, we propose that any element of b Z that can beinserted on a loop surrounding a twisted regular puncture does not contribute to the set of 4 d line defects (modulo flavor charges) either. However, notice that the only elements of b Z thatcan be inserted along a loop surrounding a single twisted regular puncture are those that areleft invariant by the Z outer-automorphism action. The set of such invariant elements wasdenoted by Inv( b Z, o ) earlier. So, according to this proposal, only the elements in the subset From this point onward, the reader should assume that an arbitrary number of untwisted regular puncturesare always present. We do not mention them in what follows since they do not enter in the computation of L and 1-form symmetry b Λ. = o o oα o · α = α α α α Figure 9: An element α ∈ b Z with the property o · α = o can be moved across a regular twistedpuncture living at the end of an o twist line. Left: α crossing a Z twist line o ending on theregular puncture. Center: A topological deformation creates a loop of α around the regularpuncture, and an α line passing nearby that does not cross the twist line. Right: The loop of α surrounding the regular twisted puncture can be collapsed using our main proposal.Inv( b Z, o ) can be moved across a twisted regular puncture, while the elements not in the subsetInv( b Z, o ) cannot be moved across a twisted regular puncture. See figure 9.Now, in order to determine the set of 4 d line defects, we collect all the twisted regularpunctures in one corner of C g as shown in figure 10. From our above proposal we deduce thatthe set of line operators L Bi originating from elements of b Z wrapping the cycle B oi can beidentified with b Z/ Inv( b Z, o ). Furthermore, we can parametrize the set of line operators L Ai,i +1 originating from the cycle A oi,i +1 with the elements of b Z inserted along the green sub-segmentof A oi,i +1 . If we insert an element α of the subgroup Inv( b Z, o ) along the green sub-segment,then the element inserted along the blue sub-segment is also α . This configuration can bemoved across the twisted regular punctures and hence trivialized. Thus, L Ai,i +1 can also beidentified with b Z/ Inv( b Z, o ). Finally, notice that any element wrapped along P i B oi can beunwrapped on the other side of C g , and hence any element wrapped along B on can be writtenin terms of elements wrapped along B oi for 1 ≤ i ≤ k − d line operators (modulo screening and flavor charges) L can be writtenas L ’ k − Y i =1 L Ai,i +1 × k − Y i =1 L Bi × b Z gA × b Z gB . (4.8)For g = A n − , we have L Ai,i +1 ’ L Bi ’ Z n . (4.9)We choose f wrapped along B oi as the generator g Bi of L Bi , and the element obtained bywrapping f along the green sub-segment of A oi,i +1 to be the generator g Ai,i +1 of L Ai,i +1 . Then,24 o B o B o A o , B ok A o , A o , A ok − ,k B A B g A g Figure 10: A Riemann surface with 2 k Z twisted regular punctures and k open twist lines oftype o , all collected in a corner of the genus g Riemann surface. From this point onward, anarbitrary number of untwisted regular punctures are always present, but are never displayed.Further we fix cycles associated with punctures and twist lines to be oriented counterclockwiseand omit orientations in future pictures. Similarly we fix the orientation of the
A, B cycles ofthe Riemann surface as shown above going forward.the non-trivial pairings mod 1 are h g Ai,i +1 , g Bi i = 1 n (4.10) h g Ai,i +1 , g Bi +1 i = − n (4.11)along with the previously discussed pairing on b Z gA × b Z gB .For g = A n , we have L Ai,i +1 ’ L Bi ’ Z n +1 (4.12)We choose f wrapped along B oi as the generator g Bi of L Bi , and the element obtained bywrapping f along the green sub-segment of A oi,i +1 to be the generator g Ai,i +1 of L Ai,i +1 . Then,the non-trivial pairings are h g Ai,i +1 , g Bi i = 22 n + 1 (4.13) h g Ai,i +1 , g Bi +1 i = − n + 1 . (4.14) The pairing is a product of an intersection number and the value of the bihomomorphism (2.2). Intersectionsare taken to be positive if complementing the direction of the first and second argument by a vector pointingoutward from the page results in a right handed basis. g = D n , we have L Ai,i +1 ’ L Bi ’ Z . (4.15)We choose s wrapped along B oi as the generator g Bi of L Bi , and the element obtained bywrapping s along the green sub-segment of A oi,i +1 to be the generator g Ai,i +1 of L Ai,i +1 . Then,the non-trivial pairings are h g Ai,i +1 , g Bi i = 12 (4.16) h g Ai,i +1 , g Bi +1 i = 12 . (4.17)For g = E , we have L Ai,i +1 ’ L Bi ’ Z . (4.18)We choose f wrapped along B oi as the generator g Bi of L Bi , and the element obtained bywrapping f along the green sub-segment of A oi,i +1 to be the generator g Ai,i +1 of L Ai,i +1 . Then,the non-trivial pairings are h g Ai,i +1 , g Bi i = 13 (4.19) h g Ai,i +1 , g Bi +1 i = − . (4.20)With these pairings, an absolute 4 d N = 2 theory is chosen by a maximal isotropic subgroupΛ ⊂ L ’ k − Y i =1 L Ai,i +1 × k − Y i =1 L Bi × b Z gA × b Z gB (4.21)The 4 d theory carries a 1-form symmetry b Λ. In the rest of this subsection, we substantiateour proposal by discussing a few Lagrangian examples.
Sphere with 2 regular twisted punctures : The following 4 d N = 2 theories can be pro-duced by compactifying (2 ,
0) theories on a sphere with 2 regular twisted punctures and 2regular untwisted punctures:• sp ( n −
1) + 2 n F by compactifying D n (2 ,
0) theory.• sp (3) + 3 F + Λ , sp (3) + F + Λ , so (8) + 8 F + S , so (7) + 4 F + S by compactifying D (2 ,
0) theory [25].• so (10) + S + 6 F , so (9) + S + 5 F by compactifying D (2 ,
0) theory [25].• so (12) + S + 8 F , so (11) + S + 7 F by compactifying D (2 ,
0) theory [25].26 so (13) + S + 7 F by compactifying D (2 ,
0) theory [25].• su (4) + 3 Λ + 2 F , sp (2) + 2 Λ + 2 F by compactifying A (2 ,
0) theory [24].In such a compactification, our proposal predicts that L is trivial, which can be verifiedby computing the set of line operators (modulo screening and flavor charges) in all of theabove gauge theories. Hence, the 1-form symmetry is trivial and the gauge group in all theseexamples must be the simply connected one. Sphere with 4 regular twisted punctures : The following 4 d N = 2 theories can be pro-duced by compactifying (2 ,
0) theories of type g = D on a sphere with 4 regular twistedpunctures:• so (2 n ) + (2 n − F , so (2 n −
1) + (2 n − F by compactifying D n (2 ,
0) theory.• su (4) + 4 Λ , sp (2) + 3 Λ by compactifying A (2 ,
0) theory [24].In both of these cases we have
L ’ L A , × L B ’ Z × Z (4.22)with the generators g A , and g B having the pairing h g A , , g B i = 12 , (4.23)which can be verified from the perspective of Wilson-’t Hooft line operators in all of the abovementioned 4 d gauge theories. The 1-form symmetry b Λ for any choice of Λ ⊂ L is b Λ ’ Z , (4.24)which can also be easily verified. For any consistent choice of gauge group and discrete thetaparameters in the above gauge theories, the 1-form symmetry of the gauge theory is Z .Let us consider another example, which is of the 4 d N = 2 quiver gauge theory su (2 n ) su ( n ) · · · su (2 n ) su ( n ) su ( n ) k su ( n ) (4.25)where an edge between two nodes denotes a bifundamental hyper between the correspondinggauge algebras. This theory can be produced by compactifying A n − (2 ,
0) theory on a sphere27ith 4 regular twisted punctures and k + 3 regular untwisted punctures [24]. In this case, ourproposal predicts that L ’ L A , × L B ’ Z n × Z n , (4.26)with the generators g A , and g B having the pairing h g A , , g B i = 1 n , (4.27)which can be verified from the point of view of the 4 d gauge theory as well. For example,choosing one of the Z n factors in (4.26) as polarization leads to the choice of simply connectedgauge group for all gauge algebras involved in the 4 d gauge theory. The 1-form symmetry b Λ ’ Z n (4.28)can then be identified from the gauge theory viewpoint as follows. Each bifundamental hyperbetween two SU (2 n ) groups, only preserves the diagonal part of the two Z n centers, while abifundamental hyper between an SU ( n ) group and an SU (2 n ) group preserves the diagonal Z n of the obvious Z n × Z n subgroup of the center Z n × Z n . Thus, in total, only a diagonal Z n of the Z k +4 n subgroup of the Z n × Z k n center of the total gauge group acts trivially on allthe matter content. Torus with 6 regular twisted punctures : The 4 d N = 2 quiver so (4 n + 2) sp (2 n ) sp (2 n ) so (4 n + 2) sp (2 n ) so (4 n + 2) (4.29)can be constructed by compactifying D n +1 (2 ,
0) theory on a torus with 6 regular twistedpunctures [22]. Our proposal would predict that for this gauge theory we have
L ’ L A , × L A , × L B × L B × b Z A × b Z B (4.30)with L A , ’ L A , ’ L B ’ L B ’ Z (4.31)and b Z A ’ b Z B ’ Z (4.32)The non-trivial pairings on L are defined in terms of generators g Ai,i +1 , g Bi , g A , g B of L Ai,i +1 , L Bi , b Z A , b Z B respectively h g A , , g B i = 12 , h g A , , g B i = 12 , h g A , , g B i = 12 , h g A , g B i = 14 . (4.33)28et us reproduce this result by explicitly studying the line operators of the 4 d gauge theory.Before accounting for the matter content, the Wilson lines for all the gauge algebra factorsform the group Z W ’ Y i =1 ( Z ) i × Y i =1 ( Z ) i , (4.34)where ( Z ) i is associated to gauge algebra so (4 n + 2) i , and ( Z ) i is associated to gauge algebra sp (2 n ) i . We choose generators W so i for ( Z ) i and W sp i for ( Z ) i . The matter content impliesthat the set of Wilson lines (modulo screening) can be generated by W so − W so , W so − W so , 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 flavorcharges) is L W ’ Z × Z × Z (4.35)with the generators identified above. On the other hand, before accounting for the mattercontent, the ’t Hooft lines for all the gauge algebra factors form the group Z H ’ Y i =1 ( Z ) i × Y i =1 ( Z ) i , (4.36)where ( Z ) i is associated to gauge algebra so (4 n + 2) i , and ( Z ) i is associated to gauge algebra sp (2 n ) i . We choose generators H so i for ( Z ) i and H sp i for ( Z ) i . The matter content requiresus to choose the subset L H of Z H which is mutually local with the matter content. We canchoose the generators for L H to be 2 H so , H so , P i ( H so i + H sp i ). The first two generators areof order two, and the last generator is of order four. Thus, the contribution of ’t Hooft linesto the set of line operators (modulo screening and flavor charges) is L H ’ Z × Z × Z (4.37)with the generators identified above. We thus see that clearly L W × L H ’ L (4.38)and the generators can be identified as g Ai,i +1 = W so i − W so i +1 , g Bi = 2 H so i , g A = W so , g B = X i ( H so i + H sp i ) . (4.39)It is straightforward to check that the Dirac pairing between Wilson and ’t Hooft lines repro-duces the pairing (4.33) with the above identification.29 ba aa aa a a b ab ab a b a ba b
1) 2) 3) 4)5) 6) 7) 8)
Figure 11: Various kinds of irreducible configurations of open twist lines valued in S . Wename these 1-8 as follows: open b line, open ab line, open a b line, meson, baryon, anti-baryon,anti-mixed configuration, mixed configuration. These configurations are distinct except for theanti-mixed and mixed configuration, see figure 16. S -twisted Regular Punctures Now we consider incorporating more general regular twisted punctures in the D (2 ,
0) theory.We can have the following various irreducible configurations of twisted regular punctures asshown in figure 11:• Two punctures joined by a Z twist line implementing the transformation b ∈ S as onecrosses it. We refer to this configuration as the open b line.• Two punctures joined by a Z twist line implementing the transformation ab ∈ S asone crosses it. We refer to this configuration as the open ab line.• Two punctures joined by a Z twist line implementing the transformation a b ∈ S asone crosses it. We refer to this configuration as the open a b line.• Two punctures joined by an oriented Z twist line implementing the transformation a ∈ S as one crosses it in a particular direction (which is left to right in the fourthconfiguration 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 apoint 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 originatefrom a common point. We refer to this configuration as an “anti-baryon”.30 b a b b aba b b = aba b b = ab ( ab )( a b )( ab ) b = abb Figure 12: A configuration involving three different types open Z twist lines can be topolog-ically deformed to a configuration involving only two different types of open Z twist lines.Going from the third configuration to the last configuration involves fusing the closed ab loopwith the open a b line, which conjugates a b by ab , resulting in an open b line. ab ba aab b = aab a = b ab = ba ( ab ) a − Figure 13: A configuration involving two different types open Z twist lines and a meson canbe topologically deformed to a configuration involving a meson and open Z twist lines of asingle type only. Going from the third configuration to the last configuration involves fusingthe closed a loop with the open ab line, which conjugates ab by a , resulting in an open b line.• Two punctures emitting a b and b Z twist lines which combine to form an a twist linewhich 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 b and b Z twist lines. Each Z 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:• Consider a situation where we have an open b line, an open ab line and an open a b line.We can pass the open a b line through the ab line to convert the open a b line into anopen b line. See figure 12. At the end of this process, we obtain a situation in which wehave two open b lines and one open ab line.• Consider a situation where we have an open a line, an open b line and an open ab line.We can pass the open ab line through the a line to convert the open ab line into an open31 = a aa a aab = b babb = bbabbab aa a Figure 14: A baryon can be converted into an anti-baryon by passing it through a b twist line. a aa aa a a aa aa a == a aa aa aa = a aa aa aa = a a a aa aa Figure 15: A configuration involving a baryon and an anti-baryon is topologically equivalentto a configuration involving three mesons. ab a b = aa b b = bab ba b = a ba b Figure 16: An anti-mixed configuration is topologically equivalent to a mixed configuration.32 ba b a ba b = a ba b a a bbaba b a b a b = a b a b = bbaa b = a bba a ba b a b = bba Figure 17: Two mixed configurations can be converted into a meson and two different kindsof open Z twist lines. Using figure 13, this is equivalent to a meson plus two open b lines ofthe same type, since the last configuration can be further transformed according to figure 13.33 ba b a = aab a b = b a ba aab aa aa b = b a aa = = b a aaa − ( a b ) a Figure 18: A mixed configuration plus a meson can be converted into a baryon plus an open b line. We first convert the mixed configuration into an anti-mixed configuration. Then we fusethe open a line internal to the meson with the junction for the mixed configuration to createa combined junction for all the open twist lines. Then we move a puncture living at the endof a b line over a puncture acting as sink for a line. This converts the latter puncture into apuncture acting as source for a line (due to conjugation). Then we move this puncture overthe puncture living at the end of a b line, thus converting the latter into a puncture living atthe end of a b line. Finally we can separate a full open b line from the junction leaving behinda baryon. a ba b = a aa aba b a a a = ba b aaa = aa b baa = bbaa a = a a b Figure 19: A mixed configuration and a baryon can decomposed into two mesons and an open b line. 34 line. See figure 13. At the end of this process, we obtain a situation in which we haveone open a line and two open b lines.• Consider a situation where we have a baryon and an open b line. We can pass the baryonthrough the b line to convert the baryon into an anti-baryon. See figure 14. At the endof this process, we obtain a situation in which we have an anti-baryon and an open b line.• A baryon and an anti-baryon can be decomposed as three mesons. See figure 15.• An anti-mixed configuration can be converted into a mixed configuration. See figure 16.• Two mixed configurations can be decomposed as a meson and two open b lines. Seefigure 17.• A mixed configuration plus a meson is equivalent to a baryon plus an open b line. Seefigure 18.• A mixed configuration plus a baryon is equivalent to two mesons plus an open b line.See figure 19.Accounting for the above redundancies we can easily show that the only topologically distinctpossibilities for non-trivial S twist lines on C g are as follows:• k open b lines. This was discussed in the previous subsection.• k open b lines plus a Z closed twist line.• k open b lines plus k open ab lines. Inserting an additional Z closed twist line does notlead to a topologically distinct scenario. See figure 20.• l mesons.• l mesons plus a Z closed twist line.• k open b lines plus l mesons.• p baryons.• p baryons plus l mesons.• One baryon plus l mesons plus a Z closed twist line.• One baryon plus k open b lines. 35 a = = Figure 20: A configuration involving a closed Z twist line and two different types of open Z twist lines is topologically equivalent to a configuration involving only the two different typesof open Z twist line without the closed Z twist line. The different types of Z twist lines aredistinguished by different colors in the above figure.• One baryon plus k open b lines plus l mesons.• One mixed configuration.• One mixed configuration plus k open b lines.We now determine L for all of the above topologically distinct possibilities one-by-one.First consider the case where we have k > b lines and a closed a line. We define cycles A oi and B oi as shown in figure 21, and let the 4 d line operators (modulo screening and flavorcharges) originating from them be L Ai and L Bi respectively. As before, L Bi ’ Z which can begenerated by wrapping s along B oi . On the other hand, L Ai ’ Z as well, and the generatorcan be chosen to be s wrapped along the green sub-segment and c wrapped along the bluesub-segment. We call the generators of L Ai and L Bi as g Ai and g Bi respectively. Then we canwrite the set L of 4 d line operators (modulo screening and flavor charges) as L ’ k Y i =1 L Ai × k Y i =1 L Bi × b Z g − A × b Z g − B (4.40)with the non-trivial pairings being h g Ai , g Bi i = 12 (4.41)along with the pairing on b Z g − A × b Z g − B .Now consider the case where we have k > b lines and k > b := ab lines.See figure 22. Then the line operators arising from B b i can be generated by wrapping v along B b i . However, v wrapped along P i B b i is equivalent to v wrapped along P i B bi , which in turncan be trivialized as v can be moved across twisted regular punctures of type b . Similarly, s o B o B ok A o A o A ok a B A A g B g Figure 21: Riemann surface of genus g with k open b lines and a closed a line. Each cycle A io is broken into green and blue sub-segments lying between two difference kinds of twist linesthe cycle crosses. B b B b B bk B b B b B b k A b , A b , A bk − ,k A b , A b , A b k − ,k B A A g B g Figure 22: Riemann surface of genus g with k open b lines and k open b lines with b = b .The two different kinds of Z twisted open lines are displayed using different colors.37 a B a B al A A a , A a , A al − ,l B A g B g Figure 23: Riemann surface of genus g with l mesons.wrapped along P i B bi is trivial. Thus, we can write the set L of 4 d line operators (moduloscreening and flavor charges) as L ’ k − Y i =1 L A,bi,i +1 × k − Y i =1 L B,bi × k − Y i =1 L A,b i,i +1 × k − Y i =1 L B,b i × b Z gA × b Z gB , (4.42)where L A,bi,i +1 ’ Z and L A,b i,i +1 ’ Z are the sets of line operators descending from cycles A bi,i +1 and A b i,i +1 respectively, and L B,bi ’ Z and L B,b i ’ Z are the sets of line opera-tors descending from cycles B bi and B b i respectively. Let the corresponding generators be g A,bi,i +1 , g A,b i,i +1 , g B,bi , g
B,b i . We can define g A,bi,i +1 by inserting s on the green sub-segment of A bi,i +1 and c on the blue sub-segment of A bi,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 h g A,bi,i +1 , g B,bi i = h g A,bi,i +1 , g B,bi +1 i = h g A,b i,i +1 , g B,b i i = h g A,b i,i +1 , g B,b i +1 i = 12 (4.43)along with the pairing (3.3) on b Z gA × b Z gB .Now consider the situation containing l mesons only. See figure 23. Then the line operators L B,ai arising from the cycle B ai can be identified with Z × Z , since no element of b Z can bemoved across a Z twisted puncture. We label the line operator in L B,ai arising by wrapping s along B ai as s B,ai , and the line operator in L B,ai arising by wrapping c along B ai as c B,ai . Wechoose s B,ai and c B,ai as the generators for L B,ai . Finally, P i s B,ai = P i c B,ai = 0. Similarly, theline operators L A,ai,i +1 arising from the cycle A ai,i +1 can be identified with Z × Z . The elementof L A,ai,i +1 arising by wrapping s along the green sub-segment of A ai,i +1 is called s A,ai,i +1 , and the38 a B a B al A A a , A a , A al − ,l B bA a,bl A g B g Figure 24: Riemann surface of genus g with l mesons and a closed b line.element of L A,ai,i +1 arising by wrapping v along the green sub-segment of A ai,i +1 is called v A,ai,i +1 .We choose s A,ai,i +1 and v A,ai,i +1 as the generators for L A,ai,i +1 . In total, we have L ’ l − Y i =1 L A,ai,i +1 × l − Y i =1 L B,ai × b Z gA × b Z gB (4.44)with the non-trivial pairings being h s A,ai,i +1 , s B,ai i = h s A,ai,i +1 , s B,ai +1 i = h v A,ai,i +1 , c B,ai i = h v A,ai,i +1 , c B,ai +1 i = 12 (4.45)along with the pairing on b Z gA × b Z gB .Now let us consider l mesons in the presence of a closed Z twist line of type b . See figure24. Notice that now P i s B,ai = P i c B,ai = 0. Thus, we label the Z subgroup of L B,al generatedby s B,al as S B,al . Correspondingly, there is a new cycle A a,bl shown in figure 24 giving rise to a4 d line operator, which is obtained by wrapping c along the green sub-segment and s along theblue sub-segment of A a,bl . We label this set of line operators as L A,a,bl ’ Z and its generatordescribed above as c A,a,bl . We choose the generator of the set of line operators Proj( b Z, o ) ’ Z originating from cycle B shown in figure 24 to be c wrapping the cycle B . Then, we obtainthat the total set of 4 d line operators is L ’ l − Y i =1 L A,ai,i +1 × l − Y i =1 L B,ai × L
A,a,bl × S
B,al × Inv( b Z, o ) × Proj( b Z, o ) × b Z g − A × b Z g − B (4.46)with non-trivial pairings (4.45), along with the pairing on b Z g − A × b Z g − B and the new pairings h s A,al − ,l , s B,al i = h c A,a,bl , s
B,al i = 12 . (4.47)39 b B b B bk B a B a B al A b , A b , A bk − ,k A a , A a , A al − ,l A b,ak, B A A g B g Figure 25: Riemann surface of genus g with k open b lines and l mesons. B a B a B a p A a , A a , A a p − ,p B A A g B g Figure 26: Riemann surface of genus g with p baryons.Consider now l mesons with 2 k = 0 Z twisted regular punctures of type b . We havethe constraint that P i s B,ai = P i c B,ai = P i g B,bi . Also we have a new cycle A b,ak, as shownin figure 25 which contributes a group L A,b,ak, ’ Z of 4 d line operators which is generatedby g A,b,ak, which is obtained by wrapping c along the green sub-segment and s along the bluesub-segment of A b,ak, . In total, we have L ’ k − Y i =1 L A,bi,i +1 × L A,b,ak, × k Y i =1 L B,bi × l − Y i =1 L A,ai,i +1 × l − Y i =1 L B,ai × b Z gA × b Z gB (4.48)The non-trivial pairings are those on b Z gA × b Z gB , those given in (4.45), and those listed below h g A,bi,i +1 , g B,bi i = h g A,bi,i +1 , g B,bi +1 i = h g A,b,ak, , g B,bk i = h g A,b,ak, , s B,a i = h g A,b,ak, , c B,a i = 12 . (4.49)Consider now the case involving p baryons. See figure 26. Let L A,a i,i +1 be the set of 4 d line operators arising from the cycle A a i,i +1 and L B,a i be the set of 4 d line operators arising40 a B a B al A a , A a , A al − ,l A a,a l, B A A g B g B a B a B a p A a , A a , A a p − ,p Figure 27: Riemann surface of genus g with l mesons and p baryons.from the cycle B a i . We can wrap any element of b Z ’ Z × Z along B a i which implies that L B,a i ’ Z × Z . We choose the generators of L B,a i to be s B,a i and c B,a i , which are obtainedby wrapping s and c respectively along B a i . Similarly, L A,a i,i +1 ’ Z × Z . The element of L A,a i,i +1 arising by wrapping s along the green sub-segment of A a i,i +1 (which implies that v is wrappedalong the red sub-segment and c is wrapped along the blue sub-segment) is called s A,a i,i +1 , andthe 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 L ’ p − Y i =1 L A,a i,i +1 × p − Y i =1 L B,a i × b Z gA × b Z gB (4.50)with the non-trivial pairings being h s A,a i,i +1 , s B,a i i = h s A,a i,i +1 , s B,a i +1 i = h v A,a i,i +1 , c B,a i i = h v A,a i,i +1 , c B,a i +1 i = 12 (4.51)along with the pairing on b Z gA × b Z gB .Consider now the case involving p baryons and l mesons. See figure 27. Along with thepreviously discussed groups L A,ai,i +1 , L A,a i,i +1 , L B,ai , L B,a i , we also have a group L A,a,a l, arising fromthe cycle A a,a l, shown in figure 27. We have L A,a,a l, ’ Z × Z . The element of L A,a,a l, arisingby wrapping s along the green sub-segment of A a,a l, is called s A,a,a l, , and the element of L A,a,a l, arising by wrapping v along the green sub-segment of A a,a l, is called v A,a,a l, . We choose s A,a,a l, and v A,a,a l, as the generators for L A,a,a l, . In total, we have L ’ l − Y i =1 L A,ai,i +1 × L A,a,a l, × l Y i =1 L B,ai × p − Y i =1 L A,a i,i +1 × p − Y i =1 L B,a i × b Z gA × b Z gB (4.52)41 B bA a ,b A g B g B a B a B al A a , A a , A al − ,l A a,a l, B a Figure 28: Riemann surface of genus g with l mesons, one baryon and a closed b line.with h s A,a,a l, , s B,al i = h s A,a,a l, , s B,a i = h v A,a,a l, , c B,al i = h v A,a,a l, , c B,a i = 12 (4.53)being the new non-trivial pairings.We now consider the case having a single baryon, l mesons and a closed b line. See figure28. The only cycle not discussed above is A a ,b which gives rise to a set of 4 d line operators L A,a ,b ’ Z whose generator c A,a ,b is obtained by wrapping c along the green sub-segment of A a ,b . Moreover, we can write c B,a = P li =1 s B,ai + s B,a + P li =1 c B,ai implying that the relevantset of 4 d line operators arising from B a can be taken to be S B,a ’ Z which is generated by s B,a . In total, we have L ’ l − Y i =1 L A,ai,i +1 × L A,a,a l, × l Y i =1 L B,ai × L
A,a ,b × S B,a × b Z g − A × b Z g − B (4.54)with h c A,a ,b , s B,a i = 12 (4.55)being the only new non-trivial pairing not discussed previously.We now consider the case having a single baryon and k open b lines. See figure 29. The onlycycle not discussed above is A b,a k, which gives rise to a set of 4 d line operators L A,b,a k, ’ Z whosegenerator g A,b,a k, is obtained by wrapping c along the green sub-segment of A b,a k, . Moreover,we can write c B,a = s B,a = P ki =1 g B,bi . In total, we have
L ’ k − Y i =1 L A,bi,i +1 × L A,b,a k, × k Y i =1 L B,bi × b Z gA × b Z gB (4.56)with h g A,b,a k, , g B,bk i = 12 (4.57)42 b B b B bk A b , A b , A bk − ,k A b,a k, B A A g B g B a Figure 29: Riemann surface of genus g with k open b lines and one baryon.being the only new non-trivial pairing not discussed previously.Consider now the case involving a single baryon, k open b lines and l mesons. See figure30. We can quickly deduce that L ’ k − Y i =1 L A,bi,i +1 × L A,b,ak, × k Y i =1 L B,bi × l − Y i =1 L A,ai,i +1 × L A,a,a l, × l Y i =1 L B,ai × b Z gA × b Z gB (4.58)There are no new non-trivial pairings.Consider now the final case involving a single mixed configuration along with k ≥ b lines. See figure 31. We have s ( B a ) = c ( B a ) = P ki =1 g B,bi . Thus B a contributes no new4 d line operators. Moreover, we can obtain a non-trivial 4 d line operator g A,b,a k, by wrapping c along the green sub-segment of A b,a k, . In total, we have L ’ k − Y i =1 L A,bi,i +1 × L A,b,a k, × k Y i =1 L B,bi × b Z gA × b Z gB (4.59)with h g A,b,a k, , g B,bk i = 12 (4.60)being the only new non-trivial pairing not discussed previously. L is trivial for k = 0.We finish this subsection by discussing some Lagrangian examples. Some more examplesillustrating the results of this subsection appear in the next subsection on atypical punctures. Sphere with 2 Z twisted regular punctures of type b and 2 Z twistedregular punctures of type b = b : The 4 d N = 2 theory g + 4 F (carrying g gauge algebraand 4 full hypers in irrep of dimension ) can be constructed using a compactification of D a B a B al A a , A a , A al − ,l A a,a l, B A A g B g B a B b B b B bk A b , A b , A bk − ,k A b,ak, Figure 30: Riemann surface of genus g with k open b lines, l mesons and one baryon. B b B b B bk A b , A b , A bk − ,k A b,a k, B A A g B g B a b a b Figure 31: Riemann surface of genus g with k open b lines and one mixed configuration.44 a b = b bba b = ba b Figure 32: Converting a configuration of open twist lines on a sphere discussed in [23] to aconfiguration of open twist lines discussed in this paper. b b = a b ba = a b b Figure 33: Converting a configuration of open twist lines on a sphere discussed in [23] to aconfiguration of open twist lines discussed in this paper.(2 ,
0) theory on a sphere with 4 regular twisted punctures, 2 open Z twist lines of type b and1 open Z twist line of type a as shown in figure 32 [23]. Combining the two open Z twistlines, we obtain a configuration with 4 regular twisted punctures, 1 open Z twist line of type b and 1 open Z twist line of type b = b . See figure 32. Using our results above, we would thusconclude that L should be trivial. Indeed, this is easily verified from the Lagrangian g + 4 F description.The 4 d N = 2 gauge theories so (8) + 3 F + 3 S and so (7) + 2 F + 3 S can be obtained bycompactifying D (2 ,
0) theory on a sphere with 4 regular twisted punctures, 2 open Z twistlines of type b and 1 closed Z twist line of type a as shown in figure 33 [23]. We can move theclosed Z twist line such that it encircles 2 regular Z twisted punctures as shown in figure33. This corresponds to conjugating the enclosed open Z twist line b by a . Thus, we canremove the closed a twist line if we convert this open Z twist line from type b to type b . Seefigure 33. 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 4 d N = 2 gauge theories so (8) + 3 F + 3 S and so (7) + 2 F + 3 S . Atypical regular punctures can be straightforwardly included in our analysis by resolving eachatypical regular puncture into typical regular punctures. See the beginning of Section 4 for thedefinition of atypical regular punctures and further references which discuss them in detail.45
Resolution aa bb Figure 34: Left: Compactification on a sphere involving a typical untwisted regular puncture,a typical twisted regular puncture and an atypical twisted regular puncture. The atypicalpunctures are denoted by a circle super-imposed on top of a cross, while typical puncturesare denoted by a cross only. Right: The atypical puncture is resolved to two typical twistedregular punctures. The resolution results in a mixed configuration. a Resolution aa bb a bb = a bb Figure 35: Left: Compactification on a sphere involving a typical untwisted regular punctureand two atypical twisted regular punctures. Center: Each atypical puncture is individuallyresolved into two typical twisted regular punctures. Right: After a topological manipulation,we find two open Z twist lines of different types. Gauge theory fixtures of type (1 , ω, ω ): We can obtain the 4 d N = 2 gauge theory sp (2) + 6 F by compactifying D (2 ,
0) theory on a sphere with one typical untwisted reg-ular puncture, one typical twisted regular puncture acting as the sink of an a twist line, andone atypical twisted regular puncture acting as the source of an a twist line [26], as shown infigure 34. The atypical puncture can be resolved into two Z twisted typical regular punctures.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 , whichmatches the result obtained using the sp (2) + 6 F gauge theory description.As another example, we can obtain the 4 d N = 2 gauge theory su (3) F su (2) sp (2) FΛ (4.61)by compactifying D (2 ,
0) theory on a sphere with one typical untwisted regular puncture,one atypical twisted regular puncture acting as the sink of an a twist line, and one atypical twisted regular puncture acting as the source of an a twist line [26]. See figure 35. The atypicalpuncture acting as the source can be resolved into two Z twisted typical regular punctures,and the atypical puncture acting as the sink can be resolved into two Z twisted typical regular46 esolution a bb aa = baaaa Figure 36: Left: Compactification on a sphere involving two typical twisted regular puncturesand one atypical twisted regular puncture. Center: The atypical puncture is resolved intotwo typical twisted regular punctures. Right: After a topological manipulation, described inFigure 19, we end up with an open b line and a meson.punctures plus one untwisted typical regular puncture. See figure 35. After this resolution, weobserve that we have a sphere containing two different kinds of open Z twist lines: an open b line and an open b := a b line. Thus, from our analysis in section 4.3 we expect to obtain atrivial L , which matches the result obtained using the above gauge theory description, as thereader can readily verify. Gauge theory fixtures of type ( ω, ω, ω ): We can obtain the 4 d N = 2 gauge theory sp (3)+2 Λ by compactifying D (2 ,
0) theory on a sphere with three twisted regular punctures actingas sources of a twist lines, thus forming a “baryon-like” configuration [26]. See figure 36.Two out of these three punctures are typical, while one of them is atypical. The atypicalpuncture can be resolved into two Z twisted typical regular punctures. See figure 36. Afterthis resolution, we observe that we have a sphere containing a configuration that we dealtwith in figure 19. From the result of that figure, we know that this is equivalent to a spherecontaining a meson-like configuration and an open b line. Thus, from our analysis in section4.3 we expect to obtain L ’ Z × Z , (4.62)which matches the result obtained using the above gauge theory description, as the reader canreadily verify.As another example, we can obtain the 4 d N = 2 gauge theory with gauge algebra sp (2) ⊕ sp (2) and hypermultiplet content ( Λ , F ) + ( Λ ,
1) + (1 , F ) by compactifying D (2 ,
0) theoryon a sphere with three twisted regular punctures acting as sources of a twist lines, thus forminga “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 Z twistedtypical regular punctures. See figure 37. After this resolution, we can perform some topologicalmoves, as shown in figure 37, and reduce to a mixed configuration plus an open b line. Thus,47 esolution = aaa b ba ba b a b ba ba bb ba ba = = b ba ba Figure 37: Before the arrow: Compactification on a sphere involving one typical twisted regularpuncture and two atypical twisted regular punctures. After the arrow: Each atypical punctureis resolved into two typical twisted regular punctures. After some topological manipulations,we end up with an open b line and a mixed configuration.from our analysis in section 4.3 we expect to obtain L ’ Z × Z , (4.63)which matches the result obtained using the above gauge theory description, as the reader canreadily verify. The analysis of this paper has focused on compactifications of 6 d (2 ,
0) theories involving onlyregular (either untwisted or twisted) punctures. In this section, we discuss how our analysiscan be generalized to incorporate irregular punctures, which are the punctures where theHitchin field has poles of higher-order than simple poles.A class of irregular punctures for A n − (2 ,
0) theory were discussed in [39]. The Hitchinfield at an irregular puncture of type P k in this class can be written as ϕ = 1 z n − k diag(0 , · · · , , Λ , Λ ω, · · · , Λ ω n − k − ) dz + 1 z diag( m , m , · · · , m k , m k +1 , m k +1 , · · · , m k +1 ) dz + · · · (5.1)where 0 ≤ k ≤ n − P k +1 i =1 m i = 0. Here ω is an n -th rootof unity and Λ denotes the dynamically generated scale. For irregular puncture of type P n − ,we can instead write ϕ = 1 z diag(Λ , Λ , · · · , Λ , − ( n − dz + 1 z diag( m , m , · · · , m n ) dz + · · · (5.2)48 ≤ k ≤ n − P ≤ k ≤ n − = α α Figure 38: A line carrying α ∈ b Z can be deformed across an irregular puncture of type P k for1 ≤ k ≤ n − W H
Figure 39: A sphere two irregular punctures, both of type P k . Two cycles W and H on thispunctured sphere have been displayed.where P ni =1 m i = 0.We would now like to understand how these irregular punctures impact the determinationof 1-form symmetry. In particular, we would like to understand whether an element α of b Z ’ Z n can be moved across an irregular puncture of type P k (where k can take values in { , , · · · , n − } ). To answer this question, we consider compactifying A n − (2 ,
0) theory ona sphere with two irregular punctures both of same type P k . This leads to the 4 d N = 2asymptotically free gauge theory su ( n ) + 2 k F [39]. We know from the gauge theory viewpointthat L is trivial if k >
0, from which we can bootstrap that an element α of b Z ’ Z n wrappedalong the cycle W displayed in figure 39 can be contracted to a trivial loop. In other words,we learn that any element α of b 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 symmetryare concerned, an untwisted irregular puncture of type P k for k > k = 0, we obtain the 4 d N = 2 pure su ( n ) gauge theory. Thisgauge theory has L = Z n × Z n (5.3)49 P = α α Figure 40: A line carrying 0 = α ∈ b Z cannot be deformed across an irregular puncture of type P .with the first Z n factor arising from Wilson line operators, and the second Z n factor aris-ing from ’t Hooft line operators. The Z n factor associated to Wilson line operators can beunderstood as arising from 6 d surface operators wrapping the cycle W shown in figure 39.This identification can be made by observing that when W is very small, we first reduce the6 d theory to 5 d N = 2 su ( n ) SYM and then reduce this 5 d theory to the 4 d N = 2 pure su ( n ) gauge theory due to the presence of boundary conditions associated to the two irregularpunctures. In this reduction the 6 d surface operators wrapping W become line operators inthe 5 d theory and hence can be identified with the Wilson line operators of the 5 d theory, andthen subsequently as Wilson line operators of the 4 d theory. This means that no element α of b Z ’ Z n wrapped along the cycle W can be contracted to a trivial loop. Hence, an untwistedirregular puncture of type P does not allow any element α of b Z ’ Z n to be moved across it,as shown in figure 40.Now one can ask what is the interpretation of the Z n factor associated to ’t Hooft lineoperators from the point of view of the compactification of the 6 d theory. We propose thatthis is associated to elements of b Z inserted along the oriented segment labeled H in figure 39.One can then observe that h f ( W ) , f ( H ) i = 1 n . (5.4)That is, the pairing between elements of L obtained by wrapping generator f of b Z along W and H is precisely the Dirac pairing between fundamental Wilson and ’t Hooft operators inthe gauge theory.Notice that our above proposal for ’t Hooft line operators implies that a 6 d surface operatorcan end on the codimension-2 defect associated to an irregular puncture of type P . This isour first example of a puncture having this property. One would imagine that more generalirregular punctures discussed in [40–42] allow a subgroup of b Z to end on them, depending onthe type of puncture. We defer a more thorough analysis to a future work, but finish this50ection by substantiating our proposal for the properties of punctures of type P k by studyingthe following example. Example : Consider compactifying A (2 ,
0) theory on C g with n regular punctures, n punctures of type P and n punctures of type P . From our above analysis, we expect L ’ (cid:0) Z n − × Z g (cid:1) A × (cid:0) Z n − × Z g (cid:1) B . (5.5)In a particular degeneration limit, we obtain the following 4 d N = 2 asymptotically free gaugetheory su (2) su (2) · · · su (2) so (3)2 F n − su (2) · · · so (4) su (2) so (4)2 g − su (2) su (2) F su (2) su (2) F · · · su (2) su (2) su (2) F n su (2) su (2) ··· su (2) su (2) su (2) n F (5.6)where a trivalent vertex denotes a half-hyper in trifundamental representation, and n , n count the number of such half-trifundamentals. From the above gauge theory description onecan verify that L is indeed given by (5.5). 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 class51 [43]. Type IIB on a canonical singularity gives rise to N = 2 SCFTs, and more generallycan provide a way to engineer gauge theories. The Calabi-Yau X geometries that realize classS theories, can be constructed as ALE-fibrations over a curve (cid:94)C / Γ ADE , → X → C g,n , (6.1)where the resolutions parametrized for the ALE-fiber are encoded in a Higgs field ϕ . Theconnection is made through the Higgs bundle, [44, 45]. The Higgs field ϕ is a meromorphic1-form valued in the respective ADE Lie algebra g . We consider the 6d (2,0) theory of typeADE on C g,n , with the standard topological twist that retains N = 2 supersymmetry in 4d,i.e. SO (5) → SU (2) × U (1) R and SO (6) → SO (4) × U (1) L twisting the U (1) L by combiningit with the U (1) R R-symmetry transformation. The scalars give rise to the (1 ,
0) and (0 , ϕ and ¯ ϕ . These define together with the gauge field components (along the curve) theHiggs bundle, satisfying the Hitchin equations. The spectral equation defines the SW curveinside the co-tangent bundle of C g,n det( ϕ − λ Id) = 0 . (6.2)We assume that the Higgs bundle is diagonalizable, i.e. ϕ = diag( λ , · · · , λ r ). The spectraldata encodes a local Calabi-Yau, which defines an ALE-fibration over C . Each sheet is labeledby a fundamental weight of g . For simplicity let us focus on the A N − case. There are N sheets, associated to the L i , i = 1 , · · · , N fundamental weights, with the simple roots realizedas α i = L i − L i +1 . The Higgs field eigenvalues λ i encode the volumes of the rational curve inthe ALE-fibration, where each simple root is associated to a rational curve P i , whose volumeis determined by Z P i Ω = λ i − λ i +1 . (6.3)When λ i = 0 for all i , the full SU ( N ) symmetry is restored. More precisely, the spectral curveallows us to construct three-cycles as follows: if b α are the branch points of the spectral curve,where two sheets of the cover collide, we can construct an S by considering the ALE-fiberover the line ‘ α,β connecting two branch-points in C . At each of the branch-points a 2-spherecollapses, and thus we obtain an S . These three-spheres are Lagrangian and give rise in IIBto the hypermultiplets in 4d. Other three-cycles with the topology of S × S are obtained byconsidering the rational curves fibered over closed 1-cycles in the base, which correspond tovectors.Regular, untwisted punctures correspond to simple poles of ϕ . In the ALE-fibration, thismaps to sending the volumes of (some) P s to infinity. The punctures are labeled by partitions52 Σ oC g,n Figure 41: The outer automorphism (6.4) acting on the spectral cover.of N = P n i h i , where n i is the multipliticy of the box of height h i in the Young tableaux. Theflavor symmetry is G F = S ( Q i U ( n i )). E.g. the full punctures corresponding to the partition1 N the flavor symmetry is SU ( N ), corresponds to sending all N sheets to infinity with thesame rate, parameterized by the residue of the pole of ϕ .Open and closed twist lines alter the global structure of the ALE geometry. Open twistlines are inserted between punctures and closed twist lines are wrapped along a 1-cycle B of the base C , both are labelled by an element o of the outer automorphism group. Whenencircling a puncture or traversing a 1-cycle intersecting B the Higgs field is acted on theby the outer automorphism o , see figure 41. In the ALE-fibration, rational curves P locallysweeping out distinct three-cycles are identified reducing the total number of 3-cycles in X .The Poincaré dual of these three-cycles are used to expand the supergravity four-form andconstruct the gauge bosons of the effective 4d theory. The gauge algebra of the theory istherefore determined by the initial choice of ADE gauge group and twist line structure. Example:
Consider the 6d (2 , A n − theory compactified on the torus C g = T with aclosed b twist line along the B cycle. The spectral cover Σ is a 2 n -sheeted cover of the torus T . Each sheet can be thought of as associated to a fundamental weight L i , i = 1 , · · · , n ,and the outer automorphism acts as o : L i ←→ − L n +1 − i , (6.4)which induces an action on the simple roots α i = L i − L i +1 ↔ α n − i . The root α n is fixed.There are n P fibers whichdetermine the root system of the 4d gauge algebra. These 3-cycles intersect linearly with the53-cycle corresponding to the fixed P lying at the end of this chain. The root originating fromthis P is shorter than than the remaining n − B n . Overall we find the gauge group to reduce from SU (2 n ) to Spin(2 n + 1) when introducingthe twist line, the center of Spin(2 n + 1) is Z . The line operators in this context are realized in terms of wrapped D3-branes, on non-compactthree-cycles, modulo screening by particles, which are D3-branes wrapped on compact three-cycles. 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 arethereby realized in terms of L = H ( X, ∂X, Z ) H ( X, Z ) . (6.5)Chasing this through the long exact sequence in relative homology, · · · −→ H ( X, Z ) q −→ H ( X, ∂X, Z ) ∂ −→ H ( ∂X, Z ) ι −→ H ( X, Z ) −→ · · · , (6.6)we find that L = H ( X, ∂X, Z ) H ( X, Z ) = H ( X, ∂X, Z )Im( q ) = Im( ∂ ) = Ker( ι ) . (6.7)In particular we can write it as L = { ‘ ∈ H ( ∂X, Z ) | ‘ is a 2-cycle in ∂X which becomes trivial in X } . (6.8)The pairing on L governing the mutual non-locality of 4 d line operators descends straightfor-wardly from the linking pairing on H ( ∂X, Z ).The boundary ∂X receives contributions B F and B k from the fibers and punctures respec-tively ∂X C g,n = B F ∪ n [ k =1 B k , (6.9)where the topology of B k is given by (cid:94)C / Γ ADE , → B k → S , (6.10)and the topology of B F is given by S / Γ ADE , → B F → C g,n . (6.11)The contribution of (6.11) part of ∂X C g,n to H ( ∂X C g,n , Z ) is obtained by choosing anelement α ∈ H ( S / Γ ADE ), which is then fibered over a loop L in C g,n . We have H ( S / Γ ADE , Z ) ’ b Z ( G ) , (6.12)54 = = Figure 42: Consider an untwisted regular puncture and a boundary cycle (
L, α ) ∈ H ( B F , Z ),with α ∈ H ( S / Γ ADE ). We illustrate how the untwisted puncture does not affect this contri-bution to the defect group. Left: A line L , associated to ( L, α ). Right: A line L associatedto ( L , α ). Center-left: Limiting configuration as L is moved towards an untwisted regularpuncture. Center-right: Limiting configuration as L is moved towards the puncture.where G is the simply connected Lie group associated to the ADE Lie algebra g associated toΓ ADE . Moreover, an outer-automorphism of g acts on H ( S / Γ ADE , Z ) in precisely the sameway as it acts on b Z ( G ). When the loop L crosses an outer-automorphism twist line o , α istransformed to o · α . Moreover, any such element ( α, L ) ∈ H ( B F , Z ) ⊂ H ( ∂X, Z ) is clearlytrivial, when embedded into X since α is contractible when embedded into C / Γ ADE . Thus,contributions of type ( α, L ) give rise to a non-trivial subgroup L F ⊆ L , (6.13)where L is defined in (6.8).Now, notice that the above contributions of the kind ( α, L ) are precisely the contributionswe have been considering throughout the paper. Let us label the group of line operatorsobtained using the earlier considerations in the paper as L . Then we clearly have L ⊆ L F . (6.14)Thus, the only way for our previous calculation L and the Type IIB calculation L to matchis if L = L F = L . (6.15)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 s in (cid:94)C / Γ ADE , but none ofthese 2-cycles is trivial when embedded into X , and hence do not contribute to L .Next, we need to show that ( L, α ) and ( L , α ) give rise to the same element in H ( ∂X, Z ) if L is obtained from L by passing it over an untwisted regular puncture. Consider the limitingconfiguration of L approaching an untwisted regular puncture k , say from the left in figure55 = = oooo Figure 43: Consider again (
L, α ) , ( L , α ) ∈ H ( B F , Z ) with α ∈ H ( S / Γ ADE ). Left: A line L associated to ( L, α ). Right: A line L associated to ( L , α ) along the blue subsegment and o · α along the green subsegment. Center-left: Limiting configuration as L is moved towardsan untwisted regular puncture. Center-right: Limiting configuration as L is moved towardsthe puncture. The central equality only holds for α = o · α .42. We hit the boundary component B k at a particular point p ∈ S . The fiber component α embeds into the fiber (cid:0) (cid:94)C / Γ ADE (cid:1) p of B k at p via the inclusion map ι p : S / Γ ADE , −→ (cid:0) (cid:94)C / Γ ADE (cid:1) p . (6.16)Similarly, the limiting configuration of L approaching an untwisted regular puncture k , sayfrom the right in figure 42, hits the boundary component B k at a particular point p ∈ S .The fiber component α embeds into the fiber (cid:0) (cid:94)C / Γ ADE (cid:1) p of B k at p via the inclusionmap described above. Since the two embeddings of α into (cid:0) (cid:94)C / Γ ADE (cid:1) p and (cid:0) (cid:94)C / Γ ADE (cid:1) p respectively are homotopic to each other, we deduce that ( L, α ) = ( L , α ) as elements of H ( ∂X, Z ).Finally, we need to show that ( L, α ) and ( L , α ) give rise to the same element in H ( ∂X, Z )if L is obtained from L by passing it over an twisted regular puncture, as long as α is leftinvariant by the action of the outer-automorphism associated to the twist line emanating fromthe twisted regular puncture. The argument proceeds exactly as in the untwisted case sincethe twist line is immaterial if α is left invariant by the corresponding outer-automorphismaction. On the other hand, if α is not left invariant by the outer-automorphism, then L needsto be divided into two sub-rays (denoted by blue and green respectively in figure 43) with α inserted along the blue sub-ray and o · α inserted along the green sub-ray. In particular, thereis no consistent limiting configuration as L approaches the puncture, and the above argumentfails. Thus, L and L give rise to different elements of H ( ∂X, Z ) (and hence L ) if α is actedupon by the twist line emanating from the regular puncture.The above argument can be viewed as a justification of our key assumption used in theearlier parts of the paper: If L is a loop surrounding a regular (untwisted or twisted) puncturecarrying an element α ∈ b 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 arguing56hat closing an untwisted regular puncture does not change the defect group. It would beinteresting to develop this point of view. Here we note, that in the geometric descriptionsone could argue as folllows: regular punctures characterize base points at which fibral P ’sboth decompactify and potentially braid upon. For line operators the decompactification ofcycles 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 otherwords, regular punctures can be filled in from the perspective of line operators and do notcontribute to the defect group. It would be interesting to develop the precise dictionary, andto expand it to include irregular punctures.Generically the above procedure can be applied to any canonical singularity. E.g. even inthe case of general irregular punctures, which realize Argyres Douglas theories, that do notnecessarily admit a Lagrangian description. The theories of type AD[ G, G ] have a realizationin terms of Type IIB on a canonical singularity and for AD[ G, G ] theories, the 1-form sym-metries are non-trivial only for G = A N with N > G = D, E type, see [14, 18]. Theseresults should provide further insights into computing the one-form symmetry for irregularpunctures more generally.
Acknowledgements
We thank Fabio Apruzzi, Chris Beem, Cyril Closset, Po-Shen Hsin, Du Pei, Yuji Tachikawa,Yi-Nan Wang and Gabi Zafrir for discussions. We in particular thank Yuji Tachikawa fordetailed comments on the draft.This work is supported by ERC grants 682608 (LB and SSN) and 787185 (LB). The workof MH is supported by the Studienstiftung des Deutschen Volkes. SSN also acknowledgessupport through the Simons Foundation Collaboration on "Special Holonomy in Geometry,Analysis, and Physics", Award ID: 724073, Schafer-Nameki.
A Summary of Notation • g : Mostly denotes the A,D,E Lie algebra denoting the 6 d N = (2 ,
0) theory underconsideration. Can also denote a simple gauge algebra (simply or non-simply laced) fora 4 d N = 2 gauge theory depending on the context.• G : The simply connected group associated to a simple Lie algebra g .• Z ( G ): The center of a simply connected group G .57 b Z ( G ): The Pontryagin dual of the center of a simply connected group G . For a 6 d N = (2 ,
0) theory associated to an A,D,E Lie algebra g , b Z ( G ) captures the group ofdimension-2 surface operators modulo screenings, also known as the defect group of the6 d 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 d N = 2 theory obtained by compactifying a 6 d N = (2 ,
0) theory on a Riemann surface C g , possibly in the presence of twist lines and (untwisted and twisted) regular punctures.• h· , ·i : Often referred to as pairing . It takes two elements of b Z ( G ) or of L , and outputsan element of R / Z which captures the phase associated to mutual non-locality of thedefect operators associated to the two elements.• Λ: Often referred to as polarization or maximal isotropic subgroup . This is a maximalsubgroup of L such that the pairing h· , ·i on L restricted to this subgroup Λ vanishes. Achoice of such a Λ is correlated to the choice of an absolute d N = 2 theory.• b Λ: Pontryagin dual of polarization Λ. Captures the 1-form symmetry of the absolute 4 d N = 2 theory associated to a polarization Λ.• F : Denotes fundamental representation for g = su ( n ) , sp ( n ); vector representation for g = so ( n ); representations of dimension , , , for g = g , f , e , e respectively;and the adjoint representation for g = e .• Λ n : Denotes the n -index antisymmetric irreducible representation for g = su ( n ) , sp ( n ).• S : Denotes the 2-index symmetric representation for g = su ( n ).• S : Denotes the irreducible spinor representation for g = so ( n ).• C : Denotes the irreducible cospinor representation for g = so (2 n ).• n R : Denotes n full hypermultiplets transforming in a representation R .• n +12 R : Denotes n full hypermultiplets and a half-hypermultiplet transforming in apseudo-real representation R . 58 eferences [1] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries , JHEP (2015) 172, [ ].[2] L. Bhardwaj, M. Hubner and S. Schafer-Nameki, Under Preparation , .[3] D. Gaiotto, N=2 dualities , JHEP (2012) 034, [ ].[4] D. S. Freed and C. Teleman, Relative quantum field theory , Commun. Math. Phys. (2014) 459–476, [ ].[5] E. Witten,
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