Down the rabbit hole with theories of class S $$ \mathcal{S} $$

Razamat, Shlomo S.; Willett, Brian

doi:10.1007/jhep10(2014)099

Cited by 69 publications

(76 citation statements)

References 59 publications

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“…But one can check that, in the star-shaped quiver description, this Z N gauging maps precisely to gauging the 1-form symmetry of the central SU (N ) node, leaving the gauge group as SU (N )/Z N . This is in fact the expected result, as the duality of the star-shaped quiver of the trinion holds only when the central node is taken as SU (N )/Z N [59]. Similarly, S 2 becomes trivial after gauging the Z N at the middle node because T (G) squares to the identity [24], when the global structure is taken into account.…”

Section: Symmetries Of the Quiversupporting

confidence: 60%

Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

Eckhard

Kim

Schäfer‐Nameki

et al. 2020

J. High Energ. Phys.

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101

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By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d (2, 0) theory on a three-manifold M 3 . This generalization is applicable to both the 3d N = 2 and N = 1 supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by M 3 . This is carried out in detail for M 3 a Seifert manifold, where we compute a refined version of the Witten index.In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on M 3 , which matches the Witten index computation that takes the higher-form symmetries into account.for which such 3d-3d duals have been discussed are the squashed three-sphere S 3 b [1, 2, 6], the superconformal index on S 2 × S 1 [3], and the twisted index on Σ g × S 1 [7]. The special case W 3 = T 3 was considered in [5,7] and computes the (regularized) Witten index [8] I = Tr(−1) F . 1 However, an important characteristic of the theories has so far been largely ignored: their higher-form symmetries [12]. Namely, the theory T [M 3 ] has a higher-form symmetry, which, as with other properties of T [M 3 ], is determined by the topology of M 3 . In fact, the theory T [M 3 ] is not fully specified by the manifold, M 3 , but requires additional topological data. This is related to the fact that the 6d N = (2, 0) theory itself is a relative QFT, i.e., it is only well-defined as the boundary of a 7d TQFT [13,14]. Equivalently, its observables depend on a choice of polarization, i.e., a choice of maximal isotropic subgroup of H 3 (M 3 , Z G ), where Z G is the center of the simply connected group, G, with Lie algebra g. Naturally, we expect that T [M 3 ] also depends on the polarization, as is the case for 4d theories [15,16]. In fact, we will see that this additional information translates into the residual 0-and 1-form symmetry of the 3d theory. We propose therefore a refined definition of the theories, which specifies this dataThis theory has a discrete 0-form (ordinary) symmetry group H. Its residual 1-form symmetry is given by the complementary subgroup, Υ H , 2 inside H 1 (M 3 , Z G ). We show that the choice of H can indeed be detected by the Witten index or, more generally, by the partition function on any W 3 with non-trivial homology. Thus, the different theories in (1.2) are indeed physically distinct.The main interest of this paper is to develop a sound definition of the theories T [M 3 , g, H]in (1.2) for M 3 a graph manifold [17], a class of three-manifolds we review in section 2.1. These manifolds, which also occur as the boundary of plumbed four-manifolds, are sometimes called plumbed three-manifolds, a special case of which are Seifert manifolds. Similar Lagrangians for the 3d theories associated to these manifolds were studied in [6,[18][19][20][21][22][23].In the following we point out new feat...

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Section: Symmetries Of the Quiversupporting

confidence: 60%

Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

Eckhard

Kim

Schäfer‐Nameki

et al. 2020

J. High Energ. Phys.

Self Cite

101

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“…The specific case of a genus g > 1 Riemann surface without punctures or flux is especially appealing for several reasons. First, these class of theories are expected to have a large conformal manifold, related to complex structure deformations of the Riemann surfaces and flavor holonomies, on a generic point of which the global symmetry is completely broken [9,10]. Also, the 6d construction allows us to compute various quantities of interest like the a and c central charges, see for instance [9,10], and for this class of compactifications these are guaranteed to be rational.…”

Section: The Basic Ideamentioning

confidence: 99%

$$ \mathcal{N} $$ = 1 conformal dualities

Razamat

Zafrir

2019

J. High Energ. Phys.

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116

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We consider on one hand the possibility that a supersymmetric N = 1 conformal gauge theory has a strongly coupled locus on the conformal manifold at which a different, dual, conformal gauge theory becomes a good weakly coupled description. On the other hand we discuss the possibility that strongly coupled theories, e.g. SCFTs in class S, having exactly marginal N = 1 deformations admit a weakly coupled gauge theory description on some locus of the conformal manifold. We present a simple algorithm to search for such dualities and discuss several concrete examples. In particular we find conformal duals for N = 1 SQCD models with G2 gauge group and a model with SU (4) gauge group in terms of simple quiver gauge theories. We also find conformal weakly coupled quiver theory duals for a variety of class S theories: T4, R0,4, R2,5, and rank 2n Minahan-Nemeschansky E6 theories. Finally we derive conformal Lagrangians for four dimensional theories obtained by compactifying the E-string on genus g > 1 surface with zero flux. The pairs of dual Lagrangians at the weakly coupled loci have different symmetries which are broken on a general point of the conformal manifold. We match the dimensions of the conformal manifolds, symmetries on the generic locus of the conformal manifold, anomalies, and supersymmetric indices. The simplicity of the procedure suggests that such dualities are ubiquitous. arXiv:1906.05088v1 [hep-th]

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“…where we use notations of [12]. Taking the matter content of the dimensionally reduced theory and the map of the R-charges (R H , R C ) = (R, −r), we can compute the index of the dimensionally reduced theory without any S 2 fluxes for global symmetries to be,…”

Section: Partition Functionsmentioning

confidence: 99%

“…A simple version of the index is the Coulomb limit [12], taking t, q → 0 while keeping the ratio q 1 2 t = x fixed. In this limit the dependence on fugacities under which Higgs branch operators are charged drops out but the dependence on the fluxes remains.…”

Section: Partition Functionsmentioning

confidence: 99%

“Lagrangian” for a Non-Lagrangian Field Theory withN=2Supersymmetry

2015

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We suggest that at least some of the strongly-coupled N = 2 QFTs in 4d can have a non-conformal N = 1 Lagrangian description flowing to them at low energies. In particular, we construct such a description for the N = 2 rank one superconformal field theory with E6 flavor symmetry, for which a Lagrangian description was previously unavailable. We utilize this description to compute several supersymmetric partition functions.Introduction: In 4d a standard definition of a quantum field theory (QFT) involves the specification of a Lagrangian. Even if the theory of interest is stronglycoupled, it typically has a UV description in terms of a non-conformal Lagrangian which flows to it in the IR; a familiar example is QCD. This is due to a multitude of relevant deformations admitted by a generic Lagrangian. However, Lagrangians with a large enough symmetry might not admit relevant deformations explicitly preserving this symmetry. Important examples are N = 2 supersymmetric QFTs, which do not admit any N = 2 relevant superpotential deformations except for mass terms. This has given N = 2 intrinsically strongly-coupled theories a somewhat mysterious status of being non-Lagrangian. The situation is somewhat analogous to the pre-QCD models of strong interactions: while a lot of understanding of the IR properties of hadrons was available, no UV Lagrangian description was known. Once such a description was available, it provided a much more powerful tool for understanding the structure of the theory, even at low energies. In this note, we discuss the possibility that the N = 2 strongly-coupeld SCFTs possess a non-conformal Lagrangian description with less supersymmetry which is enhanced to N = 2 at the fixed point.We will discuss here in detail an N = 1 Lagrangian description for the N = 2 superconformal theory with E 6 flavor symmetry [1], which is a paradigmatic example of an intrinsically strongly-coupled N = 2 QFT. The strategy which will work for us in this case, and which we will argue is more general, is as follows. We will look for special points on the conformal manifold of an N = 2 theory where the flavor symmetry commuting with an N = 1 subalgebra is enhanced. Upon gauging that symmetry with an N = 1 vector multiplet, an RG flow is initiated which generically leads to a theory with N = 1 superconformal symmetry. However, we will argue that, at least in some cases, this RG flow leads to an N = 2 strongly-coupled theory.The inspiration for our construction is the structure of the exact expression for the superconformal index of the E 6 theory, which can be obtained using, a priori, formal mathematical manipulations [2]. We will argue

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Down the rabbit hole with theories of class S $$ \mathcal{S} $$

Abstract: We review some of the properties of 3d N = 4 theories obtained by dimensionally reducing theories of class S. We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.

Cited by 69 publications

References 59 publications

Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

Higher-form symmetries, Bethe vacua, and the 3d-3d correspondence

$$ \mathcal{N} $$ = 1 conformal dualities

“Lagrangian” for a Non-Lagrangian Field Theory withN=2Supersymmetry

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