3d modularity

We propose dual pairs of N = (0, 4) half-BPS boundary conditions for 3d N = 4 Abelian gauge theories related to mirror symmetry and S-duality by showing the matching of boundary 't Hooft anomalies and supersymmetric indices. We find simple N = (0, 4) mirror symmetry between 2d N = (0, 4) Abelian gauge theories and free Fermi multiplets that generalizes N = (0, 2) Abelian duality. We also propose a prescription for computing half-index of enriched Neumann boundary condition including 2d boundary bosonic matters by gauging the 2d boundary flavor symmetry of Dirichlet boundary condition. By coupling N = (0, 4) half-BPS boundary configurations of 3d N = 4 gauge theories to quarter-BPS corner configurations of 4d N = 4 Super Yang-Mills theories, we further obtain a new type of 4d-3d-2d duality that may involve 3d non-Abelian gauge symmetry.

Section: Future Workmentioning

confidence: 99%

Abelian dualities of $$ \mathcal{N} $$ = (0, 4) boundary conditions

Okazaki

2019

“…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 features related to the global structure of the gauge groups and higher-form symmetries of these theories, as well as the explicit computation of the Witten index and related observables. The approach we take is as follows: 1.…”

mentioning

confidence: 77%

“…The vanishing of the degree can always be assumed using (2.11) and (2.12). Then, the graph Ω (k i ) 23 for the Seifert manifolds that we are interested in is given by figure 10. In order to guarantee that this corresponds to a Seifert manifold we have to impose k 5 = 1. and q > 0 for simplicity.…”

Section: Seifert Quiversmentioning

confidence: 99%

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

Eckhard

Kim

Schäfer‐Nameki

et al. 2020

101

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...

“…Apart from its intrinsic importance in differential topology, the η-invariant has a number of interesting physics applications, for example, in the analysis of global gravitational anomalies [16], in fermion fractionization [17,18] , in relation to spectral flow in quantum chromodynamics [19,20], and more recently in the description of symmetry-protected phases of topological insulators (see [21] for a recent review). Similarly, apart from their intrinsic interest in number theory [22,23], mock modular forms and their cousins have come to play an important role in the physics of quantum black holes and quantum holography [2], in umbral moonshine [24,25], in the context of WRT invariants [26][27][28][29] , and more generally in the context of elliptic genera of noncompact SCFTs [30][31][32][33][34][35]. We expect our results will have useful implications in these diverse contexts.…”

Section: Introductionmentioning

confidence: 82%

“…which can be viewed as a τ 2 → 0 − limit of the false theta function for a = k and b = l using the fact that sgn ( + 2kn) = sgn(n) for positive k and 0 ≤ l < 2k. In this limit, the functions f (τ 1 ) have appeared in the computation of topological invariants of Seifert manifolds with three singular fibers [29]. See [29] for a discussion of quantum modular forms and the relation to mock and false theta functions in the context of Chern-Simons theory and WRT invariants [26-28, 60, 61].…”

Section: The η-Invariant and Quantum Modular Formsmentioning

confidence: 99%

APS η-invariant, path integrals, and mock modularity

Dabholkar

Jain

Rudra

2019

We show that the Atiyah-Patodi-Singer η-invariant can be related to the temperature dependent Witten index of a noncompact theory and give a new proof of the APS theorem using scattering theory. We relate the η-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the η-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.(Dedicated to the memory of Michael Atiyah)