3-Manifolds and 3d indices

We show that specializations of the 4d N = 2 superconformal index labeled by an integer N is given by Tr M N where M is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras A N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S 2 × T 2 where we turn on 1 2 N units of U(1) r flux on S 2 .

Section: Insertion Of Wilson Loop Line Operatorsmentioning

confidence: 99%

Superconformal index, BPS monodromy and chiral algebras

Cecotti

Song

Vafa

et al. 2017

“…As in four dimensions, a suitable trace over states on S 2 in Hamiltonian quantization defines a supersymmetric index I(y, u), where y and u are (generally complex) fugacities that couple to the Hamiltonian H, which generates translations along R, and an Abelian flavor charge Q f that commutes with the supercharges [75][76][77][78]. (As in four dimensions, generic values of the chemical potentials are only consistent with two supercharges.)…”

Section: S 2 × S 1 Without Flux and The Supersymmetric Indexmentioning

confidence: 99%

The geometry of supersymmetric partition functions

Closset

Dumitrescu

Festuccia

et al. 2014

172

415

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z M . Our primary focus is the dependence of Z M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N = 1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N = 2 theories with a U(1) R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 ×S 1 or S 2 ×S 1 , which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where Z M has been calculated explicitly, our results explain many of its observed properties.

“…Notably, this quantization depends critically on the relative values of k and s, which determine the real symplectic structure of the phase space. The quantization for the k = 0 case is elaborated in [27]. Next, to discuss the two dimensional complex Toda theory with k = 0, we first note that for the k ≥ 1 cases, the previous work by Cordova and Jafferis [7] showed that the complex Toda theory is dual to the para-Toda theory plus a decoupled coset model.…”

Section: Jhep05(2017)121mentioning

confidence: 99%

“…Following the discussion of [27] by Dimofte et al, we first note that the Chern-Simons path integral on M is related to a wavefunction in a boundary Hilbert space H ∂M . And H ∂M is given by the quantization of the classical phase space associated to the boundary:…”

Section: Jhep05(2017)121mentioning

confidence: 99%

Four-dimensional N $$ \mathcal{N} $$ = 2 supersymmetric theory with boundary as a two-dimensional complex Toda theory

Luo

Tan

Vasko

et al. 2017

We perform a series of dimensional reductions of the 6d, N = (2, 0) SCFT on S 2 × Σ × I × S 1 down to 2d on Σ. The reductions are performed in three steps: (i) a reduction on S 1 (accompanied by a topological twist along Σ) leading to a supersymmetric Yang-Mills theory on S 2 × Σ × I, (ii) a further reduction on S 2 resulting in a complex Chern-Simons theory defined on Σ × I, with the real part of the complex Chern-Simons level being zero, and the imaginary part being proportional to the ratio of the radii of S 2 and S 1 , and (iii) a final reduction to the boundary modes of complex Chern-Simons theory with the Nahm pole boundary condition at both ends of the interval I, which gives rise to a complex Toda CFT on the Riemann surface Σ. As the reduction of the 6d theory on Σ would give rise to an N = 2 supersymmetric theory on S 2 × I × S 1 , our results imply a 4d-2d duality between four-dimensional N = 2 supersymmetric theory with boundary and two-dimensional complex Toda theory.