3d dualities from 4d dualities

167

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.

“…For instance, this happens for the S 3 × S 1 background discussed in section 4. In this example, the quantization of FI-terms was recently discussed in [54]. 22 …”

Section: Jhep01(2014)124mentioning

confidence: 99%

Section: Jhep01(2014)124mentioning

confidence: 99%

The geometry of supersymmetric partition functions

Closset

Dumitrescu

Festuccia

et al. 2014

167

415

“…Since the det ij factor does not affect to exponential contributions, the integrand in (4.19) for |σ| 1 behaves as 25) whereL S 1 ×Σg (a, m) is given in (4.18). The condition for convergence depends on the label m of the topological sector, like the effective potential.…”

Section: Jhep04(2017)055mentioning

confidence: 99%

“…Similarly to the Cardy formula in 2d CFT [24], this limit controls the asymptotic behavior at large energies of the weighted density of short states. Following the observations in [25][26][27], and assuming the existence of a weakly-coupled point in the space of couplings, 1 it was argued in [28] that for β → 0 the partition function has a universal behavior (see also [34][35][36][37][38])…”

Section: Introductionmentioning

confidence: 99%

Cardy formula for 4d SUSY theories and localization

Pietro

Honda

2017

119

Abstract:We study 4d N = 1 supersymmetric theories on a compact Euclidean manifold of the form S 1 × M 3 . Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold M 3 . Taking the limit of shrinking S 1 , we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of [1]. The limit is given in terms of an effective potential for the holonomies around the S 1 , whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by Tr(R), while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if Tr(R) ≤ 0.

“…Naively speaking we can derive the corresponding 3d Seiberg dual by dimensionally reducing the 4d electric and magnetic theories respectively. This method was studied in [28,29], where those authors claimed that in reducing the 4d Seiberg dual pairs to 3d, it is important to take into account the non-perturbative effects from the twisted instantons (KK-monopoles) and carefully to take some low-energy limit on both sides. In our case of G 2 , we can easily find the electric theory on a circle.…”

Section: Jhep02(2018)158mentioning

confidence: 99%

Low-energy dynamics of 3d N $$ \mathcal{N} $$ = 2 G2 supersymmetric gauge theory

Nii

Sekiguchi

2018

We study a three-dimensional N = 2 supersymmetric G 2 gauge theory with and without fundamental matters. We find that a classical Coulomb branch of the moduli space of vacua is partly lifted by monopole-instantons and the quantum Coulomb moduli space would be described by a complex one-dimensional space. Depending on the number of the matters in a fundamental representation, the low-energy dynamics of the theory shows various phases like s-confinement or quantum merging of the Coulomb and the Higgs branches. We also investigate superconformal indices as an independent check of our analysis.