BPS indices, modularity and perturbations in quantum K-theory

Abstract

We study a perturbation family of $$ \mathcal{N} $$ N = 2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

Web of Seiberg-like dualities for 3D N=2 quivers

“…While we have checked the matching of indices as strong evidence of the dualities, analytic proof of the various identities of indices presented in this paper is desirable. Also the half-index which involves the 3D bulk degrees of freedom can exhibit interesting properties under modular transformations [44,92]. (vii) The Seiberg-like dualities in 3D N ¼ 2 gauge theories are obtained from 4D dualities by compactifying 4D N ¼ 1 gauge theories on a circle [10,93].…”

“…Half-BPS boundary conditions can preserve either N ¼ ð1; 1Þ which have been described in [35,36], or N ¼ ð0; 2Þ which we focus here. In particular, N ¼ ð0; 2Þ half-BPS chiral UV boundary conditions preserving the Uð1Þ R symmetry lead to various applications [35,[37][38][39][40][41][42][43][44][45][46][47]. For pairs of UV boundary conditions, there exist dual boundary conditions which flow to the same IR boundary conditions.…”

Web of Seiberg-like dualities for 3d $\mathcal{N}=2$ quivers

“…While we have checked the matching of indices as strong evidence of the dualities, analytic proof of the various identities of indices presented in this paper is desirable. Also the half-index which involves the 3d bulk degrees of freedom can exhibit interesting properties under modular transformations [85,39].…”

“…Half-BPS boundary conditions can preserve either N = (1, 1) which have been described in [30,31], or N = (0, 2) which we focus here. In particular, N = (0, 2) half-BPS chiral UV boundary conditions preserving the U(1) R symmetry lead to various applications [32,30,33,34,35,36,37,38,39,40,41]. For pairs of UV boundary conditions, there exist dual boundary conditions which flow to the same IR boundary conditions.…”

Confluence in quantum K-theory of weak Fano manifolds and q-oscillatory integrals for toric manifolds

“…Remark 4.4.2. Note also that in Equation (2.13) of [JMNT21], another q-gamma class is introduced for Grassmannians through a different motivation.…”