Duality and Transport for Supersymmetric Graphene from the Hemisphere Partition Function

TL;DR

This paper uses localization to compute the partition function of a supersymmetric gauge theory on a hemisphere, revealing its equivalence to a Chern-Simons theory and enabling calculations of operator dimensions and conductivities at arbitrary couplings.

Contribution

It establishes a novel connection between supersymmetric gauge theories on hemispheres and Chern-Simons theories, allowing exact calculations of physical observables across couplings.

Findings

Partition function matches that of Chern-Simons theory with complexified coupling.

Allows calculation of operator dimensions and two-point functions at any coupling.

Identifies self-dual theories with specific conductivity properties.

Abstract

We use localization to compute the partition function of a four dimensional, supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter on the boundary. Our theory has eight real supercharges in the bulk of which four are broken by the presence of the boundary. The main result is that the partition function is identical to that of ${\mathcal N}=2$ abelian Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where the quantized Chern-Simons level is replaced by an arbitrary complexified gauge coupling $\tau$. The localization reduces the path integral to a single ordinary integral over a real variable. This integral in turn allows us to calculate the scaling dimensions of certain protected operators and two-point functions of abelian symmetry currents at arbitrary values of $\tau$. Because the underlying theory has conformal symmetry, the current two-point functions tell us the zero temperature conductivity of the Lorentzian versions of these theories at any value of the coupling. We comment on S-dualities which relate different theories of supersymmetric graphene. We identify a couple of self-dual theories for which the complexified conductivity associated to the U(1) gauge symmetry is $\tau/2$.