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 τ. 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 τ. 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 τ/2.
Highlights
Supersymmetry with a boundaryTwo facts about supersymmetry greatly constrain our problem. The first is that most of the localization technology available requires the existence of a continuous R-symmetry
Localization calculation for a non-abelian gauge theory on HS4 [16].1 Ref. [16] is in one respect more general than our case, but in another not general enough
The main result is that the partition function is identical to that of 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 τ
Summary
Two facts about supersymmetry greatly constrain our problem. The first is that most of the localization technology available requires the existence of a continuous R-symmetry. The second is that the presence of a boundary will break at least half of the supersymmetry present in the bulk, because supercharges square to translation generators. The vector τ = (σ1, σ2, σ3) of Pauli spin matrices generates the SU(2) R-symmetry, while v is a unit vector which determines how the SU(2) is broken down to a U(1) subgroup by the boundary. The projected spinor at the boundary is the 3d Killing spinor, projecting the algebra (2.2) to its tangential direction. As Π±ij becomes diagonal in the SU(2) basis, it is convenient to identify projectors associated with each basis element:. Because the γA commute with Π±ij (2.5), we can use the tangential 4d gamma matrices γA to generate our 3d gamma matrix algebra: ΓA = π+γA.
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