Duality and transport for supersymmetric graphene from the hemisphere partition function

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 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.

Line and surface defects for the free scalar field

“…We consider this somewhat surprising: for example, we do not expect this conclusion to hold for co-dimension one (boundaries). Indeed, for d > 2 several non-trivial boundary conditions appear possible [3,[25][26][27][28][29] and for d = 2 there exists a family of conformal boundary conditions for a free (compact) scalar [30]. Also, non-trivial defects do exist in other cases where the bulk is free, like the non-trivial co-dimension two monodromy defects for a free hypermultiplet in 4d with N = 2 [31,32] (see also [33,34] for not necessarily conformal defects in this theory) and the co-dimension four surface operators in the abelian (2, 0) theory [35][36][37][38][39][40][41].…”

“…For p = 1 the solution (B.5) is either a constant mode (|s| = 1 2 ) or below the unitarity bound and we are free to set it to zero by choosing B = 0 for all s. 28 Let us now go back to the inhomogeneous problem and fix the solution (B.6) in order to reproduce the contact term in the r.h.s. of (B.4).…”

Janus interface entropy and Calabi’s diastasis in four-dimensional $$ \mathcal{N} $$ = 2 superconformal field theories

“…in terms of supersymmetric sphere partition functions with and without an interface. We note that the combination (2.26) coincides with the "boundary free energy" considered in [33][34][35]. 6 We explain in section 6.1 that one can use the super-Weyl anomaly of [7] to prove the 2d version of the assumption (2.25).…”

Holographic and localization calculations of boundary F for $$ \mathcal{N} $$ = 4 SUSY Yang-Mills theory

“…Analogous calculations were performed for N = 2 theories on S 3 in [53], with exact evaluation of the partition function for 3-dimensional quiver gauge theories appearing in [54][55][56]. Localization calculations on manifolds with boundary in two and three dimensions first appeared in [57]; in four dimensions, the first direct calculations appeared in [58], which considered Neumann and Dirichlet boundary conditions only, followed by [59], which considered more general boundary conditions for the Abelian theory. Earlier general considerations for the case with boundaries can be found in [17,60,61].…”