SL(2,Z) Action on AdS/BCFT and Hall Conductivities

TL;DR

The paper constructs a holographic AdS/BCFT model with a boundary brane Q to study Hall transport in a 2+1D BCFT and analyzes how $SL(2,\mathbb{Z})$ duality acts on the bulk and the boundary theory. By formulating the bulk gauge dynamics with boundary terms and exploring Neumann and Dirichlet boundary conditions, it derives a rich set of transport coefficients, including novel ones $\kappa_x,\kappa_y,\gamma_x,\gamma_y$, and establishes that $\kappa_t=\sigma_{xy}$ while revealing conditions under which Onsager relations hold. The authors show that the Hall conductivity can be nonzero even without large external magnetic fields, and they illustrate generalized duality transformations of $\sigma_{xy}$ under $S$ and $T$, with explicit results for special angles and boundary conditions. A Type IIA string realization is presented, where a holographic FQHE with filling fraction $\nu = M/k$ arises and duality actions map to new conductivities in the stringy setup. Overall, the work clarifies how boundaries and dualities shape Hall physics in strongly coupled BCFTs and opens avenues for further exploration of multi-boundary, time-dependent, and string-theoretic generalizations.

Abstract

We study the response of a conserved current to external electromagnetic fields in a holographic system with boundaries using the recently proposed AdS/BCFT (boundary conformal field theory) framework. This, in particular, allows us to extract the Hall current, the Hall conductivity, plus some potentially novel transport coefficients, and relations among them. We also analyze the action of SL(2,Z) duality in the gravity bulk, which acts non-trivially on the conductivity of the BCFT. Finally we consider a type IIA string theory embedding of our setup.

Figures (1)

• Figure 1: Embedding of the defect brane corresponding to various values of the tension $T$ and hence to various angles $\theta$. The solid (blue) line is the part of AdS boundary on which the BCFT lives. The dashed (red) line the brane $Q$ at which spacetime terminates. The filled part of spacetime is excised; the unfilled part is physical. The dotted (blue) line is the part of the AdS boundary that is cut out.