# 3d Abelian Gauge Theories at the Boundary

## TL;DR

This paper explores the properties of boundary conformal field theories arising from 4d Abelian gauge fields coupled to 3d CFTs, analyzing their behavior under dualities, and computing key observables including scaling dimensions and free energies.

## Contribution

It introduces a framework for studying BCFTs from 4d Abelian gauge theories, computes boundary operator dimensions, and applies dualities to analyze specific models like the O(2) model.

## Key findings

- Boundary two-point functions are expressed in terms of boundary electric and magnetic currents.
- Scaling dimensions and hemisphere free-energy are computed up to two loops for the O(2) model.
- Extrapolation techniques provide good approximations to strong coupling observables.

## Abstract

A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a $U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling $\tau$ in the upper-half plane and by the choice of the CFT in the decoupling limit $\tau \to \infty$. Upon performing an $SL(2,\mathbb{Z})$ transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's $SL(2, \mathbb{Z})$ action [1]. In particular the cusps on the real $\tau$ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk $S$ transformation, and it also admits a decoupling limit which gives the $O(2)$ model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the $O(2)$ model. We also consider examples with other theories on the boundary, such as large-$N_f$ Dirac fermions --for which the extrapolation to strong coupling can be done exactly order-by-order in $1/N_f$-- and a free complex scalar.