3d Abelian Gauge Theories at the Boundary

TL;DR

The paper develops a unified BCFT framework for a 4d U(1) gauge field coupled to a 3d CFT on a boundary, producing a continuous family B(\tau,\bar{\tau}) of boundary conformal data related by SL(2,\mathbb{Z}) duality. It expresses bulk correlators and hemisphere free energy in terms of boundary current data c_{ij}(\tau,\bar{\tau}) and the displacement correlator C_{\hat{D}}, and analyzes strong- and weak-coupling frames, including decoupled cusps corresponding to free Dirac fermions and to the O(2) model. Focusing on a boundary Dirac fermion, the authors compute boundary operator dimensions and F_{\partial} up to two loops, then apply a duality-informed Padé resummation to extrapolate to the O(2) cusp with good agreement to known results, illustrating the method’s predictive power. They extend the analysis to other boundary theories, including large-\,N_f fermions and a free complex scalar, and describe how to realize QED3 with two flavors via an Sp(4,\mathbb{Z}) duality acting on two bulk photons. Collectively, the work provides a versatile perturbative-to-nonperturbative strategy for accessing strong-coupling BCFT data through dualities and boundary OPE techniques.

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 $τ$ in the upper-half plane and by the choice of the CFT in the decoupling limit $τ\to \infty$. 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 $τ$ 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.

Figures (19)

• Figure 1: The family of conformal boundary conditions $B(\tau,\bar{\tau})$ labeled by the variable $\tau$ in the upper-half plane and by a 3d CFT $T_{0,1}$ with $U(1)$ global symmetry. At the cusp at infinity the current $\hat{I}^a$ decouples and we are left with the local 3d theory $T_{0,1}$ on the boundary, with $U(1)$ current $\hat{J}^a$. Approaching this cusp from $T$-translations of the fundamental domain amounts to adding a CS contact term to the 3d theory, or equivalently to redefine the current $\hat{J}^a$ by multiples of the current $\hat{I}^a$ that is decoupling. This is the $T$ operation on $T_{0,1}$ in the sense of Witten:2003ya. In the favorable situation in which no phase transitions occur, the BCFT continuously interpolate to the cusps at the rational points of the real axis $\tau = -q/p$, where again the bulk and the boundary decouple and we find new 3d CFTs $T_{p,q}$. These theories are obtained from $T_{0,1}$ with a more general $SL(2,\mathbb{Z})$ transformation, that involves coupling the original $U(1)$ global symmetry to a 3d dynamical gauge field.
• Figure 2: A cartoon of a possible phase transition at strong coupling. A scalar boundary operator becomes marginal at a certain curve in the $\tau$ plane, i.e. setting $\hat{\Delta}(\tau,\bar{\tau}) = 3$ we find solutions in the upper-half plane. In conformal perturbation theory from a point on the curve, the beta function takes the form \ref{['eq:betamarg']}. We might be unable to find real fixed points for the marginal coupling. In such a situation, $B(\tau,\bar{\tau})$ can only be defined as a complex BCFT. Assuming that we were able to define $B(\tau,\bar{\tau})$ as a real BCFT in perturbation theory around $\tau \to \infty$ by continuity such a real BCFT is ensured to exist in the full region above the wall, but we might be unable to continue it beyond the wall without introducing complex couplings (or breaking conformality).
• Figure 3: Diagrams for the two-point function of the displacement operator. The leading order contribution $(a)$ is the square of the two-point function of the topological current $\hat{I}$. At next-to-leading order we have the diagrams $(b.1)$-$(b.2)$-$(b.3)$ that are also sensitive to the electric current $\hat{J}$. The shaded blobs denote insertions/correlators of $\hat{J}$ in the undeformed CFT.
• Figure 4: The upper-half plane of the gauge coupling $\tau_{DN}$, i.e. in the duality frame in which at $\tau_{DN}\to \infty$ we find the $O(2)$ model on the boundary. Thanks to particle-vortex duality, the cusp in the origin $\tau_{DN} = 0$ also gives a decoupled $O(2)$ model on the boundary. Thanks to the boson-fermion duality between $U(1)_1$ coupled to a critical scalar and a free Dirac fermion, the cusps at $\tau_{DN} = \pm 1$ give a free Dirac fermion.
• Figure 5: The upper-half plane of the gauge coupling $\tau = \tau_{NN'} -\frac{1}{2}$, i.e. in the duality frame in which at $\tau\to \infty$ we find a free Dirac fermion on the boundary. Thanks to fermionic particle-vortex duality, the cusp in the origin $\tau = 0$ also gives a free Dirac fermion on the boundary. Thanks to the boson-fermion duality between $U(1)_{\frac{1}{2}}$ coupled to a Dirac fermion and the $O(2)$ model, the cusps at $\tau= \pm \frac{1}{2}$ give the $O(2)$ model.