Boundary Conformal Field Theory and a Boundary Central Charge

TL;DR

This work analyzes conformal field theories in the presence of a planar boundary, establishing a fundamental link between the boundary central charge $b_2$ and the near-boundary displacement-operator correlator via $oxed{b_2 = rac{2\pi^4}{15}\alpha(1)}$. It shows that for free theories the well-known relation $b_2=8c$ holds because $oldsymbol{ u}$ lies at a simple boundary-block configuration with $oldsymbol{ u}= ext{±1}$ and $ ext{alpha}(1)=2 ext{alpha}(0)$, but interacting boundary theories can break this relation by modifying $ ext{alpha}(1)$ through boundary degrees of freedom. The paper then introduces several boundary-interacting models (e.g., mixed dimensional QED resembling graphene) at perturbative fixed points, deriving beta functions and showing that $b_2$ can depend on marginal couplings, hence departing from the bulk-central-charge intuition. Collectively, these results illuminate the rich structure of bCFTs, the role of boundary anomalies, and the potential for boundary dynamics to alter central charges in four dimensions, with implications for both field theory and condensed-matter realizations.

Abstract

We consider the structure of current and stress tensor two-point functions in conformal field theory with a boundary. The main result of this paper is a relation between a boundary central charge and the coefficient of a displacement operator correlation function in the boundary limit. The boundary central charge under consideration is the coefficient of the product of the extrinsic curvature and the Weyl curvature in the conformal anomaly. Along the way, we describe several auxiliary results. Three of the more notable are as follows: (1) we give the bulk and boundary conformal blocks for the current two-point function; (2) we show that the structure of these current and stress tensor two-point functions is essentially universal for all free theories; (3) we introduce a class of interacting conformal field theories with boundary degrees of freedom, where the interactions are confined to the boundary. The most interesting example we consider can be thought of as the infrared fixed point of graphene. This particular interacting conformal model in four dimensions provides a counterexample of a previously conjectured relation between a boundary central charge and a bulk central charge. The model also demonstrates that the boundary central charge can change in response to marginal deformations.

Figures (4)

• Figure 1: Crossing symmetry for two-point functions in bCFTs.
• Figure 2: For the mixed dimensional Yukawa theory: (a) scalar one loop propagator correction; (b) fermion one loop propagator correction; (c) one loop vertex correction.
• Figure 3: For the mixed dimensional QED: (a) photon one loop propagator correction; (b) fermion one loop propagator correction; (c) one loop vertex correction.
• Figure 4: For the mixed dimensional scalar theory: (a) a 4d bulk scalar one loop self energy correction; (b) a 3d boundary scalar one loop self energy correction; (c) one loop vertex correction.