Conformal anomalies of CFT's with boundaries

TL;DR

This work extends the four-dimensional conformal anomaly to manifolds with boundaries, showing the integral anomaly acquires boundary terms parameterized by two charges, $q_1$ and $q_2$, in addition to the bulk $a$ and $c$ charges. Using heat-kernel methods for a Laplace-type operator, the author derives a universal decomposition ${\\cal A} = -2a\\chi_4 - c i + q_1 j_1 + q_2 j_2$, with the boundary piece $A_4^{\\text{bd}} = \\eta (q_1 j_1 + q_2 j_2 - 2 a S_4)$, and computes the explicit boundary charges for conformal scalars, Dirac spinors, and gauge bosons under conformally invariant boundary conditions. The results show that $q_2$ depends on boundary conditions whereas $q_1$ exhibits a universal relation with the bulk $c$ via $q_1 = 8 c$ across the models studied; these findings motivate further study of boundary RG flows and extensions to higher dimensions. The work provides a framework for linking bulk and boundary conformal data and suggests a richer structure of anomalies when boundaries are present.

Abstract

The trace anomaly of conformal field theories in four dimensions is characterized by '$a$' and '$c$'-functions. The scaling properties of the effective action of a CFT in the presence of boundaries is shown to be determined by $a$, $c$ and two new functions (charges) related to boundary effects. The boundary charges are computed for different theories and different boundary conditions. One of the boundary charges depends on the bulk $c$ charge.