# From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects

## TL;DR

This paper explores the relationships between Weyl anomalies, entanglement entropy, and thermal entropy in 2D boundaries and defects of higher-dimensional CFTs, revealing new constraints and computations for central charges.

## Contribution

It extends the understanding of Weyl anomalies and entanglement entropy to 2D boundaries and defects in higher-dimensional CFTs, introducing new bounds and holographic calculations.

## Key findings

- The defect's conformal dimension $d_2$ is non-negative by the ANEC.
- The entanglement entropy has a defect-specific logarithmic term determined by $b$ and $d_2$.
- No universal Cardy formula relates central charges to thermal entropy.

## Abstract

We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of $D \geq 3$ CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, $b$, obeys a c-theorem, as in 2D CFT. For a 2D defect, we show that another, $d_2$, interpreted as the defect's `conformal dimension,' must be non-negative by the Averaged Null Energy Condition (ANEC). We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by $b$ and $d_2$. Using this and known holographic results, we compute $b$ and $d_2$ for 1/2-BPS surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with $b$'s c-theorem. Via free field and holographic examples we show that no universal `Cardy formula' relates the central charges to thermal entropy.