Determinant and Weyl anomaly of Dirac operator: a holographic derivation

TL;DR

This paper derives a holographic formula connecting bulk fermion determinants in AdS space to boundary two-point functions, enabling new insights into spectral theory, conformal geometry, and anomalies.

Contribution

It introduces a holographic relation between bulk fermion determinants and boundary two-point functions, advancing the understanding of spectral and conformal geometric properties.

Findings

Computed the type-A Weyl anomaly holographically.

Expressed the Dirac operator determinant on spheres via Barnes' multiple gamma function.

Provided insights into a conjecture by Bär and Schopka.

Abstract

We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background, and the determinant of the two-point function of the dual operator at the conformal boundary. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk partition function to a subleading large-N contribution in the boundary partition function. We use this holographic picture to address questions in spectral theory and conformal geometry. As an instance, we compute the type-A Weyl anomaly and the determinant of the iterated Dirac operator on round spheres, express the latter in terms of Barnes' multiple gamma function and gain insight into a conjecture by B\"ar and Schopka.