# The anomaly formula of the analytic torsion on CR manifolds with $S^1$ action

## TL;DR

This paper derives an anomaly formula for the Quillen metric on CR manifolds with an $S^1$-action, linking geometric changes to spectral invariants in the context of Kohn-Rossi cohomology.

## Contribution

It introduces the Quillen metric on the determinant line of Fourier components of Kohn-Rossi cohomology for CR manifolds with $S^1$-action and establishes an anomaly formula for it.

## Key findings

- Derived the anomaly formula for the Quillen metric under metric variations
- Connected the Quillen metric behavior to the $S^1$-action on CR manifolds
- Extended understanding of spectral invariants in CR geometry

## Abstract

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n\ge 1$ with a transversal CR $S^1$-action on $X$. In this paper we introduce the Quillen metric on the determinant line of the Fourier components of the Kohn-Rossi cohomology on $X$ with respect to the $S^1$-action. We study the behavior of the Quillen metric under the change of the metrics on the manifold $X$ and on the vector bundle over $X$. We obtain an anomaly formula for the Quillen metric on $X$ with respect to the $S^1$-action.