Warped Schwarzian theory

TL;DR

This paper develops the warped Schwarzian theory based on the warped Virasoro group, computes its symplectic form, constructs its Euclidean action, evaluates the path integral, and compares its thermodynamics with the complex SYK model.

Contribution

It introduces the warped Schwarzian theory, computes its symplectic structure, and analyzes its quantum and thermodynamic properties, extending the Schwarzian framework to warped conformal symmetries.

Findings

Computed the Kirillov-Kostant-Souriau symplectic form on coadjoint orbits.

Constructed the Euclidean action for the warped Schwarzian theory.

Evaluated the one-loop exact path integral and discussed thermodynamics.

Abstract

We consider the (twisted) warped Virasoro group Diff($S^1$)$\ltimes$ C$^\infty$($S^1$) in the presence of its three cocycles. We compute the Kirillov-Kostant-Souriau symplectic 2-form on coadjoint orbits. We then construct the Euclidean action of the `warped Schwarzian theory' associated to the orbit with SL(2,$\mathbb{R}$)$\times$U(1) stabilizer as the effective theory of the reparametrization over the base circle and evaluate the corresponding one-loop-exact path integral. We further discuss thermodynamics of the wSch theory in comparison with the complex SYK model.