On Modular Invariance and 3D Gravitational Instantons

TL;DR

This paper investigates the modular transformation properties of Euclidean 3D gravity solutions with torus topology, focusing on their boundary conformal field theory correspondence, holonomies, and implications for the grand canonical partition function.

Contribution

It demonstrates the conditions under which a modular invariant grand canonical partition function exists for 3D gravity solutions, linking bulk dynamics with boundary CFT transformations.

Findings

Existence of modular invariant partition function under specific holonomy transformations

Inclusion of BTZ black hole solutions as saddle points in the spectrum

Classical bulk dynamics can describe modular transformations as foliation changes

Abstract

We study the modular transformation properties of Euclidean solutions of 3D gravity whose asymptotic geometry has the topology of a torus. These solutions represent saddle points of the grand canonical partition function with an important example being the BTZ black hole, and their properties under modular transformations are inherited from the boundary conformal field theory encoding the asymptotic dynamics. Within the Chern Simons formulation, classical solutions are characterised by specific holonomies describing the wrapping of the gauge field around cycles of the torus. We find that provided these holonomies transform in an appropriate manner, there exists an associated modular invariant grand canonical partition function and that the spectrum of saddle points naturally includes a thermal bath in $AdS_3$ as discussed by Maldacena and Strominger. Indeed, certain modular transformations can naturally be described within classical bulk dynamics as mapping between different foliations with a "time" coordinate along different cycles of the asymptotic torus.