Finite Temperature at Finite Places

TL;DR

This paper explores p-adic AdS/CFT on the Tate curve, deriving boundary correlators and demonstrating its usefulness as a toy model for finite-temperature conformal field theories.

Contribution

It generalizes Zabrodin's result to the p-adic setting, explicitly derives boundary duals, and computes perturbative correlators on the Tate curve.

Findings

Boundary duals of free massive theories are explicitly derived.

Two-point and three-point correlators are computed at one-loop and tree level.

p-adic AdS/CFT on the Tate curve models finite-temperature CFTs effectively.

Abstract

This paper studies AdS/CFT in its $p$-adic version (at the ``finite place") in the setting where the bulk geometry is made up of the Tate curve, a discrete quotient of the Bruhat-Tits tree. Generalizing a classic result due to Zabrodin, the boundary dual of the free massive bulk theory is explicitly derived. Introducing perturbative interactions, the Wittens diagrams for the two-point and three-point correlators are computed for generic scaling dimensions at one-loop and tree level respectively. The answers obtained demonstrate how $p$-adic AdS/CFT on the Tate curve provides a useful toy model for real CFTs at finite temperature.