Holographic Ward identities for symmetry breaking in two dimensions

TL;DR

This paper explores how symmetry breaking and Ward identities manifest in two-dimensional holographic field theories, revealing conditions under which Goldstone bosons appear and clarifying the holographic renormalization process.

Contribution

It establishes the correct holographic renormalization procedure for symmetry breaking in 2D theories, aligning Ward identities with higher-dimensional cases.

Findings

Goldstone bosons can exist in 2D holographic theories at large N.

Proper boundary conditions are crucial for accurate Ward identities.

The renormalization process matches higher-dimensional results.

Abstract

We investigate symmetry breaking in two-dimensional field theories which have a holographic gravity dual. Being at large N, the Coleman theorem does not hold and Goldstone bosons are expected. We consider the minimal setup to describe a conserved current and a charged operator, and we perform holographic renormalization in order to find the correct Ward identities describing symmetry breaking. This involves some subtleties related to the different boundary conditions that a vector can have in the three-dimensional bulk. We establish which is the correct prescription that yields, after renormalization, the same Ward identities as in higher dimensions.