Weighed average over Narain moduli space as $T \bar T$ deformation of CFT target space

TL;DR

This paper explores how a weighted average over Narain moduli space, inspired by $T ar T$ deformation, affects the partition function of a 2D CFT with target space $T^2$, revealing a marginal deformation related to target space rescaling.

Contribution

It introduces a novel weighted averaging method over Narain moduli space using a kernel linked to $T ar T$ deformation, showing it results in a marginal rescaling of the target space.

Findings

The weighted average preserves the form of the partition function as a sum over Boltzmann factors.

The $T ar T$ inspired kernel leads to a marginal deformation of the target space.

Smearing over moduli space can generally break the sum-over-states structure, but not with this kernel.

Abstract

We consider the weighted average of a two dimensional CFT, whose target space is $T^2$, over its Narain moduli space. We take as the weighing function the integral kernel which gives rise to $T \bar T$ deformation when applied to the world sheet moduli data of the partition function viewed as vacuum amplitude when the world sheet is a torus. We compute the smeared partition function where this kernel is applied to the target space moduli. Smearing the partition function over the parameter space of a field theory generally leads to the breakdown in the ability to write the partition function as a sum over Boltzmann factor with unit coefficients. The weight function inspired by the $T \bar T$ deformation appears to be an exception to this general expectation. We show that this smearing leads to a marginal deformation corresponding to the overall rescaling of the target space $T^2$.