Burgers Equation vs. Large $N$ Limit in $T\bar{T}$-deformed $O(N)$ Vector Model

TL;DR

This paper investigates the thermal free energy of a $Tar{T}$-deformed $O(N)$ vector model, demonstrating that the large $N$ limit reproduces the exact result and revealing a cancellation of $1/N$ corrections.

Contribution

It provides a field-theoretical derivation of the large $N$ limit result, showing the exact free energy expression is recovered with leading order contributions.

Findings

The exact thermal free energy density is obtained for any N.

Large N limit reproduces the exact free energy, with $1/N$ corrections canceled.

The cancellation mechanism of corrections is non-trivial.

Abstract

We study a $T\bar{T}$-deformed $O(N)$ vector model, which is classically equivalent to the Nambu-Goto action with static gauge. The thermal free energy density can be computed exactly by using the Burgers equation as a special property of $T\bar{T}$-deformation. The resulting expression is valid for an arbitrary value of $N$. One may consider a large $N$ limit while preserving this expression. We try to derive this result in the field-theoretical approach directly by employing the large $N$ limit. As a result, the leading contribution coincides with the exact one. That is, the $1/N$ corrections are cancelled out through a non-trivial mechanism.