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

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.