Weighted average over the Narain moduli space as a TT¯ deformation of the CFT target space

Abstract

We consider the weighted average of a two dimensional conformal field theory (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 $T\overline{T}$ 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 $T\overline{T}$ 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 ${T}^{2}$.