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

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 $T^2$.

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