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

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