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$T\overline{T}$-deformed two-dimensional quantum Maxwell theory on the torusis examined, taking into account nonperturbative effects in the deformationparameter $\mu$. We study the deformed partition function solving the relevantflow equation at the level of individual flux sectors. Summing exactly the"instanton" series, we obtain a well-defined expression for the partitionfunction at arbitrary $\mu$. For $\mu>0$, the quantum spectrum of the theoryexperiences a truncation, the partition function reducing to a sum over afinite set of positive-energy states. For $\mu<0$ instead, the appearance ofnonperturbative contributions in $\mu$ drastically modifies the structure ofthe partition function, regularizing its naive divergences throughinstanton-like subtractions. For each flux sector, we show that thesemiclassical contribution is dominated by the deformed classical action. Thetheory is observed to undergo infinite-order phase transitions for certainvalues of $\mu$, associated with the vanishing of Polyakov-loop correlators.

Exact $T\overline{T}$ deformation of two-dimensional Maxwell theory