The free energy of the large-$N$ fermionic Chern$\text{unicode}x2013$Simons theory in the 'temporal' gauge

Most of the computational evidence for the Bose$\text{unicode}x2013$Fermi duality of fundamental fields coupled to $U(N)$ Chern$\text{unicode}x2013$Simons theories originates in large-$N$ calculations performed in the light-cone gauge. This gauge is ill-suited to computations in curved spacetimes, like the evaluation of the partition function on ${\mathrm{\Sigma }}_g\times S^1$ for arbitrary genus $g$. In this paper, we use another gauge, the ‘temporal’ gauge, to set up the computation of this partition function. In the large-$N$ limit, and the special case ${\mathrm{\Sigma }}_g={\mathbb{R}}^2$, we take the computation through to the end by setting up and solving the gap equations, generalizing tricks explored in arXiv:1410.0558 to finite temperature. Our final results are in perfect agreement with earlier light-cone gauge results, providing a consistency check of both the formalism developed in this paper as well as previously performed light-cone gauge computations. In a follow-up paper, we will report on using our formalism to explicitly compute the partition function on $S^2\times S^1$ for a finite-sized sphere.