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Electric-magnetic duality ($S$-duality) is a well-known property of pure$u(1)$ gauge theory in 3+1 dimensions. In this paper, we investigate thecompatibility of this duality with time-reversal symmetry. We consider twotheories obtained by coupling a Dirac fermion with an "inverted" sign of themass $m$ to a $u(1)$ gauge field. Time-reversal in the two theories isimplemented respectively via the $T$ and $CT$ symmetries of the Dirac fermion.It was recently conjectured (C. Wang and T. Senthil (arXiv:1505.03520), and M.Metlitski and A.Vishwanath (arXiv:1505.05142)) that in the $|m| \to \infty$limit these two theories are $S$-dual to each other. We provide support forthis conjecture by studying partition functions of the two theories onnon-orientable manifolds as a way to probe the realization of time-reversal.Upon integrating out the Dirac fermion, topological terms in the actions of thetwo theories are generated. While on an orientable manifold topological termsin both theories reduce to a $\theta$-term with $\theta = \pi$, on anon-orientable manifold they are distinct. We explicitly compute partitionfunctions of the two theories on the manifold $\mathbb{RP}^4$ and show thatthey are equal; this result combined with certain physical arguments issufficient to establish the duality. The two theories can be viewed as a gaugedtopological insulator in class AII and a gauged topological superconductor inclass AIII, and the bulk duality allows us to derive previously conjecturednon-trivial symmetric gapped surface states of these phases.

$S$-duality of $u(1)$ gauge theory with $\theta =\pi$ on non-orientable manifolds: Applications to topological insulators and superconductors