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

TL;DR

This work tests the compatibility of S-duality with time-reversal symmetry for a U(1) gauge theory at θ = π by comparing two time-reversal implementations (T and CT) across orientable and non-orientable four-manifolds. Central to the analysis is the computation of bulk partition functions and their topological contributions, including η-invariants on Pin_c structures and the role of analytic and Reidemeister torsions on non-orientable spaces like RP^4. The results establish a robust S-duality between the gauged TI (class AII) and gauged TSc (class AIII) pictures on orientable manifolds and on RP^4, with implications for symmetric gapped surface states and a bordism-based classification of interacting TI/ TSc phases. The findings connect dual bulk descriptions to surface phenomena (including T-Pfaffian states) and to a mathematical framework based on Pin_c bordism, strengthening the interplay between high-energy dualities and condensed-mmatter SPT classifications.

Abstract

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 the compatibility of this duality with time-reversal symmetry. We consider two theories obtained by coupling a Dirac fermion with an "inverted" sign of the mass $m$ to a $u(1)$ gauge field. Time-reversal in the two theories is implemented 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 for this conjecture by studying partition functions of the two theories on non-orientable manifolds as a way to probe the realization of time-reversal. Upon integrating out the Dirac fermion, topological terms in the actions of the two theories are generated. While on an orientable manifold topological terms in both theories reduce to a $θ$-term with $θ= π$, on a non-orientable manifold they are distinct. We explicitly compute partition functions of the two theories on the manifold $\mathbb{RP}^4$ and show that they are equal; this result combined with certain physical arguments is sufficient to establish the duality. The two theories can be viewed as a gauged topological insulator in class AII and a gauged topological superconductor in class AIII, and the bulk duality allows us to derive previously conjectured non-trivial symmetric gapped surface states of these phases.

Figures (1)

• Figure 1: Lattice of dyon excitations in a $u(1)$ gauge-theory with fermionic matter and $\theta = \pi$. $q$ and $m$ denote the electric and magnetic charges. Left and right sides correspond to different implementations of time-reversal symmetry. Left side corresponds to the theory ${\cal L}_{CT}$, where time-reversal acts on the Dirac fermion via Eq. (\ref{['eq:CTintro']}), so $CT: q \to - q, m \to m$. Blue arrows mark the dyons $d_{\pm}: (q = \pm 1/2, m = 1)$, which are partners under $CT$. Right side corresponds to the theory ${\cal L}_T$, where time-reversal acts on the Dirac fermion via Eq. (\ref{['eq:Tintro']}), so $T: q \to q, m \to -m$. Red arrows mark the dyons $\tilde{d}_{\pm}: (q = 1/2, m = \mp 1)$, which are partners under $T$. The duality (\ref{['eq:Sfdyonsintro']}) maps ${\cal L}_{CT}$ to ${\cal L}_T$, sending $d_{\pm} \to \tilde{d}_{\pm}$. In particular, the Kramers doublet fermion $d_+ d_-: (q = 0, m = 2)$ is mapped to the Kramers doublet fermion $\tilde{d}_+ \tilde{d}_-: (q = 1, m = 0)$.