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

Max A. Metlitski|arXiv (Cornell University)|Oct 19, 2015

Black Holes and Theoretical Physics37 citations

TL;DR

This paper establishes S-duality between a U(1) gauge theory with θ=π and time-reversal symmetry realized via T or CT symmetry in Dirac fermions on non-orientable manifolds. By computing partition functions on RP⁴, it shows equivalence between the two theories, confirming a conjecture that a gauged topological insulator (class AII) and a gauged topological superconductor (class AIII) are S-dual, enabling derivation of non-trivial gapped surface states.

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 $|m| o \infty$ 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.

Motivation & Objective

• To investigate the compatibility of S-duality with time-reversal symmetry in U(1) gauge theories with θ=π.
• To test the conjecture that two U(1) gauge theories—differing in how time-reversal is implemented (T vs. CT)—are S-dual in the |m|→∞ limit.
• To derive non-trivial symmetric gapped surface states of topological insulators and superconductors using bulk duality.
• To compute partition functions on non-orientable manifolds (specifically RP⁴) to probe time-reversal realization and confirm duality.

Proposed method

• Computes the partition function of a Dirac fermion with mass m on non-orientable manifolds, distinguishing between T and CT time-reversal symmetries.
• Integrates out the Dirac fermion to generate topological terms in the effective action, which differ on non-orientable manifolds despite both reducing to θ=π on orientable ones.
• Uses the manifold RP⁴ to compute and compare partition functions of the two theories, showing equality under S-duality.
• Applies Reidemeister torsion and analytic torsion techniques to relate partition functions to topological invariants.
• Relies on the Cheeger-Müller theorem to equate Reidemeister torsion with analytic torsion via orthonormal and integer-coefficient cohomology bases.
• Derives the duality by showing that the partition functions match under S-duality, supported by physical consistency and topological invariance.

Experimental results

Research questions

• RQ1 Are the U(1) gauge theories with θ=π and time-reversal realized via T or CT symmetry S-dual to each other in the |m|→∞ limit?
• RQ2 How do topological terms in U(1) gauge theories differ on non-orientable manifolds when time-reversal is implemented via T or CT symmetry?
• RQ3 Can the partition function on RP⁴ be used to test S-duality in fermionic U(1) gauge theories with θ=π?
• RQ4 What is the role of Reidemeister torsion in computing partition functions for non-orientable manifolds in this context?
• RQ5 How does the duality between topological insulators and superconductors emerge from bulk S-duality on non-orientable spacetimes?

Key findings

• The partition functions of the two U(1) gauge theories with θ=π, differing in time-reversal realization (T vs. CT), are equal when computed on RP⁴.
• This equality provides strong evidence for the conjecture that the two theories are S-dual, despite their distinct topological terms on non-orientable manifolds.
• The duality confirms that a gauged topological insulator in class AII is S-dual to a gauged topological superconductor in class AIII.
• The computation shows that the effective action on RP⁴ captures the correct topological invariants, including torsion in cohomology, via Reidemeister torsion.
• The result implies the existence of non-trivial symmetric gapped surface states for these topological phases, consistent with earlier conjectures.
• The equivalence of partition functions under S-duality is established through a precise match in analytic and Reidemeister torsion, validated by the Cheeger-Müller theorem.