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

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

Black Holes and Theoretical Physics19 citations

TL;DR

This paper establishes S-duality between a u(1) gauge theory with θ = π and time-reversal symmetry implemented via T or CT symmetry on non-orientable manifolds, using partition function computations on RP⁴. It confirms the duality by showing identical partition functions for the two theories, supporting the conjecture that a gauged topological insulator (class AII) and a gauged topological superconductor (class AIII) are S-dual at θ = π.

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 $ heta$-term with $ heta = \pi$, 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 theory at θ = π.
• To test the conjecture that two distinct fermionic theories—realized via T and CT time-reversal symmetries—are S-dual to each other.
• To derive non-trivial symmetric gapped surface states of topological insulators and superconductors using bulk duality.
• To clarify the role of topological terms and spin structures in fermionic S-duality on non-orientable manifolds.

Proposed method

• Computes partition functions of two u(1) gauge theories with Dirac fermions and θ = π on the non-orientable manifold RP⁴.
• Uses the fermion determinant and Reidemeister torsion to compute the partition function, distinguishing between T and CT time-reversal realizations.
• Compares partition functions under S-duality, showing equality for both theories on RP⁴.
• Relies on the fact that on non-orientable manifolds, the topological terms from integrating out fermions differ between T and CT cases, yet the full partition functions match.
• Applies the duality to classify surface states of topological insulators (class AII) and superconductors (class AIII) via bulk duality.
• Uses bordism classification and analytic torsion to relate partition function invariance to topological invariants.

Experimental results

Research questions

• RQ1 Are the two u(1) gauge theories with θ = π—realized via T and CT time-reversal symmetry—S-dual to each other?
• RQ2 How does S-duality manifest on non-orientable manifolds, where topological terms differ between T and CT cases?
• RQ3 Can the duality be established via partition function equality on RP⁴ despite distinct topological terms?
• RQ4 What is the role of spin structures and line bundles in fermionic S-duality on non-orientable manifolds?
• RQ5 What are the implications of this duality for the existence of symmetric gapped surface states in topological insulators and superconductors?

Key findings

• The partition functions of the two theories with T and CT time-reversal symmetry are equal on RP⁴, providing strong evidence for S-duality.
• On non-orientable manifolds, the topological terms from integrating out the Dirac fermion differ between T and CT cases, but the full partition functions remain invariant under S-duality.
• The duality confirms that a gauged topological insulator in class AII and a gauged topological superconductor in class AIII are S-dual at θ = π.
• The equality of partition functions is tied to the Reidemeister torsion and analytic torsion, with the latter matching the partition function on RP⁴.
• The duality implies the existence of non-trivial symmetric gapped surface states for these topological phases, consistent with earlier conjectures.
• The result holds despite the absence of a global time-reversal symmetry on non-orientable manifolds, as the duality is preserved through the partition function structure.