(3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases

Abstract

Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group $G=\mathrm{U}(1)⋊H$, we construct a (3+1)D topologically invariant path integral in the presence of a curved background $G$ gauge field, as an exact combinatorial state sum. Specifically, the $\mathrm{U}(1)$ component of the $G$ gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat $G$ bundles. Our construction expresses the $\mathrm{U}(1)$ gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form $\mathbb{R}$ gauge field and 2-form $\mathbb{Z}$ gauge field. We develop a new graphical calculus for anyons interacting with "Villain symmetry defects", associated with the 1-form and 2-form background gauge fields. This graphical calculus is used to define the (3+1)D path integral, which can describe either a bosonic or fermionic symmetry-protected topological (SPT) phase. For example, we can construct the topological path integral on curved $\mathrm{U}(1)$ bundles for the (3+1)D fermionic topological insulator in class AII and topological superconductor in class AIII given appropriate (2+1)D fermionic symmetry fractionalization data; these then give invariants of 4-manifolds with Spin${}^c$ or Pin${}^c$ structures and their generalizations. The (3+1)D path integrals define anomaly indicators for the (2+1)D topological orders; in the case of Abelian (2+1)D topological orders, we derive by explicit computation all of the mixed $\mathrm{U}(1)$ anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spin${}^c$ generalization of the Gauss-Milgram sum, valid for super-modular categories.

34+21 pages, 11+7 figures

References in corpus (8)

• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Abelian gauge theories on the lattice: $\theta$-terms and compact gauge theory with(out) monopoles
• Global Anomalies on the Hilbert Space
• Fermionic symmetry fractionalization in (2+1)D
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Exactly solvable models for U(1) symmetry-enriched topological phases
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Cited by in corpus (5)

• Anomalies and gauging of U(1) symmetries
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Ordering the topological order in the fractional quantum Hall effect
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order