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arXiv:2104.14567 · Communications in Mathematical Physics

Anomalies in (2+1)D Fermionic Topological Phases and (3+1)D Path Integral State Sums for Fermionic SPTs

Sri Tata, Ryohei Kobayashi, Daniel Bulmash, Maissam Barkeshli

Published Dec 5, 2022• 33 citations
arXivHTMLPDF
DOI
10.1007/s00220-022-04484-w

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

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Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group $G = \mathrm{U}(1) \rtimes H$, 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.

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

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The authors construct here a lattice path integral for a (3+1)D topological phase with U(1) symmetry, realizing (2+1)D topological order on its boundary. The U(1) background gauge field can possibly have a nontrivial Chern number. This allows the derivation of various formulas for anomalies of topological order involving U(1) symmetry, and leads to Walker-Wang construction of a topological insulator and superconductor. Mathematically, they correspond to new combinatorial state sums for invariants of topologically nontrivial, curved U(1) bundles over 4-manifolds, including spin${}^{c}$, pin${}^{c}$ structures.

Physical review. B./Physical review. B•Oct 17, 2024•5 citations
Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group G=U(1)⋊HG = \mathrm{U}(1) \rtimes HG=U(1)⋊H, we construct a (3+1)D topologically invariant path integral in the presence of a curved background GGG gauge field, as an exact combinatorial state sum. Specifically, the U(1)\mathrm{U}(1)U(1) component of the GGG gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat GGG bundles. Our construction expresses the U(1)\mathrm{U}(1)U(1) gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form R\mathbb{R}R gauge field and 2-form Z\mathbb{Z}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 U(1)\mathrm{U}(1)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 Spinc^cc or Pinc^cc 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 U(1)\mathrm{U}(1)U(1) anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spinc^cc generalization of the Gauss-Milgram sum, valid for super-modular categories.
Abstract We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the Z8\mathbb{Z}_8Z8​ classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the Z16\mathbb{Z}_{16}Z16​ classification.
The authors construct here a lattice path integral for a (3+1)D topological phase with U(1) symmetry, realizing (2+1)D topological order on its boundary. The U(1) background gauge field can possibly have a nontrivial Chern number. This allows the derivation of various formulas for anomalies of topological order involving U(1) symmetry, and leads to Walker-Wang construction of a topological insulator and superconductor. Mathematically, they correspond to new combinatorial state sums for invariants of topologically nontrivial, curved U(1) bundles over 4-manifolds, including spinc{}^{c}c, pinc{}^{c}c structures.