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arXiv:2111.14827

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

Ryohei Kobayashi, Maissam Barkeshli

Published Nov 29, 2021• 16 citations
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Abstract

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.

DOI
10.48550/arxiv.2111.14827

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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.
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.
We study anomalies in time-reversal-(Z2T{\mathbb{Z}}_{2}^{T}Z2T​) and U(1)U(1)U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)(2+1)(2+1)-dimensional system but can be realized on the surface of a (3+1)(3+1)(3+1)-dimensional [(3+13+13+1)D] symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators, functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e., $U(1)\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{T}$ and $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of (3+1)D(3+1)\mathrm{D}(3+1)D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk (3+1)D(3+1)\mathrm{D}(3+1)D SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1)U(1)U(1) symmetry.
Abstract We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is Z/4T×Z/2f\mathbb {Z}/4^T\times \mathbb {Z}/2^fZ/4T×Z/2f Z / 4 T × Z / 2 f time-reversal symmetry with symmetry algebra T2=(−1)FC{\mathcal {T}}^2 = (-1)^FCT2=(−1)FC T 2 = ( - 1 ) F C , which many fermionic topological orders enjoy, including the U(1)5\textrm{U}(1)_5U(1)5​ U ( 1 ) 5 spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including Z/4Tf\mathbb {Z}/4^{Tf}Z/4Tf Z / 4 Tf time-reversal symmetry with symmetry algebra T2=(−1)F\mathcal T^2 = (-1)^FT2=(−1)F T 2 = ( - 1 ) F , and other symmetry groups in the 10-fold way involving Lie group symmetries.
We develop a theory of anomalies of fermionic topological phases of matter in (2+12+12+1)D with a general fermionic symmetry group Gf{G}_{f}Gf​. In general, Gf{G}_{f}Gf​ can be a nontrivial central extension of the bosonic symmetry group Gb{G}_{b}Gb​ by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the Gf{G}_{f}Gf​ symmetry, which we dub the anomaly cascade: (i) An H1(Gb,ZT){\mathcal{H}}^{1}({G}_{b},{\mathbb{Z}}_{\mathbf{T}})H1(Gb​,ZT​) obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An H2(Gb,kerr){\mathcal{H}}^{2}({G}_{b},kerr)H2(Gb​,kerr) obstruction to extending the Gb{G}_{b}Gb​ group structure of the symmetry permutations to the fermion parity gauged theory, where rrr is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An H3(Gb,Z2){\mathcal{H}}^{3}({G}_{b},{\mathbb{Z}}_{2})H3(Gb​,Z2​) obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known H4(Gb,U(1[))]{\mathcal{H}}^{4}({G}_{b},U(1[))]H4(Gb​,U(1[))] obstruction to developing a consistent theory of Gb{G}_{b}Gb​ symmetry defects for the fermion parity gauged theory. We describe how the H2{\mathcal{H}}^{2}H2 obstruction can be canceled by anomaly inflow from a bulk (3+13+13+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is nontrivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.
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A bstract We use the intrinsic one-form and two-form global symmetries of (3+1) d bosonic field theories to classify quantum phases enriched by ordinary (0-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the 0-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in (2 + 1) d . We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative ’t Hooft anomaly for different couplings. We discuss several examples including the gapless pure U(1) gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in (3 + 1) d for SU(2) gauge theory with two adjoint Weyl fermions.