Report on scipost_202212_00004v1

Symmetry acting on a (2+1)D topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)D on-site symmetry.In this paper, we develop a (3+1)D topological quantum field theory to calculate the anomaly indicators of a (2+1)D topological order with a general symmetry group G, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not.These anomaly indicators are partition functions of the (3+1)D topological quantum field theory on a specific manifold equipped with some G-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions.Our framework is applied to derive the anomaly indicators for various symmetry groups, including, where Z2 and Z T 2 denote a unitary and anti-unitary order-2 group, respectively, and O(N ) T denotes a symmetry group O(N ) such that elements in O(N ) with determinant -1 are anti-unitary.In particular, we demonstrate that some anomaly of O(N ) T and SO(N ) × Z T 2 exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order.As a byproduct, for SO(N ) symmetric topological orders, we derive their SO(N ) Hall conductance. Contents I. Introduction1 A. Relation to prior work 3 B. Outline and summary 4 II.Review of topological order with symmetry G 4 A. Review of UMTC notation 4 B. Global symmetry 5 III.(3+1)D TQFT with finite group symmetry G 7 A. Characterizing the anomaly by bulk-boundary correspondence 8 B. General construction of TQFT 8 C. Handle decomposition 11 D. Recipe for calculating the partition function 12 IV.Examples: finite group symmetry 15 A. No symmetry 15 B. Z T 2 16 C.Z 2 × Z 2 17 D. Z T 2 × Z T 2 19 1. All-fermion Z 2 topological order 20 V. Generalization to connected Lie group symmetry 23 A. Example: SO(N ) 24 1.Anomaly indicator for N 5 26 2. SO(N ) Hall conductance 26 VI.Other symmetry groups 27 A. O(N ) T 28 B. SO(N ) × Z T 2 30 VII. Discussion 31 A. Derivation of Eq. (44) 33 1.Vector Spaces 33 2. Partition functions 34 3. Inner Products 35 4. Requirement from Invertibility 36 B. An explicit expression of the η-factor 36 C. Consistency check of TQFT 37 1.Independence on the handle decomposition 37 2. Invariance under change of defects 41 3. Gauge invariance 43 4. Cobordism invariance 44 5. Invertibility 45 6. Generalization to connected Lie groups 45 D. Identifying the manifold M from bordism 46 E.More information about handle decomposition of manifolds 47 1.CP 2 48 2. RP 4 48 3. RP 3 × S 1 48 4. RP 2 × RP 2 49

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