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 $$\mathbb {Z}/4^T\times \mathbb {Z}/2^f$$ Z / 4 T × Z / 2 f time-reversal symmetry with symmetry algebra $${\mathcal {T}}^2 = (-1)^FC$$ T 2 = ( - 1 ) F C , which many fermionic topological orders enjoy, including the $$\textrm{U}(1)_5$$ 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 $$\mathbb {Z}/4^{Tf}$$ Z / 4 Tf time-reversal symmetry with symmetry algebra $$\mathcal T^2 = (-1)^F$$ T 2 = ( - 1 ) F , and other symmetry groups in the 10-fold way involving Lie group symmetries.
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 Z / 4 T × Z / 2 f time-reversal symmetry with symmetry algebra T2=(−1)FC T 2 = ( - 1 ) F C , which many fermionic topological orders enjoy, including the U(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 Z / 4 Tf time-reversal symmetry with symmetry algebra T2=(−1)F T 2 = ( - 1 ) F , and other symmetry groups in the 10-fold way involving Lie group symmetries.
We study anomalies in time-reversal-(Z2T) and U(1)-symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly (2+1)-dimensional system but can be realized on the surface of a (3+1)-dimensional [(3+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 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 SPT phase on a particular manifold and in the presence of a particular background gauge field for the U(1) symmetry.
Abstract We study time-reversal symmetry in (2+1)D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by Z2×Z2 symmetry-protected topological (SPT) phases in (3+1)D, and can be diagnosed via partition functions on manifolds such as RP4 and CP2. These partition functions are related by the anomaly formula Z(RP4)Z(CP2)=θM, where θM is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges ξn, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP2)=ξ1, we propose a generalization of the anomaly formula that involves both the higher central charges ξn and a new time-reversal invariant ηn. Introducing a distinguished subset Mn⊂A of anyons, we establish the relation ηn⋅ξn=∣∑a∈Mnθ(a)n∣∑a∈Mnθ(a)n,which generalizes the known anomaly formula. We analyze the algebraic structure of Mn, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.
We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group Gf. In general, Gf can be a nontrivial central extension of the bosonic symmetry group Gb by fermion parity ${(\ensuremath{-}1)}^{F}$. We encounter four layers of obstructions to gauging the Gf symmetry, which we dub the anomaly cascade: (i) An H1(Gb,ZT) obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An H2(Gb,kerr) obstruction to extending the Gb group structure of the symmetry permutations to the fermion parity gauged theory, where r is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An 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[))] obstruction to developing a consistent theory of Gb symmetry defects for the fermion parity gauged theory. We describe how the H2 obstruction can be canceled by anomaly inflow from a bulk (3+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.
Anomaly of (2+1)-dimensional symmetry-enriched topological order from (3+1)-dimensional topological quantum field theory
Symmetry acting on a (2+1) D D topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1) D D on-site symmetry. In this paper, we develop a (3+1) D D topological quantum field theory to calculate the anomaly indicators of a (2+1) D D topological order with a general symmetry group G 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 D topological quantum field theory on a specific manifold equipped with some G 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 \mathbb{Z}_2\times\mathbb{Z}_2 ℤ 2 × ℤ 2 , \mathbb{Z}_2^T\times\mathbb{Z}_2^T ℤ 2 T × ℤ 2 T , SO(N) S O ( N ) , O(N)^T O ( N ) T , SO(N)\times \mathbb{Z}_2^T S O ( N ) × ℤ 2 T , etc, where \mathbb{Z}_2 ℤ 2 and \mathbb{Z}_2^T ℤ 2 T denote a unitary and anti-unitary order-2 group, respectively, and <
Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group G=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 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 U(1) gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form R gauge field and 2-form 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) 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 or Pinc 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) anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spinc generalization of the Gauss-Milgram sum, valid for super-modular categories.
Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensional systems; instead, they can only be realized on the surface of three-dimensional symmetry-protected topological phases. We propose two quantities, which we call anomaly indicators, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, η_{2}, applies to bosonic systems while the second indicator, η_{f}, applies to fermionic systems in the DIII class. We conjecture that η_{2}, together with a previously known indicator η_{1}, can detect the two known Z_{2} anomalies in the bosonic case, while η_{f} can detect the Z_{16} anomaly in the fermionic case.