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Anomaly constraint on chiral central charge of (2+1)d TQFT

Ryohei Kobayashi

Published Jan 4, 2021• 0 citations
OpenAlex

Abstract

In this short paper, we argue that the chiral central charge $c_-$ of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with $T^2=(-1)^F$ labeled as $\nu\in\mathbb{Z}_{16}$, the TQFT must have $c_-=1/4$ mod $1/2$ for odd $\nu$, while $c_-=0$ mod $1/2$ for even $\nu$. This generalizes the fact that the bosonic TQFT with $T$ anomaly in a particular class must carry $c_-=4$ mod $8$ to fermionic cases. We also study such a constraint for fermionic TQFT with $U(1)\times CT$ symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.

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Anomaly constraint on chiral central charge of (2+1)d topological order

Ryohei Kobayashi

In this short paper, we argue that the chiral central charge c -of a (2 + 1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry.For example, if a (2 + 1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T 2 = (-1) F labeled as ν ∈ Z 16 , the TQFT must have c -= 1/4 mod 1/2 for odd ν, while c -= 0 mod 1/2 for even ν.This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c -= 4 mod 8 to fermionic cases.We also study such a constraint for fermionic TQFT with U (1) × CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.

arXiv:2101.01018•Physical Review Research•May 10, 2021•5 citations

Time-reversal and spatial-reflection symmetry localization anomalies in (2+1)-dimensional topological phases of matter

Maissam Barkeshli, Meng Cheng

We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a ${\mathbb{Z}}_{2}$ symmetry which can be associated with either time reversal (denoted ${\mathbb{Z}}_{2}^{\mathbf{T}})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when ${\mathbb{Z}}_{2}^{\mathbf{T}}$ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include $\mathrm{USp}{(4)}_{2}$ Chern-Simons (CS) theory and $\mathrm{SO}{(4)}_{4}$ CS theory. The theories that we find with these ${\mathbb{Z}}_{2}^{\mathbf{T}}$ anomalies can all be obtained by gauging the unitary ${\mathbb{Z}}_{2}$ subgroup of a different TQFT with a ${\mathbb{Z}}_{4}^{\mathbf{T}}$ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_{4}^{\mathbf{T}}$, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the ${\mathbb{Z}}_{2}^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_{2}^{\mathbf{T}}$ if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical ${\mathbb{Z}}_{2}$ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.

arXiv:1706.09464•Physical review. B./Physical review. B•Sep 17, 2018•48 citations

Generalization of Anomaly Formula for Time-Reversal Symmetry in (2+1)D Abelian Bosonic Topological Quantum Field Theories

Ippo Orii

Abstract We study time-reversal symmetry in $(2+1)$D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by $\mathbb {Z}_2 \times \mathbb {Z}_2$ symmetry-protected topological (SPT) phases in $(3+1)$D, and can be diagnosed via partition functions on manifolds such as $\mathbb {RP}^4$ and $\mathbb {CP}^2$. These partition functions are related by the anomaly formula $Z(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}$, where $\theta _\mathcal {M}$ is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $\xi _n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb {CP}^2) = \xi _1$, we propose a generalization of the anomaly formula that involves both the higher central charges $\xi _n$ and a new time-reversal invariant $\eta _n$. Introducing a distinguished subset $\mathcal {M}^n \subset \mathcal {A}$ of anyons, we establish the relation $\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},$which generalizes the known anomaly formula. We analyze the algebraic structure of $\mathcal {M}^n$, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.

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Sixteen-Fold Way for Fermionic Topological Orders

Ryohei Kobayashi, Abhinav Prem, Matthew Yu

Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.

cond-mat.str-elCross-listedarXiv:2606.28682•Jun 30, 2026•0 citations

Topological orders with global gauge anomalies

Yi‐Zhuang You, Cenke Xu

By definition, the physics of the $d$-dimensional (dim) boundary of a $(d+1)$-dim symmetry protected topological (SPT) state cannot be realized as itself on a $d$-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a $d$-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the $3d$ spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a $4d$ topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the $4d$ boundary of a $5d$ topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped ${\mathbb{Z}}_{2N}$ topological order with topological degeneracy, but this ${\mathbb{Z}}_{2N}$ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure $3d$ space.

arXiv:1502.07752•Physical Review B•Aug 6, 2015•6 citations

On time-reversal anomaly of 2+1d topological phases

Yuji Tachikawa, Kazuya Yonekura

We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter $ν$ mod 16 of the bulk 3+1d topological superconductor with $\mathsf{T}^2=(-1)^F$ on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.

arXiv:1610.07010•Progress of Theoretical and Experimental Physics•Jan 30, 2017•64 citations
In this short paper, we argue that the chiral central charge c−c_-c−​ of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with T2=(−1)FT^2=(-1)^FT2=(−1)F labeled as ν∈Z16\nu\in\mathbb{Z}_{16}ν∈Z16​, the TQFT must have c−=1/4c_-=1/4c−​=1/4 mod 1/21/21/2 for odd ν\nuν, while c−=0c_-=0c−​=0 mod 1/21/21/2 for even ν\nuν. This generalizes the fact that the bosonic TQFT with TTT anomaly in a particular class must carry c−=4c_-=4c−​=4 mod 888 to fermionic cases. We also study such a constraint for fermionic TQFT with U(1)×CTU(1)\times CTU(1)×CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
In this short paper, we argue that the chiral central charge c -of a (2 + 1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry.For example, if a (2 + 1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T 2 = (-1) F labeled as ν ∈ Z 16 , the TQFT must have c -= 1/4 mod 1/2 for odd ν, while c -= 0 mod 1/2 for even ν.This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c -= 4 mod 8 to fermionic cases.We also study such a constraint for fermionic TQFT with U (1) × CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.
We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a Z2{\mathbb{Z}}_{2}Z2​ symmetry which can be associated with either time reversal (denoted Z2T){\mathbb{Z}}_{2}^{\mathbf{T}})Z2T​) or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)2\mathrm{USp}{(4)}_{2}USp(4)2​ Chern-Simons (CS) theory and SO(4)4\mathrm{SO}{(4)}_{4}SO(4)4​ CS theory. The theories that we find with these Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ anomalies can all be obtained by gauging the unitary Z2{\mathbb{Z}}_{2}Z2​ subgroup of a different TQFT with a Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z4T{\mathbb{Z}}_{4}^{\mathbf{T}}Z4T​, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^{2}={(\ensuremath{-}1)}^{{N}_{f}}$ corresponding to fermion parity. Finally, we demonstrate that theories with the Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ localization anomaly can be compatible with Z2T{\mathbb{Z}}_{2}^{\mathbf{T}}Z2T​ if they are ``pseudorealized'' at the surface of a (3+1)D symmetry-enriched topological phase. The ``pseudorealization'' refers to the fact that the bulk (3+1)D system is described by a dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.
Abstract We study time-reversal symmetry in (2+1)(2+1)(2+1)D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by Z2×Z2\mathbb {Z}_2 \times \mathbb {Z}_2Z2​×Z2​ symmetry-protected topological (SPT) phases in (3+1)(3+1)(3+1)D, and can be diagnosed via partition functions on manifolds such as RP4\mathbb {RP}^4RP4 and CP2\mathbb {CP}^2CP2. These partition functions are related by the anomaly formula Z(RP4) Z(CP2)=θMZ(\mathbb {RP}^4)\, Z(\mathbb {CP}^2) = \theta _{\mathcal {M}}Z(RP4)Z(CP2)=θM​, where θM\theta _\mathcal {M}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\xi _nξn​, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP2)=ξ1Z(\mathbb {CP}^2) = \xi _1Z(CP2)=ξ1​, we propose a generalization of the anomaly formula that involves both the higher central charges ξn\xi _nξn​ and a new time-reversal invariant ηn\eta _nηn​. Introducing a distinguished subset Mn⊂A\mathcal {M}^n \subset \mathcal {A}Mn⊂A of anyons, we establish the relation ηn⋅ξn=∑a∈Mnθ(a)n∣∑a∈Mnθ(a)n∣,\eta _n \cdot \xi _n = \frac{\sum _{a \in \mathcal {M}^n} \theta (a)^n}{\left| \sum _{a \in \mathcal {M}^n} \theta (a)^n \right|},ηn​⋅ξn​=∣∑a∈Mn​θ(a)n∣∑a∈Mn​θ(a)n​,which generalizes the known anomaly formula. We analyze the algebraic structure of Mn\mathcal {M}^nMn, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.
Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a Z2\mathbb{Z}_2Z2​ one-form symmetry, generated in each theory by a single nontrivial Z2\mathbb{Z}_2Z2​ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single Z2\mathbb{Z}_2Z2​ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the Z2\mathbb{Z}_2Z2​ one-form symmetry, which acquires a Z16\mathbb{Z}_{16}Z16​ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.
By definition, the physics of the ddd-dimensional (dim) boundary of a (d+1)(d+1)(d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a ddd-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a ddd-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d3d3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d4d4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d4d4d boundary of a 5d5d5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N{\mathbb{Z}}_{2N}Z2N​ topological order with topological degeneracy, but this Z2N{\mathbb{Z}}_{2N}Z2N​ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d3d3d space.
We describe a method to find the anomaly of the time-reversal symmetry of 2+1d topological quantum field theories, by computing the fractional anomalous momentum on the cross-cap background. This allows us, for example, to identify the parameter ννν mod 16 of the bulk 3+1d topological superconductor with T2=(−1)F\mathsf{T}^2=(-1)^FT2=(−1)F on whose boundary a given 2+1d time-reversal-invariant topological phase can appear.