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$C$-$P$-$T$ anomaly in bosonic systems

Tin Sulejmanpašić, Yuya Tanizaki

Published Feb 6, 2018• 2 citations
OpenAlex

Abstract

We consider the $O(3)$ nonlinear sigma model with the $\theta$-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\theta$, and enjoys the periodicity in $\theta\rightarrow \theta+2\pi$. At $\theta=0,\pi$ it also has a charge-conjugation $C$-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb RP^2$ manifold, after which the $2\pi$ periodicity of $\theta$ and the $C$ symmetry at $\theta=\pi$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $\theta=\pi$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as $C$, $P$ and $T$ are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the $SO(3)$-spin symmetry), requiring that the ground state is never trivially gapped, even if $SO(3)$ spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}P^{N-1}$ models and show that the $C$-$P$-$T$ anomaly exists for even $N$.

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C−P−T anomaly matching in bosonic quantum field theory and spin chains

Tin Sulejmanpašić, Yuya Tanizaki

We consider the $O(3)$ nonlinear sigma model with the $\ensuremath{\theta}$ term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\ensuremath{\theta}$, and enjoys the periodicity in $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\theta}+2\ensuremath{\pi}$. At $\ensuremath{\theta}=0,\ensuremath{\pi}$ it also has a charge-conjugation $C$ symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable $\mathbb{R}{P}^{2}$ manifold, after which the $2\ensuremath{\pi}$ periodicity of $\ensuremath{\theta}$ and the $C$ symmetry at $\ensuremath{\theta}=\ensuremath{\pi}$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation $C$, parity $P$, and time-reversal $T$ symmetries when $\ensuremath{\theta}=\ensuremath{\pi}$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as $C,P$, and $T$ are all good symmetries of the theory. We make several consistency checks with various semiclassical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity, and time reversal [but not involving the SO(3)-spin symmetry], requiring that the ground state is never trivially gapped, even if SO(3) spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}{P}^{N\ensuremath{-}1}$ models and show that the $C\text{\ensuremath{-}}P\text{\ensuremath{-}}T$ anomaly exists for even $N$.

arXiv:1802.02153•Physical review. B./Physical review. B•Apr 9, 2018•54 citations

Anomaly and global inconsistency matching: θ angles, SU(3)/U(1)2 nonlinear sigma model, SU(3) chains, and generalizations

Yuya Tanizaki, Tin Sulejmanpašić

We discuss the $SU(3)/[U(1)\ifmmode\times\else\texttimes\fi{}U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ gauge theories and therefore allow for two topological $\ensuremath{\theta}$ angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $\ensuremath{\theta}$ angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\ensuremath{\theta}$ angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}\ensuremath{\subset}SU(3)/{\mathbb{Z}}_{3}$. In addition, the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\ensuremath{\theta}$ angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the $SU(3)$ Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U{(1)}^{N\ensuremath{-}1}$ nonlinear sigma models as well as the 't Hooft anomaly of the $SU(N)$ Wess-Zumino-Witten model, and show that they match. Finally, the $(2+1)$-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.

arXiv:1805.11423•Physical review. B./Physical review. B•Sep 14, 2018•106 citations

Global anomaly matching in the higher-dimensional CPN−1 model

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We investigate 't Hooft anomalies in the ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model in space-time dimensions higher than two and identify two types of anomalies: one is a mixed anomaly between the $\mathrm{PSU}(N)$ flavor-rotation and magnetic symmetries and the other is between the reflection and magnetic symmetries. The latter indicates that even in the absence of the flavor symmetry, the model cannot have a unique gapped ground state as long as the reflection and magnetic symmetries are respected. We also clarified the condition for the 't Hooft anomalies to survive under monopole deformations, which explicitly break the magnetic symmetry down to its discrete subgroup. Besides, we explicitly show how the identified 't Hooft anomalies match in the low-energy effective description of symmetry broken phases---the N\'eel, $\mathrm{U}(1)$ spin liquid, and the valence bond solid phases. An application to the finite-temperature phase diagram of the four-dimensional ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model is also discussed.

arXiv:2001.07373•Physical review. B./Physical review. B•Apr 13, 2020•8 citations

Nonlinear sigma models, antiperiodic boundary conditions, spin chains, and 't Hooft anomalies

N. Read, Hubert Saleur

We consider two sets of related models: initially, these are $SU(2)$ antiferromagnetic spin chains with $N$ sites of spin $S$, and the $O(3)$ nonlinear sigma model in two dimensions with topological coefficient $Θ$ a multiple of $π$ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with $Θ=2πS$. We study these models with $N$ odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the $\mathbb{Z}_2$ inversion symmetry $\vec{n}\to-\vec{n}$, and amounts to a flux of a $\mathbb{Z}_2$ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the $\mathbb{Z}_2$ inversion symmetry. We show directly that, if and only if $(-1)^{Θ/π}=-1$, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space $\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2$, and hence this model does not exist for $Θ=π$ (mod $2π$). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on $N$ modulo $4$ and $2S$ modulo $2$. For a large class of spin chains and associated sigma models we find similar results, but now $(-1)^{Θ/π}$ is replaced by the value $\pm 1$ of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.

arXiv:2606.07041•Jun 5, 2026•0 citations

Construction of bosonic symmetry-protected-trivial states and their topological invariants via G x SO(∞) nonlinear σ

Xiao-Gang Wen

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Physical Review Letters•May 1, 2015•4 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
CCC-PPP-TTT anomaly in bosonic systems
We consider the O(3)O(3)O(3) nonlinear sigma model with the θ\thetaθ-term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any θ\thetaθ, and enjoys the periodicity in θ→θ+2π\theta\rightarrow \theta+2\piθ→θ+2π. At θ=0,π\theta=0,\piθ=0,π it also has a charge-conjugation CCC-symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable RP2\mathbb RP^2RP2 manifold, after which the 2π2\pi2π periodicity of θ\thetaθ and the CCC symmetry at θ=π\theta=\piθ=π are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation CCC, parity PPP, and time-reversal TTT symmetries when θ=π\theta=\piθ=π. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as CCC, PPP and TTT are all good symmetries of the theory. We make several consistency checks with various semi-classical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity and time reversal (but not involving the SO(3)SO(3)SO(3)-spin symmetry), requiring that the ground state is never trivially gapped, even if SO(3)SO(3)SO(3) spin symmetry is explicitly and completely broken. We also consider generalizations to CPN−1\mathbb{C}P^{N-1}CPN−1 models and show that the CCC-PPP-TTT anomaly exists for even NNN.
We consider the O(3)O(3)O(3) nonlinear sigma model with the $\ensuremath{\theta}$ term and its linear counterpart in 1+1D. The model has discrete time-reflection and space-reflection symmetries at any $\ensuremath{\theta}$, and enjoys the periodicity in $\ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\theta}+2\ensuremath{\pi}$. At $\ensuremath{\theta}=0,\ensuremath{\pi}$ it also has a charge-conjugation CCC symmetry. Gauging the discrete space-time reflection symmetries is interpreted as putting the theory on the nonorientable RP2\mathbb{R}{P}^{2}RP2 manifold, after which the $2\ensuremath{\pi}$ periodicity of $\ensuremath{\theta}$ and the CCC symmetry at $\ensuremath{\theta}=\ensuremath{\pi}$ are lost. We interpret this observation as a mixed 't Hooft anomaly among charge-conjugation CCC, parity PPP, and time-reversal TTT symmetries when $\ensuremath{\theta}=\ensuremath{\pi}$. Anomaly matching implies that in this case the ground state cannot be trivially gapped, as long as C,PC,PC,P, and TTT are all good symmetries of the theory. We make several consistency checks with various semiclassical regimes, and with the exactly solvable XYZ model. We interpret this anomaly as an anomaly of the corresponding spin-half chains with translational symmetry, parity, and time reversal [but not involving the SO(3)-spin symmetry], requiring that the ground state is never trivially gapped, even if SO(3) spin symmetry is explicitly and completely broken. We also consider generalizations to $\mathbb{C}{P}^{N\ensuremath{-}1}$ models and show that the $C\text{\ensuremath{-}}P\text{\ensuremath{-}}T$ anomaly exists for even NNN.
We discuss the $SU(3)/[U(1)\ifmmode\times\else\texttimes\fi{}U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ gauge theories and therefore allow for two topological $\ensuremath{\theta}$ angles. These models provide a field theoretic description of the SU(3)SU(3)SU(3) chains. We show that, for particular values of $\ensuremath{\theta}$ angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\ensuremath{\theta}$ angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}\ensuremath{\subset}SU(3)/{\mathbb{Z}}_{3}$. In addition, the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\ensuremath{\theta}$ angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the SU(3)SU(3)SU(3) Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U{(1)}^{N\ensuremath{-}1}$ nonlinear sigma models as well as the 't Hooft anomaly of the SU(N)SU(N)SU(N) Wess-Zumino-Witten model, and show that they match. Finally, the (2+1)(2+1)(2+1)-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.
We investigate 't Hooft anomalies in the ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model in space-time dimensions higher than two and identify two types of anomalies: one is a mixed anomaly between the PSU(N)\mathrm{PSU}(N)PSU(N) flavor-rotation and magnetic symmetries and the other is between the reflection and magnetic symmetries. The latter indicates that even in the absence of the flavor symmetry, the model cannot have a unique gapped ground state as long as the reflection and magnetic symmetries are respected. We also clarified the condition for the 't Hooft anomalies to survive under monopole deformations, which explicitly break the magnetic symmetry down to its discrete subgroup. Besides, we explicitly show how the identified 't Hooft anomalies match in the low-energy effective description of symmetry broken phases---the N\'eel, U(1)\mathrm{U}(1)U(1) spin liquid, and the valence bond solid phases. An application to the finite-temperature phase diagram of the four-dimensional ${\mathbb{C}\mathbb{P}}^{N\ensuremath{-}1}$ model is also discussed.
We consider two sets of related models: initially, these are SU(2)SU(2)SU(2) antiferromagnetic spin chains with NNN sites of spin SSS, and the O(3)O(3)O(3) nonlinear sigma model in two dimensions with topological coefficient ΘΘΘ a multiple of πππ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with Θ=2πSΘ=2πSΘ=2πS. We study these models with NNN odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the Z2\mathbb{Z}_2Z2​ inversion symmetry n⃗→−n⃗\vec{n}\to-\vec{n}n→−n, and amounts to a flux of a Z2\mathbb{Z}_2Z2​ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the Z2\mathbb{Z}_2Z2​ inversion symmetry. We show directly that, if and only if (−1)Θ/π=−1(-1)^{Θ/π}=-1(−1)Θ/π=−1, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space RP2≅S2/Z2\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2RP2≅S2/Z2​, and hence this model does not exist for Θ=πΘ=πΘ=π (mod 2π2π2π). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on NNN modulo 444 and 2S2S2S modulo 222. For a large class of spin chains and associated sigma models we find similar results, but now (−1)Θ/π(-1)^{Θ/π}(−1)Θ/π is replaced by the value ±1\pm 1±1 of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.
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.