If the Standard Model is understood as the first term of an effective field theory, the anomaly-cancellation conditions have to be worked out and fulfilled order by order in the effective field-theory expansion. We bring attention to this issue and study in detail a subset of the anomalies of the effective field theories at the electroweak scale. The end result is a set of sum rules for the operator coefficients. These conditions, which are necessary for the internal consistency of the theory, lead to a number of phenomenological consequences when implemented in analyses of experimental data. In particular, they not only decrease the number of free parameters in different physical processes but have the potential to relate processes with different flavor content. Conversely, a violation of these conditions would necessarily imply the existence of undetected non-decoupling new physics associated with the electroweak energy scale.
I discuss and extend several results concerning the cancellation of discrete gauge anomalies. I show how heavy fermions do not decouple in the presence of discrete gauge anomalies. As a consequence, in general, cancellation of discrete gauge anomalies cannot be described merely in terms of low energy operators involving only the light fermions. I also discuss cancellation of discrete gauge anomalies through a discrete version of the Green-Schwarz (GS) mechanism as well as the possibility of discrete gauge R-symmetries and their anomalies. Finally, some phenomenological applications are discuss…
Higher order effects of the two-dimensional non-Abelian gauge theories, in which the vector-meson mass is generated by chiral anomalies, will be studied. The $β$ function and the topological nature of the non-linear $σ$ model in the action and unitarity of the theory will be discussed.
The possible anomaly of the tensor current divergence equation in U(1) gauge theories is calculated by means of perturbative method. It is found that the tensor current divergence equation is free of anomalies.
We review recent work on the effective field theory description and the phenomenology of axion-like and scalar particles in models characterized by gauge and/or conformal anomalies.
We discuss the various resolutions which have been suggested in the literature for the way that the equivalence theorem can be satisfied in theories with global anomalies. We provide a model-independent proof for the resolution originally suggested in the context of a toy model by Kilgore.
Supersymmetric standard model with softly broken lepton symmetry provides a suitable framework to accommodate the solar and atmospheric neutrino anomalies. This model contains a natural explanation for large mixing and hierarchal masses without fine tuning of the parameters. Neutrino spectrum is particularly constrained in the minimal messenger model (MMM) of gauge mediated SUSY breaking, since all SUSY breaking effects are controlled in MMM by a single parameter. We study the structure of neutrino masses and mixing both in MMM and in simple extensions of it in the context of solar and atmosph…
The transverse axial vector and vector anomalies in four-dimensional U(1) gauge theories studied in [10] is reexamined by means of perturbative methods. The absence of transverse anomalies for both axial vector and vector current is verified. We also show that the Pauli-Villars regularization and dimensional regularization give the same result on the transverse anomaly of both axial vector and vector current.
The possibilities of global (non-perturbative) gauge anomalies for a class of gauge groups are investigated. Intimately connected to branching rules and topological aspect of gauge groups, the results are applied to the study of unification gauge groups such as SO(10), SU(5), $E_6$, $E_8$ etc. Especially, we discuss extensively about the selection rule for generation numbers $N_f+N_{mf}=even\ge 4$ in SO(10) and supersymmetric SO(10) unification theories as originally proposed by the author$^1$, where $N_f$ and $N_{mf}$ denote the generation numbers for ordinary fermions and mirror fermions res…
We analyze the theory and phenomenology of anomalous global chiral symmetries in the presence of an extra dimension. We propose a simple extension of the Standard Model in 5D whose signatures closely resemble those of supersymmetry with gauge mediation, and we suggest a novel scalar dark matter candidate.
In light of the recent $R_{K^{(*)}}$ data on neutral current flavour anomalies in $B \to K^{(*)} \ell^+ \ell^-$ decays, we re-examine their quantitative interpretation in terms of an extended 331 gauge theory framework. We achieve this by adding two extra lepton species with novel 331 charges, while ensuring that the model remains anomaly free. In contrast to the canonical 331 models, the gauge charges of the first and second lepton families differ from each other, allowing lepton flavour universality violation. We further expand the model by adding the neutral fermions required to provide an …
In this talk, we briefly review the basic concepts of anomalous gauge theories. It has been known for some time how theories with local anomalies can be handled. Recently it has been pointed out that global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization.
We present a three-site Pati-Salam gauge model able to explain the Standard Model flavor hierarchies while, at the same time, accommodating the recent experimental hints of lepton-flavor non-universality in $B$ decays. The model is consistent with low- and high-energy bounds, and predicts a rich spectrum of new states at TeV scale that could be probed in the near future by the high-$p_T$ experiments at the LHC.
We study more extensively and completely for global gauge anomalies with some semisimple gauge groups as initiated in ref.1. A detailed and complete proof or derivation is provided for the Z_2 global gauge anomaly given in ref.1 for a gauge theory with the semisimple gauge group SU(2)\times SU(2)\times SU(2) in D=4 dimensions and Weyl fermions in the irreducible representation (IR) ω=(2,2,2) with 2 denoting the corresponding dimensions. This Z_2 anomaly was used in the discussions related to generic SO(10) and supersymmetric SO(10) unification theories^1 for the total generation numbers of fer…
We discuss the generalization to global gauge anomalies of the familiar procedure for the cancellation of local gauge anomalies in effective theories of spontaneously broken symmetries. We illustrate this mechanism in a recently proposed six-dimensional extension of the standard model.
The muon $(g-2)_μ$ and $b\to s \bar μμ$ induced $B$ anomalies as hints of new physics beyond the standard model (SM) have attracted much attention. These two anomalies indicate that there may exist new interaction specifically related to muon. A lot of theoretical ideas have been proposed to explain these anomalies. Gauged flavor specific $U(1)_{B_q-L_μ}$ is among the promising ones. The new gauge boson $Z'$ from $U(1)_{B_q-L_μ}$ interacts with muon and provides necessary ingredient to solve the $(g-2)_μ$ anomaly. The $Z'$-quark coupling can generate flavor changing interactions after diagonal…
The original version of this paper contains an error; when this is corrected the basic conclusion changes. A revised manuscript will be submitted shortly.
The anomaly of a discrete symmetry is defined as the Jacobian of the path-integral measure. Assuming that the anomaly at low energies is cancelled by the Green-Schwarz (GS) mechanism at a fundamental scale, we investigate possible Kac-Moody levels for anomalous discrete family symmetries. As the first example, we consider discrete abelian baryon number and lepton number symmetries in the minimal supersymmetric standard model with the see-saw mechanism, and we find that the ordinary unification of gauge couplings is inconsistent with the GS conditions, indicating the possible existence of furth…
A possibility of explaining the anomalies in the semileptonic $B$-meson decay $B \to K^{*} μ\barμ$ has been explored in the framework of the gauged $U(1)_{μ-τ}$ symmetry. Apart from the muon anomalous magnetic moment and neutrino sector, we formulate the model starting with a valid Lagrangian and consider the constraints from the neutral meson mixings, the bounds on direct detection and the relic density of the bosonic dark matter candidate augmented to collider constraints. We search the parameter space, which accommodates the size of the anomaly of the $B \rightarrow K^* μ\bar μ$ decay, to s…
In the presence of some forms of global anomalies, the equivalence theorem, which relates the interactions of longitudinal gauge bosons to those of the Goldstone bosons, is not always valid. This can occur when the Goldstone sector contains an anomaly which is canceled in the gauge currents by the effects of a different sector of the theory. The example of the Standard Model without Higgs particles is used to illustrate this phenomena.
It is argued that the gauge anomalies are only the artefacts of quantum field theory when certain subtleties are not taken into account. With the Berry's phase needed to satisfy certain boundary conditions of the generating path integral, the gauge anomalies associated with homotopically nontrivial gauge transformations are shown explicitly to be eliminated, without any extra quantum fields introduced. This is in contra-distinction to other quantisations of `anomalous' gauge theory where extra, new fields are introduced to explicitly cancel the anomalies.
The local cohomology of an extended BRST differential which includes global N=1 supersymmetry and Poincare transformations is completely and explicitly computed in four-dimensional supersymmetric gauge theories with super-Yang-Mills multiplets, chiral matter multiplets and linear multiplets containing 2-form gauge potentials. In particular we determine to first order all N=1 supersymmetric and Poincare invariant consistent deformations of these theories that preserve the N=1 supersymmetry algebra on-shell modulo gauge transformations, and all Poincare invariant candidate gauge and supersymmetr…
We apply the path-integral formalism to compute the anomalies in general orbifold gauge theories (including possible non-trivial Scherk-Schwarz boundary conditions) where a gauge group G is broken down to subgroups H_f at the fixed points y=y_f. Bulk and localized anomalies, proportional to δ(y-y_f), do generically appear from matter propagating in the bulk. The anomaly zero-mode that survives in the four-dimensional effective theory should be canceled by localized fermions (except possibly for mixed U(1) anomalies). We examine in detail the possibility of canceling localized anomalies by the …
A search for supersymmetry anomalies requires an examination of the BRS cohomology of supersymmetric Yang-Mills coupled to chiral matter, and the physically interesting (on-shell) anomalies are those which cannot be eliminated using the equations of motion. An analysis of this cohomology problem shows that the simplest situation where a physically interesting supersymmetry anomaly can arise is when: 1. the anomaly occurs in the renormalization of a composite antichiral spinor superfield operator constructed from the chiral matter in the theory, 2. the anomalous diagrams are one-loop triangle d…
The global chiral symmetry of a $SU(2)$ gauge theory is studied in the framework of renormalization group (RG). The theory is defined by the RG flow equations in the infrared cutoff $Ł$ and the boundary conditions for the relevant couplings. The physical theory is obtained at $Ł=0$. In our approach the symmetry is implemented by choosing the boundary conditions for the relevant couplings not at the ultraviolet point $Ł=Ł_0\to\infty$ but at the physical value $Ł=0$. As an illustration, we compute the triangle axial anomalies.
We work out in the forward limit and up to order $e^6$ in perturbation theory the collinear divergences. In this kinematical regime we discover new collinear divergences that we argue can be only cancelled using quantum interference with processes contributing to the gauge anomaly. This rules out the possibility of a quantum consistent and anomaly free theory with massless charges and long range interactions. We use the anomalous threshold singularities to derive a gravitational lower bound on the mass of the lightest charged fermion.
Some of the basic issues related to the regularization and anomalies in gauge theory are reviewed, with particular emphasis on the recent development in lattice gauge theory. The generalized Pauli-Villars regularization is discussed from a view point of the covariant regularization of currents, and the construction of a regularized effective action in terms of covariant currents is compared with the lattice formulation of chiral Abelian theory.
We embed two 4D chiral multiplets of opposite representations in the 5D N=2 $SU(N+K)$ gauge theory compactified on an orbifold $S^1/(Z_2\times Z'_2)$. There are two types of orbifold boundary conditions in the extra dimension to obtain the 4D N=1 $SU(N)\times SU(K)\times U(1)$ gauge theory from the bulk: in Type I, one has the bulk gauge group at $y=0$ and the unbroken gauge group at $y=πR/2$ while in Type II, one has the unbroken gauge group at both fixed points. In both types of orbifold boundary conditions, we consider the zero mode(s) as coming from a bulk $(K+N)$-plet and brane fields at …
We study UV-complete Froggatt-Nielsen-like models for the generation of mass and mixing hierarchies, assuming that the integrated heavy fields are chiral with respect to an abelian Froggatt-Nielsen symmetry. It modifies the mixed anomalies with respect to the Standard Model gauge group, which opens up the possibility to gauge the Froggatt-Nielsen symmetry without the need to introduce additional spectator fermions, while keeping mass matrices usually associated to anomalous flavour symmetries. We give specific examples where this happens, and we study the flavourful axion which arises from an …
We propose a predictive radiative seesaw model at one-loop level with a flavor dependent gauge symmetry $U(1)_{xB_3-xe-μ+τ}$ and Majorana fermion dark matter. For the neutrino mass matrix, we obtain an $A_1$ type texture (with two zeros) that provides us several predictions such as the normal ordering for the neutrino masses. We analyze the constraints from lepton flavor violations, relic density of dark matter, and collider physics for the new $U(1)_{xB_3-xe-μ+τ}$ gauge boson. Within the allowed region, the LHCb anomalies in $B\rightarrow K^* μ^+ μ^-$ and $B\rightarrow K \ell^+ \ell^-$ with $…
A recent proposal by Kaplan for a chiral gauge theory on the lattice is tested with background gauge fields. The spectrum of the finite lattice Hamiltonian is calculated and the existence of a chiral fermion is demonstrated. Lattice doublers are found to decouple. The flavor anomalies, which are in agreement with the continuum anomaly relation, are obeserved on a finite lattice. Non-trivial anomaly cancellation is observed in a chiral gauge current.
We study the generic features of minimal gauge extensions of the Standard Model in view of recent hints of lepton-flavor non-universality in semi-leptonic $b \to s \ell^+ \ell^-$ and $b \to c \ell ν$ decays. We classify the possible models according to the symmetry-breaking pattern and the source of flavor non-universality. We find that in viable models the $\mathrm{SU(2)}_L$ factor is embedded non-trivially in the extended gauge group, and that gauge couplings should be universal, hinting to the presence of new degrees of freedom sourcing non-universality. Finally, we provide an explicit mode…
Motivated by flavor non-universality and anomalies in semi-leptonic B-meson decays, we present a general and systematic discussion about how to construct anomaly-free $U(1)'$ gauge theories based on an extended standard model with only three right-handed neutrinos. If all standard model fermions are vector-like under this new gauge symmetry, the most general family non-universal charge assignments, $(a,b,c)$ for three-generation quarks and $(d,e,f)$ for leptons, need satisfy just one condition to be anomaly-free, $3(a+b+c)=-(d+e+f)$. Any assignment can be linear combinations of five independen…
We study whether higher-dimensional operators in effective field theories, in particular in the Standard Model Effective Field Theory (SMEFT), can source gauge anomalies via the modification of the interactions involved in triangle diagrams. We find no evidence of such gauge anomalies at the level of dimension-6 operators that can therefore be chosen independently to each others without spoiling the consistency of SMEFT, at variance with recent claims. The underlying reason is that gauge-invariant combinations of Goldstone bosons and massive gauge fields are allowed to couple to matter current…
We propose a new class of $U(1)_{μ-τ}$ gauged model that can explain recent flavor anomalies such as muon $g-2$, $b\to sμ^+ μ^-$, as well as include a scalar dark matter candidate while satisfying all the phenomenological constraints. For this purpose, we add new vectorlike quarks and leptons as well as two inert singlet scalars, one leptophilic and the other leptophobic. In our model $b\to sμ^+ μ^-$ anomalies can be explained by loop induced interactions among the SM and exotic quarks and $Z'$ gauge boson. In our numerical analysis, we show allowed region of our model to be narrow, and it wou…
We propose a generation dependent lepton/baryon gauge symmetry, $U(1)_{B_3 - x_μL_μ- x_τL_τ} \equiv U(1)_X$ (with $x_μ+ x_τ=1$ for anomaly cancellation), as a possible solution for the $b \rightarrow s μ^+ μ^-$ anomalies. By introducing two Higgs doublet fields, we can reproduce the observed CKM matrix, and generate flavor changing $Z'$ interactions in the quark sector. Thus one can explain observed anomalies in $b \to s \ell^+ \ell^-$ decay with the lepton non-universal $U(1)_X$ charge assignments. We show the minimal setup explaining $b \to s \ell^+ \ell^-$ anomalies, neutrino masses and mix…
We present a simple $U(1)_{B_3-3L_μ}$ gauge Standard Model extension that can easily account for the anomalies in $R(K)$ and $R(K^*)$ reported by LHCb. The model is economical in its setup and particle content. Among the Standard Model fermions, only the third generation quark family and the second generation leptons transform non-trivially under the new $U(1)_{B_3-3L_μ}$ symmetry. This leads to lepton non-universality and flavor changing neutral currents involving the second and third quark families. We discuss the relevant experimental constraints and some implications.
We study the impact of an additional $U(1)'$ gauge symmetry with flavour-dependent charges for quarks and leptons on the LHC flavour anomalies observed in $B \to K^* μ^+μ^-$, $R(K) = B \to K μ^+μ^-/B \to K e^+e^-$, and $h \to μτ$. In its minimal version with two scalar doublets, the resulting model naturally explains the deviations from the Standard Model observed in $B \to K^* μ^+μ^-$ and $R(K)$. The CMS access in $h \to μτ$ can be explained by introducing a third scalar doublet, which gives rise to a prediction for $τ\to 3μ$. We investigate constraints from flavour observables and direct LHC…
We study anomalies of six-dimensional gauge theories compactified on orbifolds. In addition to the known bulk anomalies, brane anomalies appear on orbifold fixpoints in the case of chiral boundary conditions. At a fixpoint, where the bulk gauge group G is broken to a subgroup H, the non-abelian G-anomaly in the bulk reduces to a H-anomaly which depends in a simple manner on the chiral boundary conditions. We illustrate this mechanism by means of a SO(10) GUT model.
We prove the Adler-Bardeen theorem in a large class of general gauge theories, including nonrenormalizable ones. We assume that the gauge symmetries are general covariance, local Lorentz symmetry and Abelian and non-Abelian Yang-Mills symmetries, and that the local functionals of vanishing ghost number satisfy a variant of the Kluberg-Stern--Zuber conjecture. We show that if the gauge anomalies are trivial at one loop, for every truncation of the theory there exists a subtraction scheme where they manifestly vanish to all orders, within the truncation. Outside the truncation the cancellation o…
After many years of investigations, our understanding of the dynamics of strongly-coupled chiral gauge theories is still quite unsatisfactory today. Conventional wisdom about strongly-coupled gauge theories, successfully applied to QCD, is not always as useful in chiral gauge theories. Recently some new ideas and techniques have been developed, which involve concepts of generalized symmetries, of gauging a discrete center symmetry, and of generalizing the 't Hooft anomaly matching constraints to include certain mixed symmetries. This new development has been applied to chiral gauge theories, l…
We explore the idea that in some class of strongly-coupled chiral $SU(N)$ gauge theories the infrared dynamics might be characterized by a bifermion condensate in the adjoint representation of the color gauge group. As an illustration, in this work we revisit an $SU(N)$ chiral gauge theory with Weyl fermions in a symmetric ($ψ$) and anti-antisymmetric ($χ$) tensor representations, together with eight fermions in the anti-fundamental representations ($η$), which we called $ψχη$ model in the previous investigations. We study the infrared dynamics of this system more carefully, by assuming dynami…
We study a class of 4-dimensional $SU(N)$ chiral gauge theories with fermions in the 2-index symmetric and antisymmetric representations and classify their infrared phases. The choice $N=4\mathbb{Z}$ corresponds to gauging the fermion number and makes the theory purely bosonic. We examine the most general background fields of the centers of the gauge, non-abelian flavor, and $U(1)$-axial groups that can be consistently activated, thereby determine the faithful global continuous and discrete symmetries of the theory. This allows us to identify new mixed 0-form/1-form `t Hooft anomalies on both …
Axial anomalies give rise to interesting new transport phenomena such as the "chiral magnetic effect". We discuss how the associated transport coefficients can be studied via Kubo formulas at weak and strong coupling, the latter via gauge gravity duality. We argue for a new "chiral gravito-magnetic" (or vortical) effect sensitive to the presence of mixed gauge-gravitational anomalies.
We study the constraints on models with extra dimensions arising from local anomaly cancellation. We consider a five-dimensional field theory with a U(1) gauge field and a charged fermion, compactified on the orbifold S^1/(Z_2 x Z_2'). We show that, even if the orbifold projections remove both fermionic zero modes, there are gauge anomalies localized at the fixed points. Anomalies naively cancel after integration over the fifth dimension, but gauge invariance is broken, spoiling the consistency of the theory. We discuss the implications for realistic supersymmetric models with a single Higgs h…
We analyse global anomalies and related constraints in the Standard Model (SM) and various Beyond the Standard Model (BSM) theories. We begin by considering four distinct, but equally valid, versions of the SM, in which the gauge group is taken to be $G=G_{\text{SM}}/Γ_n$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $Γ_n$ isomorphic to $\mathbb{Z}/n$ where $n\in\left\{1,2,3,6\right\}$. In addition to deriving constraints on the hypercharges of fields transforming in arbitrary representations of the $SU(3)\times SU(2)$ factor, we study the possibility of global anomalies in theories …
We study two well-known $SU(N)$ chiral gauge theories with fermions in the symmetric, anti-symmetric and fundamental representations. We give a detailed description of the global symmetry, including various discrete quotients. Recent work argues that these theories exhibit a subtle mod 2 anomaly, ruling out certain phases in which the theories confine without breaking their global symmetry, leaving a gapless composite fermion in the infra-red. We point out that no such anomaly exists. We further exhibit an explicit path to the gapless fermion phase, showing that there is no kinematic obstructi…
The vector leptoquark representation, $U_μ= (3,1,2/3)$, was recently identified as an exceptional single mediator model to address experimental hints on lepton flavour universality violation in semileptonic $B$-meson decays, both in neutral ($b \to s μμ$) and charged ($b \to c τν$) current processes. Nonetheless, it is well-known that massive vectors crave an ultraviolet (UV) completion. We present the first full-fledged UV complete and calculable gauge model which incorporates this scenario while remaining in agreement with all other indirect flavour and electroweak precision measurements, as…
In this note, we discuss the consistency conditions which a discrete $Z_N$ symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the $Z_N$ ${\cal R}$-symmetries as well as non-${\cal R}$ $Z_N$ symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous $Z_3$ ${\cal R}$-symm…
Using the Fujikawa and the heat-kernel methods we make a complete and detailed computation of the global, gauge and gravitational anomalies present in the Standard Model defined on a curved space time with torsion. We find new contributions coming from curvature and torsion terms to the leptonic number anomaly (so that $B-L$ is not conserved any more), to the $U(1)_Y$ gauge and to the mixed $U(1)_Y$-gravitational anomalies, but the gauge anomaly cancellation conditions on the hypercharges remain the same. We also find that the condition, usually related to the cancellation of the mixed $U(1)_Y…
We study the issue of gauge invariance in five-dimensional theories compactified on an orbifold $S^1/(\mathbb{Z}_2\times \mathbb{Z}^\prime_2)$ in the presence of an external U(1) gauge field. From the four-dimensional point We study the issue of gauge invariance in five-dimensional theories compactified on an orbifold $S^1/(\mathbb{Z}_2\times \mathbb{Z}^\prime_2)$ in the presence of an external U(1) gauge field. From the four-dimensional point of view the theory contains a tower of Kaluza-Klein Dirac fermions with chiral couplings and it looks anomalous at the quantum level. We show that this …
I introduce a model to explain the recent anomalies in the muon anomalous magnetic moment and the rare semi-leptonic $B$ meson decays. The Standard Model is extended by a $U(1)^\prime$ gauge symmetry and a complete fourth family fermions which are vector-like under the gauge symmetry. We found parameter points which can explain the anomalies consistently with the other observables. We then propose an interesting possibility to have signals with four muons or more from vector-like lepton pair production.
We show that the recent anomalies in $b \to s \ell^+ \ell^-$ transitions observed by the LHCb collaboration can be accommodated within string motivated models with a low mass $Z^{\prime}$ gauge boson. Such $Z^{\prime}$ gauge boson can be obtained in compactifications with a low string scale. We consider a class of intersecting D-brane models, in which different families of quarks and leptons are simultaneously realized at different D-brane intersections. The explanation of $b \to s \ell^+ \ell^-$ anomalies via a stringy $Z^{\prime}$ sets important restrictions on these viable D-brane construct…
The axion couplings to SM gauge bosons are derived in various models, and shown to always arise entirely from non-anomalous fermion loops. They are thus independent of the anomaly structure of the model. This fact is without consequence for vector gauge interactions like QCD and QED, but has a major impact for chiral gauge theories. For example, in the DFSZ axion model, the couplings of axions to electroweak gauge bosons do not follow the pattern expected from chiral anomalies, as we prove by an explicit calculation. The reason for this mismatch is traced back to triangle Feynman diagrams sens…
We hypothesize a new and more complete set of anomalies of certain quantum field theories (QFTs) and then give an eclectic verification. First, we propose a set of 't Hooft higher anomalies of 4d time-reversal symmetric pure SU(N)-Yang-Mills (YM) gauge theory with a second-Chern-class topological term at $θ=π$, via 5d cobordism invariants (higher symmetry-protected topological states), with N = $2, 3, 4$ and others. Second, we propose a set of 't Hooft anomalies of 2d $\mathbb{CP}^{\mathrm{N}-1}$-sigma models with a first-Chern-class topological term at $θ=π$, by enlisting all possible 3d cobo…
