Anomaly Matching Research Articles

Dynamics of strongly-coupled chiral gauge theories

We study the dynamics of SU(N) chiral gauge theories with massless fermions belonging to various combinations of the symmetric, antisymmetric or fundamental representations. We limit ourselves to the gauge-anomaly-free and asymptotically free systems. These theories have a global symmetry group with the associated ’t Hooft anomaly-matching conditions severely limiting the possible RG flows. Recent developments on the applications of the generalized symmetries and the stronger requirement of the matching of the mixed anomalies also give further indications on the possible IR dynamics. In vectorlike theories such as the quantum chromodynamics (QCD), gauge-invariant “quark-antiquark” condensates form and characterize the IR dynamics, and the anomaly matching involves the Nambu-Goldstone (NG) bosons. In some other special cases, such as the Bars-Yankielowicz (BY) or Georgi-Glashow (GG) models, a hypothetical solution was proposed in the literature, with no global symmetry breaking and with some simple set of composite massless fermions saturating all the anomalies. For the BY and GG systems, actually, a more plausible candidate for their IR physics is the dynamical Higgs phase, with a few simple bi-fermion color-flavor locked condensates, breaking the color and flavor symmetries, partially or totally. Remarkably, the ’t Hooft anomaly-matching (and generalized anomaly-matching) conditions are automatically satisfied in this phase. Another interesting possibility, occurring in some chiral gauge theories, is dynamical Abelianization, familiar from N = 2 supersymmetric gauge theories. We explore here even more general types of possible IR phases than the ones mentioned above, for wider classes of models. With the help of large-N arguments we look for IR free theories, whereas the MAC (maximal attractive channel) criterion might suggest some simple bi-fermion condensates characterizing the IR dynamics of the systems. In many cases the low-energy effective theories are found to be described by quiver-like gauge theories, some of the (nonAbelian) gauge groups are infrared-free while some others might be asymptotically free.

Topological mass generation in gapless systems

Mass generation of gauge fields can be universally described by topological couplings in gapped systems, such as the Abelian Higgs model in $(3+1)$ dimensions and the Maxwell-Chern-Simons theory in $(2+1)$ dimensions. These systems also exhibit the spontaneous breaking of higher-form $\mathbb{Z}_k$ symmetries and topological orders for level $k \geq 2$. In this paper, we consider topological mass generation in gapless systems. As a paradigmatic example, we study the axion electrodynamics with level $k$ in $(3+1)$ dimensions in background fields that hosts both gapped and gapless modes. We argue that the gapped mode is related to those in fully gapped systems in lower dimensions via dimensional reduction. We show that this system exhibits the spontaneous breaking of a higher-form $\mathbb{Z}_k$ symmetry despite the absence of the conventional topological order. In the case of the background magnetic field, we also derive the low-energy effective theory of the gapless mode with the quadratic dispersion relation and show that it satisfies the chiral anomaly matching.

General anomaly matching by Goldstone bosons

We describe how Goldstone bosons of spontaneous symmetry breaking G → H can reproduce anomalies of UV theories under the symmetry group G at the nonpertur- bative level. This is done by giving a general definition of Wess-Zumino-Witten terms in terms of the invertible field theories in d + 1 dimensions which describe the anomalies of d-dimensional UV theories. The hidden local symmetry hat{H} , which is used to describe Goldstone bosons in coset construction G/H , plays an important role. Our definition also naturally leads to generalized θ-angles of the hidden local gauge group hat{H} . We illustrate this point by SO(Nc) (or Spin(Nc)) QCD-like theories in four dimensions.

Vector mesons on the wall

A domain-wall configuration of the η′ meson bounded by a string (called a pancake or a Hall droplet) is recently proposed to describe the baryons with spin Nc/2. In order to understand its baryon number as well as the flavor quantum number, we argue that the vector mesons (the ρ and ω mesons) should play an essential role for the consistency of the whole picture. We determine the effective theory of large-Nc QCD with Nf massless fermions by taking into account a mixed anomaly involving the θ-periodicity and the global symmetry. The anomaly matching requires the presence of a dynamical domain wall on which a mathrm{U}{left({N}_fright)}_{-{N}_c} Chern-Simons theory is supported. We consider the boundary conditions that should be imposed on the edge of the domain wall, and conclude that there should be a boundary term that couples the mathrm{U}{left({N}_fright)}_{-{N}_c} gauge field to the vector mesons. We discuss the impact on physics of the chiral phase transition and the relation to the “duality” of QCD.

Topological terms and anomaly matching in effective field theories on \u211d3 \xd7 \U0001d54a1. Part I. Abelian symmetries and intermediate scales

We explicitly calculate the topological terms that arise in IR effective field theories for SU(N) gauge theories on ℝ3 × \U0001d54a1 by integrating out all but the lightest modes. We then show how these terms match all global-symmetry ’t Hooft anomalies of the UV description. We limit our discussion to theories with abelian 0-form symmetries, namely those with one flavour of adjoint Weyl fermion and one or zero flavours of Dirac fermions. While anomaly matching holds as required, it takes a different form than previously thought. For example, cubic- and mixed-U(1) anomalies are matched by local background-field-dependent topological terms (background TQFTs) instead of chirallagrangian Wess-Zumino terms. We also describe the coupling of 0-form and 1-form symmetry backgrounds in the magnetic dual of super-Yang-Mills theory in a novel way, valid throughout the RG flow and consistent with the monopole-instanton ’t Hooft vertices. We use it to discuss the matching of the mixed chiral-center anomaly in the magnetic dual.

Anomaly matching in the symmetry broken phase: Domain walls, CPT, and the Smith isomorphism

Symmetries in Quantum Field Theory may have ’t Hooft anomalies. If the symmetry is unbroken in the vacuum, the anomaly implies a nontrivial low-energy limit, such as gapless modes or a topological field theory. If the symmetry is spontaneously broken, for the continuous case, the anomaly implies low-energy theorems about certain couplings of the Goldstone modes. Here we study the case of spontaneously broken discrete symmetries, such as \mathbb{Z}_2ℤ2 and TT. Symmetry breaking leads to domain walls, and the physics of the domain walls is constrained by the anomaly. We investigate how the physics of the domain walls leads to a matching of the original discrete anomaly. We analyze the symmetry structure on the domain wall, which requires a careful analysis of some properties of the unbreakable CPTCPT symmetry. We demonstrate the general results on some examples and we explain in detail the mod 4 periodic structure that arises in the \mathbb{Z}_2ℤ2 and TT case. This gives a physical interpretation for the Smith isomorphism, which we also extend to more general abelian groups. We show that via symmetry breaking and the analysis of the physics on the wall, the computations of certain discrete anomalies are greatly simplified. Using these results we perform new consistency checks on the infrared phases of 2+12+1 dimensional QCD.

Trialities of minimally supersymmetric 2d gauge theories

We study dynamics of two-dimensional mathcal{N} = (0, 1) supersymmetric gauge theories. In particular, we propose that there is an infrared triality between certain triples of theories with orthogonal and symplectic gauge groups. The proposal is supported by matching of anomalies and elliptic genera. This triality can be viewed as a (0, 1) counterpart of the (0, 2) triality proposed earlier by two of the authors and A. Gadde. We also describe the relation between global anomalies in gauge theoretic and sigma-model descriptions, filling in a gap in the present literature.

Bosonization with a background U(1) gauge field

Bosonization is one of the most significant frameworks to analyze fermionic systems. In this work, we propose a new bosonization of Dirac fermion coupled with $U(1)$ background gauge field consistent with gauge invariance, global chiral anomaly matching and fermion-boson operator correspondence, either of which is not satisfied by previously developed bosonizations. The bilinear Dirac-mass term condensation paradox and its generalized form are resolved by our bosonization. This new bosonization approach to interacting systems also correctly captures the conformal characters of a significant class of critical lattice models, such as conformal dimensions and conformal anomaly of XXZ spin chain with twisted boundary condition. Our work clarifies the equivalence of fermionic flux insertion and bosonic background charge insertion for two-dimensional conformal field theory.

Solving Gravity Anomaly Matching Problem Under Large Initial Errors in Gravity Aided Navigation by Using an Affine Transformation Based Artificial Bee Colony Algorithm.

Gravity aided inertial navigation system (GAINS), which uses earth gravitational anomaly field for navigation, holds strong potential as an underwater navigation system. The gravity matching algorithm is one of the key factors in GAINS. Existing matching algorithms cannot guarantee the matching accuracy in the matching algorithms based gravity aided navigation when the initial errors are large. Evolutionary algorithms, which are mostly have the ability of global optimality and fast convergence, can be used to solve the gravity matching problem under large initial errors. However, simply applying evolutionary algorithms to GAINS may lead to false matching. Therefore, in order to deal with the underwater gravity matching problem, it is necessary to improve the traditional evolutionary algorithms. In this paper, an affine transformation based artificial bee colony (ABC) algorithm, which can greatly improve the positioning precision under large initial errors condition, is developed. The proposed algorithm introduces affine transformation to both initialization process and evolutionary process of ABC algorithm. The single-point matching strategy is replaced by the strategy of matching a sequence of several consecutive position vectors. In addition, several constraints are introduced to the process of evolution by using the output characteristics of the inertial navigation system (INS). Simulations based on the actual gravity anomaly base map have been performed for the validation of the proposed algorithm.

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

We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\times U(1)$ gauge theories and therefore allow for two topological $\theta$-angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $\theta$-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 $\theta$-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 $\mathbb Z_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. 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 $\theta$-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 $SU(N)/U(1)^{N-1}$ 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.

Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking

We discuss consequences of the ’t Hooft anomaly matching condition for Quantum Chromodynamics (QCD) with massless fundamental quarks. We derive the new discrete ’t Hooft anomaly of massless QCD for generic numbers of color Nc and flavor Nf , and an exotic chiral-symmetry broken phase without quark-bilinear condensate is ruled out from possible QCD vacua. We show that the U(1)B baryon number symmetry is anomalously broken when the {left({mathrm{mathbb{Z}}}_{2{N}_{mathrm{f}}}right)}_{mathrm{A}} discrete axial symmetry and the flavor symmetry are gauged. In the ordinary chiral symmetry breaking, the Skyrmion current turns out to reproduce this ’t Hooft anomaly of massless QCD. In the exotic chiral symmetry breaking, however, the anomalous breaking of U(1)B does not take the correct form, and it is inconsistent with anomaly matching. This no-go theorem is based only on symmetries and anomalies, and thus has a wider range of applicability to the QCD phase diagram than the previous one obtained by QCD inequalities. Lastly, as another application, we check that duality of mathcal{N}=1 supersymmetric QCD with Nf ≥Nc + 1 satisfies the new anomaly matching.

Small instanton transitions for M5 fractions

M5-branes on an ADE singularity are described by certain six-dimensional “conformal matter” superconformal field theories. Their Higgs moduli spaces contain information about various dynamical processes for the M5s; however, they are not directly accessible due to the lack of a Lagrangian formulation. Using anomaly matching, we compute their dimensions. The result implies that M5 fractions can recombine in several different ways, where the M5s are leaving behind frozen versions of the singularity. The anomaly polynomial gives hints about the nature of the freezing. We also check the Higgs dimension formula by comparing it with various existing conjectures for the CFTs one obtains by torus compactifications down to four and three dimensions. Aided by our results, we also extend those conjectures to compactifications of theories not previously considered. These involve class mathcal{S} theories with twisted punctures in four dimensions, and affine-Dynkin-shaped quivers in three dimensions.

Global gravitational anomalies and transport

We investigate the constraints imposed by global gravitational anomalies on parity odd induced transport coefficients in even dimensions for theories with chiral fermions, gravitinos and self dual tensors. The eta-invariant for the large diffeomorphism corresponding to the T transformation on a torus constraints the coefficients in the thermal effective action up to mod 2. We show that the result obtained for the parity odd transport for gravitinos using global anomaly matching is consistent with the direct perturbative calculation. In d = 6 we see that the second Pontryagin class in the anomaly polynomial does not contribute to the eta-invariant which provides a topological explanation of this observation in the `replacement rule'. We then perform a direct perturbative calculation for the contribution of the self dual tensor in d = 6 to the parity odd transport coefficient using the Feynman rules proposed by Gaume and Witten. The result for the transport coefficient agrees with that obtained using matching of global anomalies.

The dilaton Wess–Zumino action in six dimensions from Weyl gauging: local anomalies and trace relations

We extend a previous analysis on the derivation of the dilaton Wess–Zumino (WZ) action in d = 4, based on the method of Weyl gauging, to six dimensions. As in the previous case, we discuss the structure of the same action in dimensional regularization using six-dimensional Weyl invariants, extracting the dilaton interactions in the most general scheme, with the inclusion of the local anomaly terms. As an application, we present the WZ action for the (2,0) tensor multiplet, which has been investigated in the past in the context of the AdS7/CFT6 holographic anomaly matching. We then extend to d = 6 the investigation of fully traced correlation functions of energy–momentum tensor’s, formerly presented in d = 4, showing that their hierarchy is functionally related only to the first six correlators. We give the explicit expressions of these in the most general scheme, up to rank-4.

Holography and conformal anomaly matching

We discuss various issues related to the understanding of the conformal anomaly matching in CFT from the dual holographic viewpoint. First, we act with a PBH diffeomorphism on a generic 5D RG flow geometry and show that the corresponding on-shell bulk action reproduces the Wess-Zumino term for the dilaton of broken conformal symmetry, with the expected coefficient aUV-aIR. Then we consider a specific 3D example of RG flow whose UV asymptotics is normalizable and admits a 6D lifting. We promote a modulus \rho appearing in the geometry to a function of boundary coordinates. In a 6D description {\rho} is the scale of an SU(2) instanton. We determine the smooth deformed background up to second order in the space-time derivatives of \rho and find that the 3D on-shell action reproduces a boundary kinetic term for the massless field \tau= log(\rho) with the correct coefficient \delta c=cUV-cIR. We further analyze the linearized fluctuations around the deformed background geometry and compute the one-point functions <T\mu\nu> and show that they are reproduced by a Liouville-type action for the massless scalar \tau, with background charge due to the coupling to the 2D curvature R. The resulting central charge matches \delta c. We give an interpretation of this action in terms of the (4,0) SCFT of the D1-D5 system in type I theory.

Chiral thermodynamics of dense hadronic matter

We discuss phases of hot and dense hadronic matter using chiral Lagrangians. A two-flavored parity doublet model constrained by the nuclear matter ground state predicts chiral symmetry restoration. The model thermodynamics is shown within the mean-field approximation. A field-theoretical constraint on possible phases from the anomaly matching is also discussed.

Black holes in five-dimensional gauged supergravity with higher derivatives

We examine five-dimensional $\mathcal N=2$ gauged supergravity including terms up to four derivatives. These additional terms correspond to the supersymmetric completion of $R^2$, and were originally obtained in hep-th/0611329 using conformal supergravity techniques. Here we integrate out the auxiliary fields and obtain the on-shell action for minimal supergravity with such corrections. We then construct $R$-charged AdS black holes to linear order in the four derivative terms and investigate the effect of these corrections on their thermodynamical properties. Finally, we relate the geometrical coefficients governing the four-derivative corrections to gauge theory data using holographic anomaly matching. This enables us to obtain a microscopic expression for the entropy of the solutions.

Anomalies, unparticles, and Seiberg duality

We calculate triangle anomalies for fermions with non-canonical scaling dimensions. The most well known example of such fermions (aka unfermions) occurs in Seiberg duality where the matching of anomalies (including mesinos with scaling dimensions between 3/2 and 5/2) is a crucial test of duality. By weakly gauging the non-local action for an unfermion, we calculate the one-loop three-current amplitude. Despite the fact that there are more graphs with more complicated propagators and vertices, we find that the calculation can be completed in a way that nearly parallels the usual case. We show that the anomaly factor for fermionic unparticles is independent of the scaling dimension and identical to that for ordinary fermions. This can be viewed as a confirmation that unparticle actions correctly capture the physics of conformal fixed point theories like Banks-Zaks or SUSY QCD.

Elaborations on the string dual toN=1SQCD

In this paper we make further refinements to the duality proposed between $N=1$ SQCD and certain string (supergravity plus branes) backgrounds, working in the regime of comparable large number of colors and flavors. Using the string theory solutions, we predict different field theory observables and phenomena like Seiberg duality, gauge coupling and its running, the behavior of Wilson and 't Hooft loops, anomalous dimensions of the quark superfields, quartic superpotential coupling and its running, continuous and discrete anomaly matching. We also give evidence for the smooth interpolation between the Higgs and confining vacua. We provide several matchings between field theory and string theory computations.

Anomaly matching and syzygies in N=1 gauge theories

We investigate the connection between the moduli space of N=1 supersymmetric gauge theories and the set of polynomial gauge invariants constrained by classical/quantum relations called syzygies. We examine the existence of a superpotential reproducing these syzygies and the link with the 't Hooft anomaly matching between the fundamental fields at high energy and the gauge invariant degrees of freedom at low energy for the flavour symmetry group. We show that the anomaly matching is equivalent to the vanishing of the flavour anomaly on the normal space to the manifold defined by the syzygies. For normal spaces in a real representation of the flavour group we strengthen the connection between the 't Hooft anomaly matching and the existence of a superpotential by constructing a flavour invariant polynomial whose gradient vanishes at least on the solutions of the syzygies. This corroborates a recent definition of confining theories. We illustrate our general result by considering two examples based on the SU( N c ) and Spin(7) gauge theories. We also examine the role of syzygies in the context of non-Abelian duality. We emphasize the relevance of non-perturbative effects in the dual magnetic theories in proving the equivalence of the electric and magnetic syzygies.