We show that $2$d adjoint QCD, an $SU(N)$ gauge theory with one massless adjoint Majorana fermion, has a variety of mixed 't Hooft anomalies. The anomalies are derived using a recent mod $2$ index theorem and its generalization that incorporates 't Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of $N$, are matched by spontaneous chiral symmetry breaking. We find that massless $2$d adjoint QCD confines for $N >2$, except for test charges of $N$-ality $N/2$, which are deconfined. In other words, $\mathbb Z_N$ center symmetry is unbroken for odd $N$ and spontaneously broken to $\mathbb Z_{N/2}$ for even $N$. All of these results are confirmed by explicit calculations on small $\mathbb{R}\times S^1$. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when $N$ is even. Furthermore, for most values of $N$, $2$d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of $8$. As a result, it fits into the classification of $(1+1)$d SPT phases of interacting Majorana fermions in an interesting way.
Generalized Konishi anomaly relations in the chiral ring of N=1 supersymmetric gauge theories with unitary gauge group and chiral matter field in two-index tensor representations are derived. Contrary to previous investigations of related models we do not include matter multiplets in the adjoint representation. The corresponding curves turn out to be hyperelliptic. We also point out equivalences to models with orthogonal or symplectic gauge groups.
't Hooft anomaly matching is powerful for constraining the low energy phases of gauge theories. In 3d one common anomaly is the parity anomaly in a $T$-symmetric theory where one cannot gauge the global symmetry group without breaking the time-reversal symmetry. We find that a $T$-symmetric $\text{SU}(N)$ gauge theory with either fermionic or bosonic matter in the fundamental representation of the gauge group has a parity anomaly between the flavor group and $T$-symmetry provided that there is also a massless Majorana fermion in the adjoint representation of the gauge group. We then analyze th…
Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^μ(\equiv θ^{μν}p_ν) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.
We discuss both the UV and IR origin of the one-loop triangle gauge anomalies for noncommutative nonabelian chiral gauge theories with fundamental, adjoint and bi-fundamental fermions for U(N) groups. We find that gauge anomalies only come from planar triangle diagrams, the non-planar triangle contributions giving rise to no breaking of the Ward identies. Generally speaking, theories with fundamental and bi-fundamental chiral matter are anomalous. Theories with only adjoint chiral fermions are anomaly free.
We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin's classification of Lie subalgebras of simple Lie algebras.
We study electric-magnetic duality in the chiral ring of a supersymmetric U(N_c) gauge theory with adjoint and fundamental matter, in presence of a general confining phase superpotential for the adjoint and the mesons. We find the magnetic solution corresponding to both the pseudoconfining and higgs electric vacua. By means of the Dijkgraaf-Vafa method, we match the effective glueball superpotentials and show that in some cases duality works exactly offshell. We give also a picture of the analytic structure of the resolvents in the magnetic theory, as we smoothly interpolate between different …
Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We pr…
We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in the adjoint representation. We apply general considerations of symmetries, anomalies, and renormalization group flows to determine the possible phases of the theory as a function of the gauge group, the Chern-Simons level $k$, and the fermion mass. We propose an inherently quantum mechanical phase of adjoint QCD with small enough $k$, where the infrared is described by a certain Topological Quantum Field Theory (TQFT). For a special choice of the mass, the theory has ${\cal N}=1$ supersymmetry. There this TQFT is…
We study gauge and gravitational anomalies of fermions and 2-form fields on eight-dimensional spin manifolds. Possible global gauge anomalies are classified by spin bordism groups $Ω^{\text{spin}}_9(BG)$ which we determine by spectral sequence techniques, and we also identify their explicit generator manifolds. It turns out that a fermion in the adjoint representation of any simple Lie group, and a gravitino in $8d$ $\mathcal{N}=1$ supergravity theory, have anomalies. We discuss how a 2-form field, which also appears in supergravity, produces anomalies which cancel against these fermion anomal…
It is known that the 't Hooft anomalies of invertible global symmetries can be characterized by an invertible TQFT in one higher dimension. The analogous statement remains to be understood for non-invertible symmetries. In this note we discuss how the linking invariants in a non-invertible TQFT known as the Symmetry TFT (SymTFT) can be used as a diagnostic for 't Hooft anomalies of non-invertible symmetries. When the non-invertible symmetry is non-intrinsically non-invertible, and hence the SymTFT is a Dijkgraaf-Witten model, the linking invariants can be computed explicitly. We illustrate thi…
We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group $SU(N)$. The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits $\sim 2^{2N}$ non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all $N$ these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, $m\ll g_\mathrm{YM}$, the theory confines and the non-invertible symmetries are softly broken. We use them to compute anal…
We propose quantum dynamics for the dipole moving in cosmic string background and show that the classical scale symmetry of a particle moving in cosmic string background is still restored even in the presence of dipole moment of the particle. However, we show that the classical scale symmetry is broken due to inequivalent quantization of the the non-relativistic system. The consequence of this quantum anomaly is the formation of bound state in the interval ξ\in(-1,1). The inequivalent quantization is characterized by a 1-parameter family of self-adjoint extension parameter Σ. We show that with…
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 the dynamics of chiral SU(N) gauge theories. These contain Weyl fermions in the symmetric or anti-symmetric representation of the gauge group, together with further fermions in the fundamental and anti-fundamental. We revisit an old proposal of Bars and Yankielowicz who match the 't Hooft anomalies of this theory to free fermions. We show that there are novel and, in some cases, quite powerful constraints on the dynamics in the large N limit. In addition, we study these SU(N) theories with an extra Weyl fermion transforming in the adjoint representation. Here we show that all 21 't Ho…
Anomalies are renormalization group invariants that constrain the dynamics of quantum field theories. We show that certain anomalies for discrete global symmetries imply that the underlying theory either spontaneously breaks its generalized global symmetry or is gapless. We identify an obstruction, formulated in terms of the anomaly inflow action, that must vanish if a symmetry preserving gapped phase, i.e. a unitary topological quantum field theory, exits with the given anomaly. Our result is similar to the $2d$ Lieb-Schultz-Mattis theorem but applies more broadly to continuum theories in gen…
Consequences of gauging exact $Z_k^C$ center symmetries in several simple $SU(N)$ gauge theories, where $k$ is a divisor of $N$, are investigated. Models discussed include: the $SU(N)$ gauge theory with $N_f$ copies of Weyl fermions in self-adjoint single-column antisymmetric representation, the well-discussed adjoint QCD, QCD-like theories in which the quarks are in a two-index representation of $SU(N)$, and a chiral $SU(N)$ theory with fermions in the symmetric as well as in anti-antisymmetric representations but without fundamentals. Mixed 't Hooft anomalies between the 1-form $Z_k^C$ symme…
We consider confining strings in pure gluodynamics and its extensions with adjoint (s)quarks. We argue that there is a direct map between the set of bulk fields and the worldsheet degrees of freedom. This suggests a close link between the worldsheet $S$-matrix and parton scattering amplitudes. We report an amusing relation between the Polchinski--Strominger amplitude responsible for the breakdown of integrability on the string worldsheet and the Yang--Mills $β$-function \[ b_0={D_{cr}-D_{ph}\over 6}\;. \] Here $b_0=11/3$ is the one-loop $β$-function coefficient in the pure Yang--Mills theory, …
We explicitly calculate the topological terms that arise in IR effective field theories for $SU(N)$ gauge theories on $\mathbb{R}^3 \times S^1$ 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 ma…
We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional $\mathcal{N}=(0,1)$ supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the $B$-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQF…
Anomalies of discrete R-symmetries appearing in heterotic orbifold models are studied. We find that the mixed anomalies for different gauge groups satisfy the universal Green-Schwarz (GS) condition, indicating that these anomalies are canceled by the GS mechanism. An exact relation between the anomaly coefficients of the discrete R-symmetries and one-loop beta-function coefficients is obtained. We also find that the discrete R-symmetries have a good chance to be unbroken down to the supersymmetry breaking scale. Even below this scale a $Z_2$ subgroup is unbroken, which may be an origin of the …
We propose to induce QCD by fermions in the adjoint representation of the gauge group SU(N_c) on the lattice. We consider various types of lattice fermions: chiral, Kogut--Susskind and Wilson ones. Using the mean field method we show that a first order large-N phase transition occurs with decreasing fermion mass. We conclude, therefore, that adjoint fermions induce QCD. We draw the same conclusion for the adjoint scalar or fermion models at large number of flavors N_f when they induce a single-plaquette lattice gauge theory. We find an exact strong coupling solution for the adjoint fermion mod…
Gauge theories with fermions in adjoint and fundamental representations are relevant for many different applications including composite Higgs models and general aspects of the confinement problem. We present first results from simulations of SU(2) gauge theory with two Dirac fermions in the fundamental representation and one adjoint flavor. In this context, we also discuss applications towards simulations of supersymmetric QCD.
We explore the possibility of embedding supersymmetric GUT texture ideas into superstring models. We discuss the construction of GUT models using free fermionic strings. We find SO(10) models with adjoint scalars, three generations of chiral fermions, and a fundamental Higgs with a Yukawa coupling to only the third generation.
It is shown that anti-BRST symmetry is the quantized counterpart of local axial symmetry in gauge theories. An extended form of descent equations is worked out which yields a set of modified consistent anomalies.
We provide a rigorous construction of the Schwinger functions for the massive and massless Thirring models. We use the renormalization group approach, controlling its flow through the Ward-Takahashi identities combined with the Schwinger-Dyson equation. The role of the anomalies is discussed.
We derive analytic formulas for the zero-modes of the Dirac equation in the adjoint representation in the background field of Q=1 SU(N) calorons. Solutions with various boundary conditions are obtained, including the physically most relevant cases of periodic and antiperiodic ones. The latter are essential ingredients in a semiclassical treatment of finite temperature supersymmetric Yang-Mills theory. A detailed discussion of adjoint zero-modes in several other contexts is also presented.
