We study four-dimensional gauge theories with arbitrary simple gauge group with $1$-form global center symmetry and $0$-form parity or discrete chiral symmetry. We canonically quantize on $mathbb{T}^3$, in a fixed background field gauging the $1$-for
m symmetry. We show that the mixed $0$-form/$1$-form t Hooft anomaly results in a central extension of the global-symmetry operator algebra. We determine this algebra in each case and show that the anomaly implies degeneracies in the spectrum of the Hamiltonian at any finite-size torus. We discuss the consistency of these constraints with both older and recent semiclassical calculations in $SU(N)$ theories, with or without adjoint fermions, as well as with their conjectured infrared phases.
We study confining strings in ${cal{N}}=1$ supersymmetric $SU(N_c)$ Yang-Mills theory in the semiclassical regime on $mathbb{R}^{1,2} times mathbb{S}^1$. Static quarks are expected to be confined by double strings composed of two domain walls - which
are lines in $mathbb{R}^2$ - rather than by a single flux tube. Each domain wall carries part of the quarks chromoelectric flux. We numerically study this mechanism and find that double-string confinement holds for strings of all $N$-alities, except for those between fundamental quarks. We show that, for $N_c ge 5$, the two domain walls confining unit $N$-ality quarks attract and form non-BPS bound states, collapsing to a single flux line. We determine the $N$-ality dependence of the string tensions for $2 le N_c le 10$. Compared to known scaling laws, we find a weaker, almost flat $N$-ality dependence, which is qualitatively explained by the properties of BPS domain walls. We also quantitatively study the behavior of confining strings upon increasing the $mathbb{S}^1$ size by including the effect of virtual $W$-bosons and show that the qualitative features of double-string confinement persist.
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 Hoof
t 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 chiral-lagrangian 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.
We study the mixed anomaly between the discrete chiral symmetry and general baryon-color-flavor (BCF) backgrounds in $SU(N_c)$ gauge theories with $N_f$ flavors of Dirac fermions in representations ${cal R}_c$ of $N$-ality $n_c$, formulated on non-sp
in manifolds. We show how to study these theories on $mathbb{CP}^2$ by turning on general BCF fluxes consistent with the fermion transition functions. We consider several examples in detail and argue that matching the anomaly on non-spin manifolds places stronger constraints on the infrared physics, compared to the ones on spin manifolds (e.g.~$mathbb{T}^4$). We also show how to consistently formulate various chiral gauge theories on non-spin manifolds.
We study $SU(N_c)$ gauge theories with Dirac fermions in representations ${cal{R}}$ of nonzero $N$-ality, coupled to axions. These theories have an exact discrete chiral symmetry, which has a mixed t Hooft anomaly with general baryon-color-flavor bac
kgrounds, called the BCF anomaly in arXiv:1909.09027. The infrared theory also has an emergent $mathbb Z_{N_c}^{(1)}$ $1$-form center symmetry. We show that the BCF anomaly is matched in the infrared by axion domain walls. We argue that $mathbb Z_{N_c}^{(1)}$ is spontaneously broken on axion domain walls, so that nonzero $N$-ality Wilson loops obey the perimeter law and probe quarks are deconfined on the walls. We give further support to our conclusion by using a calculable small-circle compactification to study the multi-scale structure of the axion domain walls and the microscopic physics of deconfinement on their worldvolume.
We study the physics of quark deconfinement on domain walls in four-dimensional supersymmetric SU(N) Yang-Mills theory, compactified on a small circle with supersymmetric boundary conditions. We numerically examine the properties of BPS domain walls
connecting vacua k units apart. We also determine their electric fluxes and use the results to show that Wilson loops of any nonzero N-ality exhibit perimeter law on all k-walls. Our results confirm and extend, to all N and k, the validity of the semiclassical picture of deconfinement of Anber, Sulejmanpasic and one of us (EP), arXiv:1501.06773, providing a microscopic explanation of mixed 0-form/1-form anomaly inflow.
We consider the most general fractional background fluxes in the color, flavor, and baryon number directions, compatible with the faithful action of the global symmetry of a given theory. We call the obstruction to gauging symmetries revealed by such
backgrounds the baryon-color-flavor (BCF) t Hooft anomaly. We apply the BCF anomaly to vector-like theories, with fermions in higher-dimensional representations of arbitrary N-ality, and derive non-trivial constraints on their IR dynamics. In particular, this class of theories enjoys an independent discrete chiral symmetry and one may ask about the fate of this symmetry in the background of BCF fluxes. We show that, under certain conditions, an anomaly between the chiral symmetry and the BCF background rules out massless composite fermions as the sole player in the IR: either the composites do not form or additional contributions to the matching of the BCF anomaly are required. We can also give a flavor-symmetric mass to the fermions, smaller than or of order the strong scale of the theory, and examine the $theta$-angle periodicity of the theory in the BCF background. Interestingly, we find that the conditions that rule out the composites are the exact same conditions that lead to an anomaly of the $theta$ periodicity: the massive theory will experience a phase transition as we vary $theta$ from $0$ to $2pi$.
We discuss an exotic phase that adjoint QCD possibly exhibits in the deep infrared (IR). It is a confining phase, with a light spectrum consisting of massless composite fermions. The discrete chiral symmetry is broken, with unbroken continuous chiral
symmetry. We argue that it may give a description of the IR of adjoint QCD with three massless Weyl flavors and that it passes all consistency checks known to us.
We study the domain walls in hot $4$-D $SU(N)$ super Yang-Mills theory and QCD(adj), with $n_f$ Weyl flavors. We find that the $k$-wall worldvolume theory is $2$-D QCD with gauge group $SU(N-k)times SU(k) times U(1)$ and Dirac fermions charged under
$U(1)$ and transforming in the bi-fundamental representation of the nonabelian factors. We show that the DW theory has a $1$-form $mathbb Z_{N}^{(1)}$ center symmetry and a $0$-form $mathbb Z_{2Nn_f}^{dchi}$ discrete chiral symmetry, with a mixed t Hooft anomaly consistent with bulk/wall anomaly inflow. We argue that $mathbb Z_{N}^{(1)}$ is broken on the wall, and hence, Wilson loops obey the perimeter law. The breaking of the worldvolume center symmetry implies that bulk $p$-strings can end on the wall, a phenomenon first discovered using string-theoretic constructions. We invoke $2$-D bosonization and gauged Wess-Zumino-Witten models to suggest that $mathbb Z_{2Nn_f}^{dchi}$ is also broken in the IR, which implies that the $0$-form/$1$-form mixed t Hooft anomaly in the gapped $k$-wall theory is saturated by a topological quantum field theory. We also find interesting parallels between the physics of high-temperature domain walls studied here and domain walls between chiral symmetry breaking vacua in the zero temperature phase of the theory (studied earlier in the semiclassically calculable small spatial circle regime), arising from the similar mode of saturation of the relevant t Hooft anomalies.
We study the discrete chiral- and center-symmetry t Hooft anomaly matching in the charge-$q$ two-dimensional Schwinger model. We show that the algebra of the discrete symmetry operators involves a central extension, implying the existence of $q$ vacu
a, and that the chiral and center symmetries are spontaneously broken. We then argue that an axial version of the $q$$=$$2$ model appears in the worldvolume theory on domain walls between center-symmetry breaking vacua in the high-temperature $SU(2)$ ${cal N}$$=$$1$ super-Yang-Mills theory and that it inherits the discrete t Hooft anomalies of the four-dimensional bulk. The Schwinger model results suggest that the high-temperature domain wall exhibits a surprisingly rich structure: it supports a non-vanishing fermion condensate and perimeter law for spacelike Wilson loops, thus mirroring many properties of the strongly coupled four-dimensional low-temperature theory. We also discuss generalizations to theories with multiple adjoint fermions and possible lattice tests.
