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 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 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.
