On the baryon-color-flavor (BCF) anomaly in vector-like theories

Abstract

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 θ-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 θ periodicity: the massive theory will experience a phase transition as we vary θ from 0 to 2π. arXiv:1909.09027v2 [hep-th]

Generalized ’t Hooft anomalies on non-spin manifolds

“…Given 1 above, one expects that turning on gravitational background does not alter the conditions that are needed to find a set of composites in the IR matching all anomalies. At this point, it is instructive to compare the set of conditions (i) to (iii) in (3.7) with those that result from turning on BCF fluxes on T 4 , as was considered before 10 [20]. Although the two sets of conditions appear to be unrelated, they give the exact same no-go condition (3.8).…”

“…which is an integer for all the examples we consider below. Before moving to examples, it is instructive to compare and contrast the above results with the BCF fluxes on the four-torus T 4 that we considered before [20]. CP 2 has one twocycle CP 1 , and hence, we were able to turn on fluxes along this single cycle (the color and flavor fluxes are labeled by m c,f in (2.9)).…”

“…Summary: We continue our study [20] of the generalized 't Hooft anomalies in SU (N c ) gauge theories with N f flavors of Dirac fermions in representations R c of N c -ality n c . These theories have exact global discrete chiral symmetries.…”

“…Here, we slightly divert from our main presentation to note, for the sake of completeness, that by turning on global anomaly-free U (1) fluxes, chiral gauge theories can also be formulated on non-spin manifolds. 4 For the sake of completeness, we give Q c,f,B on T 4 [20]:…”

“…Notice that these are embedded into the Cartan subalgebras of SU (N c ) and SU (N f ) and represent a generalization of the BCF 't Hooft flux backgrounds on T 4 studied in [20]. When the U (1) background F = CK is replaced by (B.6), we obtain, instead of (B.1), for Ψ of unit charge under baryon number, in a representation of N c -ality n c and N f -ality n f , Ψ(s = 1) = e which now has integer values.…”