Condensates and anomaly cascade in vector-like theories

Abstract

We study the bilinear and higher-order fermion condensates in 4-dimensional SU(N) gauge theories with a single Dirac fermion in a general representation. Augmented with a mixed anomaly between the 0-form discrete chiral, 1-form center, and 0-form baryon number symmetries (BC anomaly), we sort out theories that admit higher-order condensates and vanishing fermion bilinears. Then, the BC anomaly is utilized to prove, in the absence of a topological quantum field theory, that nonvanishing fermion bilinears are inevitable in infrared-gapped theories with 2-index (anti)symmetric fermions. We also contrast the BC anomaly with the 0-form anomalies and show that it is the former anomaly that determines the infrared physics; we argue that the BC anomaly lurks deep to the infrared while the 0-form anomalies are just variations of local terms. We provide evidence of this assertion by studying the BC anomaly in vector-like theories compactified on a small spacial circle. These theories are weakly-coupled, under analytical control, and they admit a dual description in terms of abelian photons that determine the deep infrared dynamics. We show that the dual photons talk directly to the 1-form center symmetry in order to match the BC anomaly, while the 0-form anomalies are variations of local terms and are matched by fiat. Finally, we study the fate of the BC anomaly in the compactified theories when they are held at a finite temperature. The effective field theory that describes the low-energy physics is 2-dimensional. We show that the BC anomaly cascades from 4 to 2 dimensions.

Notes on Confinement on R3 × S1: From Yang–Mills, Super-Yang–Mills, and QCD (adj) to QCD(F)

“…The conclusion we draw from comparing (31) with (32) is that, in the long-distance abelian IR theory, in the dilute gas regime where monopole-instantons are sufficiently far away so that their cores do not overlap, the appearance of a monopole-instanton of magnetic charge q, at x ∈ R 3 , can be represented by inserting v 3 e −S 0 e iqσ(x) in the path integral of the dual-photon theory. The v 3 e −S 0 fugacity factor was already discussed in Section 2.3.…”

“…Third, the theoretically-controlled study of the symmetry realization at small-L has led to the conjectured existence of novel possible phases of various theories on R 4 , as in [30][31][32]. Admittedly, one needs dedicated numerical studies to confirm or refute these conjectured phases.…”

“…validity of semiclassics is assumed. There are further corrections, suppressed by these two small parameters, to the dimensionally reduced partition function (4.4), see [32] for a detailed discussion. Now, without much ado (see [11], also [12] for the derivation), we write the partition function and explain the ingredients and notation in some detail:…”

“…Since various topological objects play a crucial role in the calculable transition in SYM it should not come as a surprise that, in all cases, the phase transition "temperature" (c m ) al acquires topological ✓-angle dependence, due to the "topological interference" e↵ect [32] (w note that [33,34] gave earlier discussions of ✓-dependence in the deconfinement transition The ✓-dependence of the critical c m (or L cr at fixed m) was studied in [2,35] and is qualitative agreement with recent lattice studies of ✓-dependence in thermal pure YM theor see [36,37] and references therein. In [35], the ✓-dependence of another quantity was al studied-the discontinuity of the trace of the Polyakov loop at the transition, and found dependence later confirmed by the lattice [37].…”

The mixed 0-form/1-form anomaly in Hilbert space: pouring the new wine into old bottles

“…The prime example in four dimensions is the mixed anomaly between parity (or time reversal) and the 1-form center symmetry of pure Yang-Mills theory at θ = π [5,6]. A related example [4] is the mixed anomaly between discrete chiral symmetry and center symmetry in super Yang-Mills theory, also present in Yang-Mills with adjoint fermions (QCD(adj)) [7,8] and in generalizations like [9][10][11].…”