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

Abstract

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 𝕋3, in a fixed background field gauging the 1-form 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.

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

“…Fourth, the small-L theories offer an interesting arena to study the realization, at various energy scales [35][36][37][38], of the recently discovered 't Hooft anomalies involving traditional global symmetries ("0-form") as well as higher-form symmetries [15][16][17]. One's hope, then, is that this will lead to better understanding of these anomalies and their implications.…”

“…We can now take stock and discuss what the result (38) represents. We immediately notice that the dual photon field acquires a mass.…”

“…We immediately notice that the dual photon field acquires a mass. As written, the potential in (38) has a unique minimum at σ = 0 (recall that σ has a 2π periodicity). Expanding the potential around the minimum, we see that the mass of the dual photon is…”

“…To end this Section, we rewrite (38) in terms of the dual photon mass m σ , absorbing the inessential (for the present discussion) coefficient into its definition, a form that we shall use from now on:…”

Nonperturbative effects in the Standard Model with gauged 1-form symmetry

“…9 To avoid confusion, we stress that one can, even in the G 1 SM theory, introduce boundary conditions on the T 4 that correspond to nonzero twists n (j) µν , but without summation over all allowed twists in the path integral. This corresponds to probing the G 1 SM theory by the insertion of nondynamical backgrounds for the Z (1) 6 global symmetry, needed in studies of generalized 't Hooft anomalies involving center symmetry, as shown using this language in [32]. An unambiguous identification of the different SM gauge groups with the presence or absence of fractional instanton contributions requires forbidding such nondynamical insertions.…”