Anomaly interplay in U(2) gauge theories

Abstract

We discuss anomaly cancellation in U(2) gauge theories in four dimensions. For a U(2) gauge theory defined with a spin structure, the vanishing of the bordism group Ω Spin 5 (BU(2)) implies that there can be no global anomalies, in contrast to the related case of an SU(2) gauge theory. We show explicitly that the familiar SU(2) global anomaly is replaced by a local anomaly when SU(2) is embedded in U(2). There must be an even number of fermions with isospin 2r +1/2, for r ∈ Z ≥0 , for this local anomaly to cancel. The case of a U(2) theory defined without a choice of spin structure but rather using a spin-U(2) structure, which is possible when all fermions (bosons) have half-integer (integer) isospin and odd (even) U(1) charge, is more subtle. We find that the recently-discovered 'new SU(2) global anomaly' is also equivalent, though only at the level of the partition function, to a perturbative anomaly in the U(2) theory, which is this time a combination of a mixed gauge anomaly with a gauge-gravity anomaly. This perturbative anomaly vanishes if there is an even number of fermions with isospin 4r + 3/2, for r ∈ Z ≥0 , recovering the condition for cancelling the new SU(2) anomaly. Alternatively, this perturbative anomaly can be cancelled by a Wess-Zumino term, leaving a low-energy theory with a global anomaly, which can itself be cancelled by coupling to topological degrees of freedom.

The algebra of anomaly interplay

“…Two examples of this interplay were recently observed in U(2) vs. SU(2) × U(1) gauge theories in four dimensions [16], defined with or without spin structures. In the spin case, a bordism computation reveals that the U(2) theory cannot suffer from global anomalies, 4 but nonetheless a U(2) theory with a perturbative anomaly can pullback to an SU(2) theory with a global anomaly.…”

“…Note that in 2 dimensions, unlike in 4, conjugating a complex fermion does not flip its chirality; hence we cannot now take all the Weyls to be left-moving without loss of generality. 16 We remark that this /8-valued global anomaly, for a 2d theory with unitary /2 symmetry, is related to a parity anomaly for (0+1)-dimensional Majorana fermions [36]. This can be understood in terms of the Smith isomorphism [37] (see also [38]).…”

“…For our next example, we move up two dimensions and revisit the anomaly interplay between U(2) and SU(2) gauge theories in d = 4, each defined using a spin structure [16] (see also [47]). The map π : SU(2) → U(2) will denote the usual embedding of SU(2) ⊂ U(2) as the subgroup of 2-by-2 unitary matrices that have determinant one.…”

“…In the rest of the paper we discuss a number of examples exhibiting the phenomenon of anomaly interplay, ascending the ladder of increasing dimension. We begin by analyzing the interplay between anomalies in U(1) and /2 gauge theories in two dimensions, followed by the examples of U(2) gauge theories in four dimensions that were recently analyzed in [16]. We close just as we begin, with an example of anomaly interplay between U(1) and /2 gauge theories, but this time in six dimensions.…”

Revisiting Wess-Zumino-Witten terms

“…4] already contains a detailed explanation of the transgression associated to the special but important case G → EG → BG. 26 If we also consider the baryonic U(1) symmetry and consider the QCD on a manifold with spin c structure, the global anomaly of Witten instead comes from the anomaly polynomial [63]. Then its transgression can be analyzed at the level of differential forms, and results in the ungauged WZW term (2.31) discussed in the latter part of Sec.…”