A New SU(2) Anomaly

TL;DR

This paper identifies a subtle SU(2) anomaly affecting certain four-dimensional gauge theories with fermions in specific half-integer spin representations, which can be understood via five-dimensional topological invariants.

Contribution

It uncovers a new SU(2) anomaly for theories with fermions in representations of spin 4r+3/2, extending the understanding of gauge anomalies beyond the familiar cases.

Findings

The anomaly affects theories without requiring a spin structure.

Higgsing to U(1) reduces the anomaly to known all-fermion electrodynamics anomaly.

The anomaly relates to a five-dimensional topological invariant (-1)^{∫_Y w_2 w_3}.

Abstract

A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r+1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r+3/2, is inconsistent. The theory is consistent if one picks a spin or spin_c structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of "all-fermion electrodynamics." Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r+3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is $(-1)^{\int_Y w_2w_3}$. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.