A New SU(2) Anomaly

TL;DR

This work identifies a second, more subtle SU(2) anomaly in four-dimensional gauge theories with half-integer fermions and integer-spin bosons, detectable only when formulated without a spin structure (via a spin-SU(2) structure). The anomaly affects representations with isospin j=4r+3/2 and is tied to a five-dimensional mod 2 index and the cobordism invariant ∫ w_2 w_3, constraining consistent formulations to spin or spin_c structures. Through Higgsing to U(1) and to Z_2, the authors connect to all-fermion electrodynamics and to boundary states of 5D topological phases, providing explicit boundary constructions with emergent gauge fields or dynamical spin structures. They show the anomaly is a diffeomorphism-gauge combined effect rather than a pure gauge anomaly, and argue that only two independent 5D cobordism invariants (I_{1/2} and I_{3/2}) classify all such anomalies. The results have broad implications for dual formulations, UV completions, and possible generalizations to other gauge groups and time-reversal symmetric settings, potentially enabling new insights into gapped boundary states and topological phases in higher dimensions.

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.

Figures (4)

• Figure 1: A codimension two vortex line $L$, around which a spin structure does not extend. A fermion that is parallel transported around a small loop $\gamma$ that links with $L$ will come back with the opposite sign. The worldvolume of $L$ is a two-manifold $C$, discussed in the text.
• Figure 2: $Y^*$ is obtained by gluing $\widetilde{Y}$ and $\widetilde{Y}'$ along their common boundary $M$. The mod 2 index is additive in such a gluing, leading to eqn. (\ref{['woxo']}).
• Figure 3: The manifold $\widehat{Y}$ is made by gluing together $Y$ and $\widetilde{Y}$ along their common boundary $M$, and similarly $\widehat{Y}'$ is made by gluing together $Y$ and $\widetilde{Y}'$ along $M$, and $Y^*$ is made by gluing $\widetilde{Y}$ and $\widetilde{Y}'$ along $M$. The shaded region schematically represents a six-manifold $Z$ whose boundary is the union of $\widehat{Y}$, $\widehat{Y}'$ and $Y^*$. The existence of this cobordism establishes eqn. (\ref{['muyg']}).
• Figure 4: $M$ is cobordant to $n$ copies of $M_0$, for some $n$, via a five-manifold $\widetilde{Y}$ over which the $\mathrm{spin \text{-} SU(2)}$ structures of $M$ and $M_0$ extend. This is depicted here for $n=2$. Each copy of $M_0$ is the boundary of some $Y_0$, and $M$ is the boundary of $Y$. So $Y$, $\widetilde{Y}$, and $n$ copies of $Y_0$ glue together naturally to a closed five-manifold $\overline Y$, depicted here. This construction is used in the general definition of the amplitude $Z_{M,A,Y}$.