## 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 transform 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)-Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.

Original language | English (US) |
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Article number | 052301 |

Journal | Journal of Mathematical Physics |

Volume | 60 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2019 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics