Joint Theory Seminar: Simon Catterall

Title: Sneaking up on lattice chiral fermions

Abstract: It is widely believed that lattice theories cannot exhibit anomalies. I will give a counter example that refutes this folklore. The anomaly in question is a gravitational anomaly and involves a novel type of fermion - the Kaehler-Dirac fermion. In flat space the Kaehler-Dirac equation describes four degenerate Dirac fermions but this is no longer true in the presence of gravity. The key advantages of the Kaehler-Dirac equation is that it may be discretized on arbitrary curved spaces without encountering fermion doubling and while preserving a certain twisted $U(1)$ chiral symmetry. In this way it does an end run around the Nielsen-Ninomiya theorem. In fact not only is the symmetry retained upon discretization but so is a corresponding mixed gravitational anomaly which breaks $U(1)\to Z_4$. If one attempts to gauge this remaining $Z_4$ symmetry one discovers a mod 2 't Hooft anomaly which is only canceled for multiples of two Kaehler-Dirac fields. In flat space one can decompose this anomaly free model into spinors and one finds the symmetries and matter representations of the Pati-Salam GUT - a chiral gauge theory containing the Standard Model. Since the free Kaehler-Dirac equation in flat space can be mapped into staggered fermions this suggests that it may be possible to give a non-perturbative definition of this chiral gauge theory in terms of a path integral over staggered fermions. I will discuss the remaining barriers to such a construction.