# Triangle Nuclear Theory Colloquium - Fermionic Nomalies and Topological Phases on the Lattice: Ginsparg-Wilson Relation and its Generalizations

Abstract:

The Standard Model of particle physics has an embarrassing problem -- we cannot formulate it nonperturbatively. This comes from the difficulty of putting a chiral gauge theory on the lattice. One of the important developments motivated by this question is the Ginsparg-Wilson (GW) relation, which encodes how the anomalous chiral symmetry "optimally" manifests on the lattice.

On the other hand, developments in condensed-matter have uncovered a deep connection between anomalies and topological insulators/superconductors. At the boundary of such topological materials, you can have massless fermions with an anomalous symmetry. Domain-wall fermions used in lattice QCD simulations can be understood as a special case of this. Since domain-wall fermions are a solution to the GW equation, one may wonder -- can the GW relation and its solutions be generalized to other topological insulators and superconductors?

In this talk, we will review the story of chiral fermions on the lattice and the importance of the Ginsparg-Wilson relation. We will then discuss how it can be generalized to boundary theories of various classes of topological insulators and superconductors. Interestingly, perturbative and even some global anomalies (à la Witten) show up in this formalism in an elementary fashion from the Jacobian of the fermionic measure.

Based on https://arxiv.org/abs/2309.08542.