Speaker
Description
Topological invariants and their associated anomalies have played a crucial role in understanding low-energy phenomena in quantum field theories. In lattice gauge theory, the standard $\mathbb{Z}$-valued Atiyah–Singer index is formulated via the overlap Dirac operator through the Ginsparg–Wilson relation, but extensions to more general topological invariants have remained limited. In this work, we propose a lattice formulation of the Arf–Brown–Kervaire (ABK) invariant, which takes values in $\mathbb{Z}_8$. The ABK invariant arises in Majorana fermion partition functions with reflection symmetry on two-dimensional unoriented manifolds, and its definition involves an infinite sum over Dirac eigenvalues that must be properly regularized. By carefully treating the boundary conditions, with and without a domain-wall mass term, we demonstrate that the ABK invariant can be extracted from Pfaffians of the Wilson Dirac operator. We further provide numerical verification on two-dimensional lattices, showing that the $\mathbb{Z}_8$-valued results on the torus, Klein bottle, real projective plane, and Möbius strip agree with those in the continuum theory.
| Parallel Session (for talks only) | Theoretical developments and applications beyond Standard Model |
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