The Arf–Brown–Kervaire (ABK) Invariant in Lattice Fermion Systems

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Abstract

Topological invariants in fermionic systems provide sharp probes of symmetry and anomaly. In this talk, we study how to formulate such a topological invariant that is valued in Z_8, known as the Arf-Brown-Kervaire (ABK) invariant, for the lattice fermion systems. The ABK invariant is a two-dimensional invariant that is encoded in the complex phase of the Majorana fermion partition function, and it plays a role analogous to topological terms such as the instanton number in 4D Dirac fermion settings. We employ massive Wilson Dirac operators and numerically demonstrate that the ABK invariant emarges on the partition function. To capture the ABK invariant fully, it is essential to consider various types of manifold including non-orientable ones such as the real projective plane and the Klein bottle. In addition, manifolds with boundaries are also important for understanding anomaly and anomaly inflow. We discuss how to realize the geometries of these manifolds on the lattice and verify numerically (and partly analytically) that our formulation reproduces the known values in continuum theory.

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