η invariant of massive Wilson Dirac operator and the index

2025/01/06 by Aoki, Shoto, Fukaya, Hidenori, Furuta, Mikio +3
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Lattice (hep-lat) #High Energy Physics - Theory (hep-th) #K-Theory and Homology (math.KT) #Strongly Correlated Electrons (cond-mat.str-el)

paper · doi:10.48550/arxiv.2501.02873

Abstract

We revisit the lattice index theorem in the perspective of K-theory. The standard definition given by the overlap Dirac operator equals to the η invariant of the Wilson Dirac operator with a negative mass. This equality is not coincidental but reflects a mathematically profound significance known as the suspension isomorphism of K-groups. Specifically, we identify the Wilson Dirac operator as an element of the K¹ group, which is characterized by the η-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this η-invariant equals to the index of the continuum Dirac operator. Our results indicate that the Ginsparg-Wilson relation and the associated exact chiral symmetry are not essential for understanding gauge field topology in lattice gauge theory.

Related

• The index of lattice Dirac operators and K-theory
• K-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators
• The index theorem of lattice Wilson--Dirac operators via higher index theory
• A comment on the index of the lattice Dirac operator and the Ginsparg-Wilson relation
• Dirac operator index and topology of lattice gauge fields