Tadpoles and commutative spectral triples

TL;DR

This paper demonstrates the absence of tadpoles in spectral actions for compact spin manifolds without boundary, using noncommutative geometry techniques, and extends analysis to cases with chiral boundary conditions.

Contribution

It provides a rigorous proof of the nonexistence of tadpoles in spectral triples and explores boundary condition effects using pseudodifferential methods.

Findings

No tadpoles of any order for compact spin manifolds without boundary

Zero terms in spectral actions are identified and tracked

Analysis extended to chiral boundary conditions

Abstract

Using the Chamseddine--Connes approach of the noncommutative action on spectral triples, we show that there are no tadpoles of any order for compact spin manifolds without boundary, and also consider a case of a chiral boundary condition. Using pseudodifferential techniques, we track zero terms in spectral actions.