Spectral action for a one-parameter family of Dirac-type operators on SU(2) and its inflation model

TL;DR

This paper computes the spectral action for a family of Dirac operators on SU(2), deriving an inflation model from the asymptotic expansion, and provides a method applicable to other Lie groups.

Contribution

It explicitly calculates the spectrum of Dirac Laplacians on SU(2) and derives an inflationary potential from the spectral action, connecting spectral geometry with cosmological models.

Findings

Explicit spectrum computation for Dirac Laplacians on SU(2)

Full asymptotic expansion of the spectral action obtained

Generated inflation potential and slow-roll parameters from spectral data

Abstract

We analyze the Dirac Laplacian of a one-parameter family of Dirac operators on a compact Lie group, which includes the Levi-Civita, cubic, and trivial Dirac operators. More specifically, we describe the Dirac Laplacian action on any Clifford module in terms of the action of the Lie algebra's Casimir element on finite-dimensional irreducible representations of the Lie group. Using this description of the Dirac Laplacian, we explicitly compute spectrum for the one-parameter family of Dirac Laplacians on SU(2), and then using the Poisson summation formula, the full asymptotic expansion of the spectral action. The technique used to explicitly compute the spectrum applies more generally to any Lie group where one can concretely describe the weights and corresponding irreducible representations, as well as decompose tensor products of an irreducible representation with the Weyl representation into irreducible components. Using the full asymptotic expansion of the spectral action, we generate the inflation potential and slow-roll parameters for the corresponding pure gravity inflationary theory.