Bounding the Heat Trace of a Calabi-Yau Manifold

TL;DR

This paper investigates bounds on the heat trace of the Laplacian on Calabi-Yau manifolds, aiming to establish spectral bounds that could constrain the moduli space of such manifolds in string theory.

Contribution

It proposes a method to find uniform bounds on the heat kernel trace for Calabi-Yau manifolds, especially near conifold singularities, to support the SCHOK bound in conformal field theories.

Findings

Eigenfunctions near conifold singularities involve confluent Heun functions.

Large curvature regions do not prevent the existence of heat trace bounds.

Spectral continuity is expected for manifolds with conical singularities.

Abstract

The SCHOK bound states that the number of marginal deformations of certain two-dimensional conformal field theories is bounded linearly from above by the number of relevant operators. In conformal field theories defined via sigma models into Calabi-Yau manifolds, relevant operators can be estimated, in the point-particle approximation, by the low-lying spectrum of the scalar Laplacian on the manifold. In the strict large volume limit, the standard asymptotic expansion of Weyl and Minakshisundaram-Pleijel diverges with the higher-order curvature invariants. We propose that it would be sufficient to find an a priori uniform bound on the trace of the heat kernel for large but finite volume. As a first step in this direction, we then study the heat trace asymptotics, as well as the actual spectrum of the scalar Laplacian, in the vicinity of a conifold singularity. The eigenfunctions can be written in terms of confluent Heun functions, the analysis of which gives evidence that regions of large curvature will not prevent the existence of a bound of this type. This is also in line with general mathematical expectations about spectral continuity for manifolds with conical singularities. A sharper version of our results could, in combination with the SCHOK bound, provide a basis for a global restriction on the dimension of the moduli space of Calabi-Yau manifolds.