The complex genera, symmetric functions and multiple zeta values

TL;DR

This paper explores the coefficients of complex genera, especially the Todd-related genera, and their connections to symmetric functions and multiple zeta values, with applications to hyper-Kähler and Calabi-Yau manifolds.

Contribution

It introduces a unified framework for Hoffman-type formulas and transition matrices among symmetric function bases, linking complex genera to multiple zeta values.

Findings

Derived new formulas for multiple zeta values.

Connected complex genera to geometric structures like hyper-Kähler manifolds.

Provided a general approach to symmetric function bases transition matrices.

Abstract

We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the $\text{Td}^{\frac{1}{2}}$-genus, the $\Gamma$-genus as well as the Todd genus. Some related geometric applications to hyper-K\"{a}hler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework.