The complex genera, symmetric functions and multiple zeta values (Q6543054)

Let \(\lambda=(\lambda_1,\dots,\lambda_l)\) be an integer partition. Denote by \(l=l(\lambda)\) and \(|\lambda|=\sum_{i=1}^l\lambda_i \) respectively the length and weight of the partition \(\lambda\). In this paper under review, the author studies the coefficients in front of Chern numbers for complex genera, and pay special attention to the \(Td^{\frac{1}{2}}\)-genus whose formal power series is \[Q(x)=\left(\frac{x}{x-e^{-x}}\right)^{\frac{1}{2}}, \] the square root of the usual Todd genus. More precisely, it is shown that the coefficients \(b_{2\lambda}(Td^{\frac{1}{2}})\) of the Chern numbers \(C_{2\lambda}(\cdot)\) in the \(Td^{\frac{1}{2}}\)-genus are given by\N\[\Nb_{2\lambda}(Td^{\frac{1}{2}})=\frac{(-1)^{|\lambda|-l(\lambda)}}{(2\pi)^{2|\lambda|}\prod_i m_i(\lambda)!}\cdot \zeta_S^\star(2\lambda_1,\dots,2\lambda_{l(\lambda)}), \N\] \Nwhere \(\zeta^\star_S\) is the multiple-star zeta values (MSZVs) and \(m_i(\lambda)\) denotes the multiplicity of \(i\) which appears among the parts \(\lambda_i\).\N\NFinally, the author gives a closed formula for the coefficients \(b_\lambda(\Gamma)\) of the Chern numbers \(C_\lambda[\cdot]\) in the \(\Gamma\)-genus when \(m_1(\lambda)=0\) \[ b_\lambda(\Gamma)=\frac{\zeta_S(\lambda_1,\dots,\lambda_{l(\lambda)})}{\prod_i m_i(\lambda)!}\] which is always positive. In particular, these determine all the coefficients for Calabi-Yau manifolds.