The truncated Witten genus
In this very interesting paper, the author studies the truncated Witten genus. To describe his results, let \(M\) be a closed, connected, spin manifold of even dimension \(d\), \(p\) be a positive integer, and \(C_p \subseteq \mathbb{S}^1\) be the group of \(p\)-th roots of unity. The author shows that the space of maps \(\text{Map}(C_p, M)\) has a spin structure to which the natural \(C_p\)-action on \(\text{Map}(C_p,M)\) lifts. In particular, the Dirac operator \(\eth_p\) on \(\text{Map}(C_p,M)\) is \(C_p\)-equivariant and Fredholm. The Witten genus of \(M\) truncated at \(p\), \(W_p(M)\), is defined as follows: Given a generator \(\xi \in C_p\), \[ W_p(M) := \text{Index}_\xi\, \eth_p = \text{Tr}\, \xi| _{\ker \eth_p} - \text{Tr}\, \xi| _{\text{coker}\, \eth_p} . \] In [Lect. Notes Math. 1326, 161--181 (1988; Zbl 0679.58045)], \textit{E. Witten} applied the equivariant Atiyah-Singer index theorem to the smooth loop space \(\text{Map}(\mathbb{S}^1,M)\) and showed that assuming there is a Dirac operator on this loop space, then {formally} \[ \text{``} \text{Index}_\xi\, \eth_{\text{Map}(\mathbb{S}^1,M)} \text{''} = (-1)^{d/2} \xi^{-d/24} \left< \widehat{\mathcal{A}}(TM) \, \text{ch}\, \left( \bigotimes_{k > 0} (\Lambda_{-\xi^k} T_{\mathbb{C}} M)^{-1} \right) \;, \;[M] \right> . \] In the case when \(p = 2m + 1\) is odd, the author proves the following theorem, which gives a deep and precise statement of Witten's result. Theorem 1. Let \(\xi\) be a primitive \(2m + 1\)-th root of unity. The Witten genus truncated at \(2m + 1\), \(W_{2m + 1}(M)\), is given by the formula \[ (-1)^{(m+1)d/2} \xi^{dm(m+1)/4} \left< \widehat{\mathcal{A}}(TM) \, \text{ch}\, \left( \bigotimes_{k = 0}^m (\Lambda_{-\xi^k} T_{\mathbb{C}} M)^{-1} \right) \;, \;[M] \right> . \]
scientific article; zbMATH DE number 2156064