A note on torus actions and the Witten genus

TL;DR

This paper proves that the Witten genus of certain string manifolds vanishes under specific torus actions and explores the classification of quasitoric manifolds with particular cohomology rings, connecting group actions to topological invariants.

Contribution

It establishes a vanishing result for the Witten genus under torus actions and classifies quasitoric manifolds with given cohomology rings, advancing understanding of symmetries in topology.

Findings

Witten genus vanishes if a torus action's dimension exceeds the second Betti number.

Finitely many quasitoric manifolds exist with the same cohomology ring as a connected sum of complex projective spaces.

Application of torus action methods to study group actions on product manifolds.

Abstract

We show that the Witten genus of a string manifold $M$ vanishes, if there is an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply this result to study group actions on $M\times G/T$, where $G$ is a compact connected Lie group and $T$ a maximal torus of $G$. Moreover, we use the methods which are needed to prove these results to the study of torus manifolds. We show that up to diffeomorphism there are only finitely many quasitoric manifolds $M$ with the same cohomology ring as $#_{i=1}^k \pm\mathbb{C} P^n$ with $k<n$.