Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds (Q2822026)

The generalized Witten genus of a spin c manifold is the index of a certain Dirac operator twisted by certain formal power series of vector bundles. When the manifold is actually spin, this recovers the classical Witten genus which can be considered as the index of a Dirac operator on the loop space and which is proved to be vanishing in many cases.NEWLINENEWLINEThis paper presents the cohomology ring of a partial flag manifold in terms of a certain potential function, and then reduces integrals over this manifold to residue calculations, generalizing \textit{E. Witten}'s work [in: Geometry, topology and physics for Raoul Bott. Lectures of a conference in honor of Raoul Bott's 70th birthday, Harvard University, Cambridge, MA, USA 1993. Cambridge, MA: International Press. 357--422 (1995; Zbl 0863.53054)]. By using such residue calculations, the author proves the vanishing theorem for the generalized Witten genus of generalized complete intersections satisfying string c condition in partial flag manifolds.