The Cayley Plane and the Witten Genus

TL;DR

This paper introduces the Cayley plane genus, demonstrating its factorization through elliptic and Witten genera, and characterizes the Witten genus as universal for Cayley plane bundles, revealing deep algebraic and geometric connections.

Contribution

It defines a new genus, the Cayley plane genus, and shows its relation to existing genera, providing new insights into their algebraic structure and geometric significance.

Findings

Cayley plane genus factors through elliptic and Witten genera.

Image characterized as a specific algebraic variety involving these genera.

Witten genus is universal for Cayley plane bundles.

Abstract

This paper defines a new genus, the Cayley plane genus. By definition it is the universal multiplicative genus for oriented Cayley plane bundles. The main result (Theorem 2) is that it factors (tensor Q) through the product of the Ochanine elliptic genus and the Witten genus---revealing a synergy between these two genera---and that its image is the homogeneous coordinate ring Q[Kum,HP^2,HP^3,CaP^2]/(CaP^3).(HP^3,CaP^2-(HP^2)^2) of the union of the curve of Ochanine elliptic genera and the surface of Witten genera meeting with multiplicity 2 at the point CaP^2=HP^3=HP^2=0 corresponding to the \^A-genus. This all remains true if the word "oriented" is replaced with the word "spin" (Theorem 3). This paper also characterizes the Witten genus (tensor Q) as the universal genus vanishing on total spaces of Cayley plane bundles (Theorem 1, a result proved independently by Dessai in [Des09].)