Witten's top Chern class via K-theory

TL;DR

This paper introduces a simplified K-theory-based construction of Witten's top Chern class, avoiding complex intersection theory, and demonstrates its natural lift to the K-theory ring, advancing the algebraic understanding of higher spin curves.

Contribution

It provides a more straightforward K-theory approach to constructing Witten's top Chern class, bypassing complex intersection theory methods.

Findings

Constructed Witten's top Chern class via K-theory

Showed the class admits a natural K-theory lift

Simplified the algebraic construction process

Abstract

The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand-Dikii hierarchies to higher spin curves. In math.AG/0011032, Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward construction via K-theory. In this way we short-circuit the passage through bivariant intersection theory and the use of MacPherson's graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the K-theory ring.