Witten’s top Chern class on the moduli space of higher spin curves. (English) Zbl 1105.14010

Polishchuk, Alexander

Hertling, Claus (ed.) et al., Frobenius manifolds. Quantum cohomology and singularities. Proceedings of the workshop, Bonn, Germany, July 8–19, 2002. Wiesbaden: Vieweg (ISBN 3-528-03206-5/hbk ). Aspects of Mathematics E 36, 253-264 (2004).

This article is a sequel to [A. Polishchuk, A. Vaintrob, in: Advances in algebraic geometry motivated by physics. Proc. AMS spec. sess. Univ. Mass. Lowell, MA, USA, 2000. Contemp. Math. 276, 229–249 (2001; Zbl 1051.14007)]. Its goal is to verify that the virtual top Chern class \(c^{1/r}\) in the Chow group of the moduli space of higher spin curves \(\overline{\mathcal M}_{g,n}^{1/r}\) constructed in the above mentioned article satisfies all the axioms of a spin virtual class as given in [T. J. Jarvis, T. Kimura, and A. Vaintrob, Compos. Math. 126, 157–212 (2001; Zbl 1015.14028)]. The only non-trivial axioms to be verified are the “Vanishing axiom” and the “Ramond factorization axiom”. Both properties are shown in the article under review.
For the entire collection see [Zbl 1062.14001].

Reviewer: Martin Schlichenmaier (Luxembourg)

MSC:

14C17	Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J81	Relationships between surfaces, higher-dimensional varieties, and physics
14H10	Families, moduli of curves (algebraic)
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Keywords:

moduli space of curves; Witten conjecture; spin virtual class