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Algebraic construction of Witten’s top Chern class. (English) Zbl 1051.14007
Previato, Emma (ed.), Advances in algebraic geometry motivated by physics. Proceedings of the AMS special session on enumerative geometry in physics, University of Massachusetts, Lowell, MA, USA, April 1–2, 2000. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2810-X/pbk). Contemp. Math. 276, 229-249 (2001).
The article deals with the compactified moduli space $$\overline{\mathcal M}_{g,n}^{1/r}$$ of stable algebraic curves of genus $$g$$ with $$n$$ marked points and an $$r$$-spin structure, (i.e. a choice of an $$r$$-th root of the canonical bundle of the curve). Conjecturally [E. Witten, Topological methods in modern mathematics, 235–269 (1993; Zbl 0812.14017)], its intersection theory should be related to the Gelfand-Dickey hierarchy. To this end there should exists a certain virtual top Chern class $$c^{1/r}$$, with respect to it the intersection numbers are calculated. T. J. Jarvis, T. Kimura and A. Vaintrob [Compos. Math. 126, 157–212 (2001; Zbl 1015.14028)] formulated axioms to be fulfilled by such a class. This class is called spin virtual class. For the case $$g=0$$ (and $$r$$ arbitrary) and $$r=2$$ (and $$g$$ arbitrary) they also showed its existence.
In the article under review the authors give an algebraic construction of a Chow cohomology class which they propose as candidate for the spin virtual class. Their class fulfills all axioms of the above mentioned article, besides possibly the vanishing axiom. It is shown that for the cases considered by Jarvis, Kimura, and Vaintrob it gives the same answer. A modification of MacPherson’s graph construction for the case of 2-periodic (unbounded) complexes is given. It is used to define an analogue of the localized top Chern class for an orthogonal bundle with an isotropic section. This is applied to orthogonal bundles related to families of $$r$$-spin structures, yielding the required virtual class. Some geometric consequences of the vanishing of the virtual top Chern class in the case of 2-spin structures are given.
For the entire collection see [Zbl 0966.00024].

MSC:
 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 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 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 14H70 Relationships between algebraic curves and integrable systems 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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