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Algebraic orbifold quantum products. (English) Zbl 1067.14055
Adem, Alejandro (ed.) et al., Orbifolds in mathematics and physics. Proceedings of a conference on mathematical aspects of orbifold string theory, Madison, WI, USA, May 4–8, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2990-4/pbk). Contemp. Math. 310, 1-24 (2002).
Near the end of the 1990’s, D. Abramovich and A. Vistoli developed some very natural and powerful algebraic techniques for compactifying the space of stable maps into a Deligne-Mumford (DM) stack, or orbifold, by allowing the source curves of the maps to be orbifolds themselves [J. Am. Math. Soc. 15, No. 1, 27–75 (2002; Zbl 0991.14007)]. Such maps are called twisted stable maps. At roughly the same time, W. Chen and Y. Ruan, inspired by physicists’ orbifold string theories, began to develop an analytic form of orbifold Gromov-Witten theory [in: Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math. 310, 25–85 (2002; Zbl 1091.53058)]. A key part of that theory was the recognition that the usual cohomology of the underlying space was inadequate for the theory. What is needed instead is a larger ring $$H^{*}_{\text{orb}}$$, called the orbifold cohomology, which, as a vector space, is simply the direct product of the cohomology of various “sectors” (roughly, the fixed point loci of the isotropy groups) of the orbifold. Chen and Ruan developed a “stringy product” on this larger space $$H^{*}_{\text{orb}}$$ which, even in degree zero, is an important invariant of the orbifold.
This paper gives an overview of the authors’ work defining an algebraic version of Chen and Ruan’s orbifold Gromov-Witten theory, with special focus on the stringy, or orbifold, quantum product. It is based primarily on the algebraic theory of twisted stable maps of the first paper cited above. The results are valid for DM stacks over any field of characteristic 0. A surprising consequence of this work is the construction of a stringy product in degree zero with integer coefficients, rather than rational coefficients. Although the paper is just an overview, it is currently the only source for much of this material. Fortunately, the authors have been very careful in their constructions and arguments, and most of the important steps are clearly sketched so the reader may fill in the details.
For the entire collection see [Zbl 1003.00015].

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14A20 Generalizations (algebraic spaces, stacks) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
##### Keywords:
stack; quantum product; moduli; stringy; stable map
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