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Ruan’s conjecture and integral structures in quantum cohomology. (English) Zbl 1231.14046
Saito, Masa-Hiko (ed.) et al., New developments in algebraic geometry, integrable systems and mirror symmetry. Papers based on the conference “New developments in algebraic geometry, integrable systems and mirror symmetry”, Kyoto, Japan, January 7–11, 2008, and the workshop “Quantum cohomology and mirror symmetry”, Kobe, Japan, January 4–5, 2008. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-62-4/hbk). Advanced Studies in Pure Mathematics 59, 111-166 (2010).
This is an expository article on the current status of the generalized Crepant Resolution conjecture of Ruan. The conjecture says basically that small (orbifold) quantum cohomologies of two birational varieties $$X_1$$, $$X_2$$ in a “crepant” relationship (crepant resolutions, flops, etc.) are isomorphic under a change of coordinates and analytic continuation. The underlying picture is that of a conjectural Kähler moduli space $$\mathcal{M}$$ with “cusps” at the classical cohomologies of $$X_1$$, $$X_2$$ connected by a path of quantum deformations. Orbifold quantum cohomology is reviewed in the first part of the paper.
The second part expands upon a stronger version of the conjecture postulating a meromorphic flat connection over $$\mathcal{M}$$, called global quantum $$D$$–module, that restricts to Dubrovin connections in a neighborhood of the cusps and carries more information than deformed cohomology rings. Together with additional structure determined by local monodromies around the cusps this connection defines a “flat structure” that turns $$\mathcal{M}$$ into a Frobenius manifold. Different cusps may induce different flat structures necessitating the change of coordinates absent in the original version of the Crepant Resolution conjecture. The paper proves a characterization, announced in [T. Coates, H. Iritani, H.-H. Tseng, Geom. Topol. 13, No. 5, 2675–2744 (2009; Zbl 1184.53086)], of flat structures induced by different cusps, and shows that generalized Hard Lefschetz condition is sufficient for them to match.
The paper further conjectures existence of integral local systems underlying the global quantum $$D$$–module, which come from the $$K$$–theory of $$X_i$$. From the physical point of view this is because flat sections of the quantum $$D$$–module (chiral fields in the A–model) are paired with vector bundles over $$X_i$$ (D–branes in the B–model), i.e. with elements of the $$K$$–group. A path in $$\mathcal{M}$$ connecting the cusps should therefore induce an isomorphism of $$K$$–groups, and this isomorphism contains complete information on the relationship between small quantum cohomologies of $$X_i$$. Based on this conjecture, the author predicts required coordinate changes in the Crepant Resolution conjecture for Calabi-Yau orbifolds.
For the entire collection see [Zbl 1200.14002].

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 83E30 String and superstring theories in gravitational theory
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