Quantum Cohomology by K. Behrend, C. Gomez, V. Tarasov, G. Tian, P. de Bartolomeis

By K. Behrend, C. Gomez, V. Tarasov, G. Tian, P. de Bartolomeis

The e-book gathers the lectures given on the C.I.M.E. summer season university "Quantum Cohomology" held in Cetraro (Italy) from June thirtieth to July eighth, 1997. The lectures and the next updating conceal a wide spectrum of the topic at the box, from the algebro-geometric viewpoint, to the symplectic method, together with contemporary advancements of string-branes theories and q-hypergeometric capabilities.

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Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. 10. W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. In Algebraic geometry—Santa Cruz 1995, pages 45-96. Amer. Math. Soc, Providence, RI, 1997. 11. A. Givental. Equivariant Gromov-Witten invariants. Internat. Math. Res. Notices, 1996(13):613-663. 12. T. Graber a n d R . Pandharipande. Localization of virtual classes. , 135:487-518, 1999. 13. M. Kontsevich. Enumeration of rational curves via torus actions. In The moduli space of curves (Texel Island, 1994), pages 335-368.

No edge connects stable vertices, 7 is constant on edges, if v is a stable vertex then j(v) = y(i), for all i 6 FT(v), if v is an unstable vertex then a) 7(4) G 7(1;), for all i G FT{v), b) all 7(4), for i G FT(v) are distinct. Fix such a marked modular graph (r, d, 7). The following stacks will be important in what follows: 1. ~M(TS) = YlvevT (pr x wr)E-, where the vertical maps are evaluation maps, 3. M(Fr,r,d;ry), which is the substack of M(Fr,r,d) defined by requiring that /a(j) (xi) = j(i) 6 {PQ, .

Introduction The great challenge of high energy theoretical physics is finding a consistent theory of quantum gravity. For the time being, string theory is the best candidate at hand. Many phisicists think that the solution to quantum gravity will have little, if any, practical implications in our daily way of doing physics; others, more optimistic, or simply with a less practical approach to science, hope that the forthcoming theory of quantum gravity will provide a new way of thinking of quantum physics.

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