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Tensor products of Frobenius manifolds and moduli spaces of higher spin curves. (English) Zbl 0988.81120
Dito, Giuseppe (ed.) et al., Conférence Moshé Flato 1999: Quantization, deformations, and symmetries, Dijon, France, September 5-8, 1999. Volume II. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 22, 145-166 (2000).
Summary: We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the $$\tau$$ function of a semiclassical limit of the $$r$$-th Gelfand-Dickey (or $$\text{KdV}_r$$) hierarchy. Additionally, we prove that tensor products of the Frobenius manifolds associated to such hierarchies admit a geometric interpretation in terms of moduli spaces of higher spin structures. We also elaborate upon the analogy with Gromov-Witten invariants of a smooth, projective variety.
For the entire collection see [Zbl 0949.00040].

##### MSC:
 81T45 Topological field theories in quantum mechanics 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.) 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 81T70 Quantization in field theory; cohomological methods 83C45 Quantization of the gravitational field
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