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Intersections in genus 3 and the Boussinesq hierarchy. (English) Zbl 1048.37060
The author studies a special case of the so-called enlarged Witten conjecture originating in E. Witten’s work on two-dimensional quantum gravity [in: Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243–310 (1991; Zbl 0757.53049); in: L. R. Goldberg (ed.) et al.; Topological methods in modern mathematics, Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held in the State University of New York at Stony Brook, USA, June 14-June 21, 1991, Houston, TX: Publish Perish, Inc., 235–269 (1993; Zbl 0812.14017); and in: Nucl. Phys., B 371, 191–245 (1992)].
This conjecture states that certain intersection numbers of Mumford-Morita-Miller classes on \(\overline{M_{g,n}}\), a compactification of the moduli space of genus-\(g\) algebraic curves with it marked points, are given by an asymptotic expansion of a specific string solution of the integrable Korteweg-de Vries (KdV) hierarchy. Since this was proved by M. Kontsevich [Commun. Math. Phys. 147, 1–23 (1992; Zbl 0756.35081)] for the case of 2-KdV using methods of differential topology and a specific matrix model, the general problem for \(n\)-KdV with arbitrary genus is still under investigation. In the article under review, the author presents calculations for intersection numbers in genus 3 for the special case of the 3-KdV (Boussinesq) hierarchy. The consideration is given in the framework of the axiomatic approach suggested by T. J. Jarvis, T. Kimura and A. Vaintrob [Compos. Math 126, 157–212 (2001; Zbl 1015.14028)] and is in general based on the recursion relations for the auxiliary intersection numbers obtained recently by the present author [Int. Math. Res. Not. 2003, 2051–2094 (2003; Zbl 1070.14030)]. By expressing the correlators in terms ot these auxiliary intersection numbers the author establishes a relation between the correlators of genus 3 and genus 0 and thus proves Witten’s conjecture for genus 3 in the case of the Boussinesq hierarchy.
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
14H10 Families, moduli of curves (algebraic)
14H81 Relationships between algebraic curves and physics
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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