By Barmak J.A., Minian E.G.

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S. T. Yau, "A Note on ODEs from mirror Symmetry", hep-th/9407192. S. Katz and C. Phys. B497 (1997) 146, hep-th/9606086. THE TODA CONJECTURE EZRA GETZLER RIMS, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 602, Japan Northwestern University, Evanston, IL 60208, USA Consider the Gromov-Witten potential oo g=0 of CP . Eguchi and Yang [4] have conjectured that Z = exp(e _ 2 F) is a T-function of the Toda hierarchy. (Similar ideas were also proposed by Dubrovin, cf. ) In this paper, we will explore this conjecture using the bihamiltonian method in the theory of integrable systems.

The Toda conjecture consists of the Toda equation d2QF = qeu, (1) whose implications have been studied by Pandharipande [25], and the recursion («V + [2] dQ)((Tk-hQ)) = (k + l ) V « r f c , Q » . (2) 1 The large phase space of CP may be identified with the jet-space of the space with coordinates u and v, that is, it has coordinates {dnu, 9"v} n >oThe Toda conjecture implies that, in these coordinates, the flows dk,Q are the flows of the Toda lattice hierarchy*. Eguchi and Yang also give a matrix integral representation of the GromovWitten potential of CP 1 .

Phys. B431 (1994) 484, hep-th/9408099. 3. N. Seiberg, "Five dimensional SUSY field theories, non-trivial fixed points and string dynamics", Phys. Lett. B 388 (1996) 753, hep-th/9608111. 4. R. Morrison and N. Seiberg, "Extremal transitions and fivedimensional supersymmetric theories", Nucl. Phys. B 483 (1997) 229, hep-th/9609070. 5. R. Douglas, S. Katz and C. Vafa, "Small instantons, del Pezzo surfaces and type F theory", Nucl. Phys. B 497 (1997) 155, hep-th/9609071. 6. A. Klemm, W. Lerche, P. Mayr, C.