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By J.N. Coldstream

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9. S. K. Donaldson, I. Smith, in preparation. 10. N. Mather, How to stratify mappings and jet spaces, Singularites d'Applications Difterentiables (Plans-sur-Bex 1975), Lecture Notes in Math. 535, Springer (1976), 128-176. 30 11. V. Munoz, F. Presas, I. Sols, Almost holomorphic embeddings in Grassmannians with applications to singular symplectic submanifolds, preprint. LOCAL M I R R O R S Y M M E T R Y A N D FIVE-DIMENSIONAL G A U G E THEORY TOHRU EGUCHI Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan 113 We propose an exact solution of 5-dimensional 5(7(2) gauge theory on M4 x S1 for arbitrary values of the radius R of S1.

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 .

F + 12q3F + 65q*F +220qF+ •••)+0(q%)] . (30) (31) 37 3 Small and Large Radius Limits Let us next examine the small and large radius limits R —> 0, oo of (30). We first consider the 4-dimensional limit R -> 0 following 7 . Due to the relations (20), (21) small R corresponds to the base of F2 becoming large (ts ->• 00) and the fiber becoming small (£F -* 0) while the ratio e~ t B jtF is kept fixed. As explained by Katz, Klemm and Vafa 7 , quantum parts of F are suppressed because of the powers of qB « -R4 and the surviving contributions come from the divergence of the series J2 4 , H I 9 F as qp = e~iRA —• 1.

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