By Barus C.

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M. Hull and M. Rocek, Nucl. Phys. B248, 157 (1984); T. L. Curtright and C. K. Zachos, Phys. Rev. Lett. 53, 1799 (1984); P. Oi Vecchia, V. G. Knizhnik, J. L. Petersen and P. Rossi, Nucl. Phys. B253, 701 (1985); E. Abdalla and M. C. B. Abdalla, Phys. Lett. -L. Chau and H. C. Yen, Phys. Lett. B177, 368 (1986) 8. A. Belavin and V. E. Zakharov, Phys. Lett. 73B, S3 (1978); K. Pohlmeyer. Comm. Math. Phys. 72, 37 (1980); G. -L. Chau, M. K. Prasad and A. Sinha, Phys. Rev. 023, 2321-cT981); ibid 024, 1574 (1981) 44 Lax, Comm.

Sj(x) E C [[xl] for all j 2: 1, then S of (10) corresponds bijectively to a point of the big-cell G+(O,O). Unfortunately this correspondence is not canonical, because it depends on the choice of a basis for V. The corresondence becomes completely canonical if one uses G+(O, -1) instead of G+(O, 0). Thus one obtains the following. 1. Let r m be the set of all monic zeroth order pseudodifferential operators of the form (10) such that all the coefficients are in C [[xl]. Then there is a canonical bijection THEOREM rm ~ G+(O, -1) .

Then there is a canonical bijection between Bl and M l; SECOND THEOREM OF BURCHNALL-CHAUNDY. These theorems were forgotten for more than half a century. Meantime, in the middle of 1970's, Krichever [K] re-discovered these theorems without knowing the earlier work of Burchnall and Chaundy. A complete proof of these theorems using the modern language of algebraic geometry was given by Mumford [Mum]. Here comes a natural question: which kind of geometric data correspond to Br for r ;:::: 2 ? 19 An obvious candidate is the moduli space Mr of the data (C, p, F, v), where F is now a vector bundle of rank r and degree reg - 1) such that HO(C, F) = o.

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