Volume 257, number 3,4 PHYSICS LETTERS B 28 March 1991 Geometric interpretation of the partition function of 2D gravity V. Kac Massachusetts Institute of Technology, Cambridge, MA 02139, USA and A. Schwarz University of California at Davis, Davis, CA 95616, USA Received 30 November 1990; revised manuscript received 31 December 1990 We construct explicitly the subspace in the infinite dimensional grassmannian corresponding to the r-function of the 2D topo- logical gravity. This allows us to give a simple proof of some conjectures on the equations defining this function. It has been discovered recently that the reduction to N× Nmatrix models, N-~oo, can be used to obtain the complete non-perturbative solution of 2D gravity and of string theory coupled with matter fields hav- ing central charge c<l [ 1-3]. The answer for the partition function was expressed in terms of r-func- tions. Later is was shown that the same partition function arises in 2D topological gravity [4,5]. Namely the generating function of amplitudes in top- ological gravity r(tj, t3, ...) and the partition function Z(t, t3, ...) of the one-matrix model at the k= 1 crit- ical point are connected by the formula Z(I,,13 .... )=r2(/,,t3 .... ) • (1) The function rub t3,---, t2n+j, ... ) satisfies the so called string equation and the following equations of the KdV hierarchy: Ou k c~ at2k+~-(-2) ~Rk(u), 3 2 u=-2 O~12 In r. Here the R,(u) are defined by the formula ¢~ Supported in part by NSF grant DMS-8802489 and DOE grant DE-FG02-88ER25066. (xl exp{ - t[ - a2/ax2+ u(x) ]} Ix) 1 ~ k__Eo t'R,(.) (2) for t-, 0 (i.e. Rk ( u ) are Seeley coefficients for the op- erator - 02/0x 2 + u (x)). Recall that the r-function of the KP hierarchy can be defined as a function of an infinite number of variables tl, t2.... satisfying bilin- ear Hirota equations. Such a function can be consid- ered as a r-function of the KdV hierarchy if it does not depend on the even variables t2, t4, .... Every r- function of the KP hierarchy corresponds to a point of the Sato infinite-dimensional grassmannian Gr [6]. (The points of Gr are linear subspaces of the space H consisting of the formal Laurent series Eanz", a~ = 0 for n >> 0. The subspace V ~ H belongs to Gr if the natural projection zc+ of V into the space H+ spanned by z n, n/> 0, is a Fredholm operator. The big cell of Gr consists of those V for which 7r+ is an iso- morphism; we denote it by Gr °. ) The r-functions of the KdV hierarchy correspond to the points of Gr ob- eying z2V ~ V; we denote this subset of Gr by Gr(2) and let Gr~2) -- Gr(2~ c~Gr °. Our aim is to describe the point of Gr correspond- ing to the r-function arising in 2D gravity. The de- scription will be based on refs. [ 7,8 ]. It is proved in 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 329