Nuclear Physics B176 (1980) 199-215 © North-Holland Publishing Company NON-LINEAR STRINGS IN TWO-DIMENSIONAL U(~) GAUGE THEORY V.A. KAZAKOV L.D. Landau Institute for Theoretical Physics, Academy of Science, 117334, Moscow, USSR I.K. KOSTOV Moscow State University, 117234, Moscow, USSR Received 4 March 1980 A non-singular version of the Makeenko-Migdal equation for the Wilson loop average in two-dimensional U(N) gauge theory is derived. In the limit N--~ oo the exact solution is obtained for an arbitrary (with any self-intersections) closed loop. 1. Introduction Recently [1], great expectations have been placed upon the string-type equations of motion in the loop space for the Wilson loop average W(C l ..... C~)= H tr Pexp , (1) k=l where P exp56ckA~,(x)dx~, is the ordered integral along the closed contour C k = {x~(~) : ~ < • < ~'}. The W(C I..... Cn) make up a complete set of gauge-invariant colorless quanti- ties, i.e., if one believes that confinement takes place, they contain all the informa- tion about the observables in gluodynamics. Closed equations for the functionals (1) with correct treatment of the intersec- tions of the contours were first obtained on a lattice by Migdal [2]. The continuum limit was found in the works of Makeenko and Migdal [4, 5]. In a somewhat different form it was discussed in Polyakov's paper [6]. The derivation of the equation is based on the well-known Mandelstam formula [7] a%(x) (2) 199