S-duality in Supersymmetric Gauge Theory and String Theory

In the year 1996 Eguchi constructed a large class of 4 dimensional superconformal field theories starting from the Seiberg-Witten exact solutions of N=2 supersymmertic gauge theories and adjusting their parameters to critical values.He then classified these conformal theories based on their critical exponents and has shown that the classification follows after the pattern of the A-D-E Lie algebra.In the year1997 Eguchi studied the topological string amplitudes on arbitrary Kahler manifolds M and has shown that they are annihilated by an infinite number of differential operators L_nL_nZ=0，n=-1，0，1，2，...(1)and these operoro Virgasn_m&lt；L_L_n&lt；L_L_m&lt；L_L_m&lt；L_L_n&lt；L_L_m&lt；L_L_n&lt；L_L_n；L_L_n&lt；L_L_n&lt；m&lt；L_L_n&lt；n-1，0，1，2，...(1)and these operororo Virtororo Virm_n&lt；L_n&lt；L_L_nZ=0，n=-1，0，1，2，...(1)and these operoro a Virlo a，n&lt；L_n&lt；n&lt；N m，0&gt；(2)Here X(M)is the Euler number of M.By representing the Virasoro operators in terms of free fields he further showed that the topological string amplitudes are described by bL_&lt；even&gt；free bosons and bL_&lt；edd&gt；free fermions。Here b_&lt；**&gt；(M)are the Betti numbers of M.

In the year 1996 Eguchi constructed a large class of 4 dimensional superconformal field theories starting from the Seiberg-Witten exact solutions of N=2 supersymmertic gauge theories and adjusting their parameters to critical values. He then classified these conformal theories based on their critical exponents and has shown that the classification follows after the pattern of the A-D-E Lie algebra.In the year 1997 Eguchi studied the topological string amplitudes on arbitrary Kahler manifolds M and has shown that they are annihilated by an infinite number of differential operators L_nL_nZ=0, n=-1,0,1,2, ・・・ (1)and these operators form a Virasoro algebra[L_n, L_m]= (n-m) L_<n+m>+(X (M) )/ (n^3-n) delta_<n+m, 0> (2)Here X (M) is the Euler number of M.By representing the Virasoro operators in terms of free fields he further showed that the topological string amplitudes are described by bL_<even> free bosons and bL_<edd> free fermions. Here b_<**> (M) are the Betti numbers of M.