Nonperturbative Relations in N=2 Supersymmetric Yang-Mills Theory and the Witten-Dijkgraaf-Verlinde-Verlinde Equation.

Abstract

We find the nonperturbative relation between $\langle {\rm tr} \phi^2 \rangle$, $\langle {\rm tr} \phi^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle \phi_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space. The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups.