Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation

TL;DR

This paper uncovers nonperturbative relations in N=2 supersymmetric SU(3) Yang-Mills theories, linking key observables, deriving nonlinear differential equations including the WDVV equation, and exploring the theory's topological and modular properties.

Contribution

It introduces new nonperturbative relations and differential equations for N=2 SU(3) SYM, connecting the prepotential with topological field theory structures and modular invariants.

Findings

Derived nonlinear differential equations for the prepotential including WDVV

Established relations between vevs and the prepotential in SU(3) SYM

Explored the structure of the quantum moduli space and beta function

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.