Seiberg--Witten Duality in Dijkgraaf--Vafa Theory

TL;DR

The paper demonstrates how a specific rescaling in matrix models reveals the duality group of N=2 SU(2) SYM theory, connecting matrix model variables with gauge theory dualities and suggesting new approaches for strong coupling analysis.

Contribution

It identifies a rescaling of matrix model variables that makes the duality group of N=2 SYM manifest and relates matrix model free energy to gauge theory prepotentials.

Findings

Rescaling matrix model couplings reveals the duality group as (2).

The rescaled free energy relates to the gauge theory prepotential.

The approach suggests a new perturbative method for strong coupling regimes.

Abstract

We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair $(S_D,S)$ which makes them dual variables with $\Gamma(2)$ monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the $u$--modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups.