Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model

TL;DR

This work tests the genus-one correction to the Seiberg-Witten prepotential for $SU(2)$ with $N_f=2,3$ by comparing a topologically twisted field-theory derivation with a Dijkgraaf-Vafa Penner-type matrix-model computation. The field-theory side expresses ${\cal F}_1$ as $\mathcal{F}_1 = b(u) - \frac{2}{3} c(u)$ with $b(u)=\frac{1}{2}\log\left(\frac{du}{da}\right)$ and $c(u)=\frac{1}{8}\log(\Delta_{SW})$, yielding explicit weak-coupling expansions in $\zeta=1/u$ for the two flavor cases. On the matrix-model side, the genus-one free energy is computed from a universal two-cut formula involving branch points $x_i$, their discriminant $\Delta$, and the complete elliptic integral $K(\ell)$, with flavor decoupling implemented by scaling limits from the $N_f=4$ case; the resulting $\mathcal{F}_1$ matches the field-theory expansions. The main result is perfect agreement between the DV matrix model and Nekrasov-based field theory at genus one, supporting the $Z_{\mathrm{DV}}=Z_{\mathrm{Nekrasov}}$ correspondence beyond leading order and suggesting avenues to extend the check to $N_f=4$.

Abstract

We find perfect agreements on the genus-one correction to the prepotential of SU(2) Seiberg-Witten theory with N_f=2, 3 between field theoretical and Dijkgraaf-Vafa-Penner type matrix model results.