Cubic curves from instanton counting
Sergey Shadchin
TL;DR
The paper develops a practical method to extract Seiberg-Witten curves from Nekrasov’s prepotential in non-hyperelliptic cases, notably cubic (trigonal) curves. By turning saddle-point equations into product equations and applying Vieta relations, it constructs explicit cubic curves for SU(N) with symmetric/antisymmetric matter and for SU(N1)×SU(N2) with bifundamentals, and validates them against instanton counting up to two instantons. It further connects these results to M-theory curves and discusses the consistency of instanton corrections with direct computations, demonstrating the method’s reliability for group-product theories. The work broadens the toolkit for obtaining SW geometries beyond hyperelliptic cases and highlights directions for future generalizations to more complex gauge-group structures.
Abstract
We investigate the possibility to extract Seiberg-Witten curves from the formal series for the prepotential, which was obtained by the Nekrasov approach. A method for models whose Seiberg-Witten curves are not hyperelliptic is proposed. It is applied to the SU(N) model with one symmetric or antisymmetric representations as well as for SU(N_1)xSU(N_2) model with (N_1,N_2) or (N_1,\bar{N}_2) bifundamental matter. Solutions are compared with known results. For the gauge group product we have checked the instanton corrections which follow from our curves against direct instanton counting computations up to two instantons.