Instanton Calculus and Nonperturbative Relations in N=2 Supersymmetric Gauge Theories

TL;DR

The paper tests a key nonperturbative relation in N=2 SU(2) SYM by performing explicit instanton calculations up to winding number $k=2$. Using the ADHM construction and constrained instanton methods, the authors compute the k=1 and k=2 contributions to $\langle \mathrm{Tr}\phi^2\rangle$ and relate them to the instanton coefficients ${\cal F}_k$ of the Seiberg–Witten prepotential ${\cal F}$. They demonstrate that ${\cal G}_k = 2k{\cal F}_k$ holds for $k=1,2$, obtaining $\langle \phi^{a} \phi^{a} \rangle = \frac{2}{g^{4}}\frac{\Lambda^{4}}{a^{2}}$ at $k=1$ and $\langle \mathrm{Tr}\phi^{2}\rangle = -\frac{5}{4}\frac{\Lambda^{8}}{g^{8} a^{6}}$ at $k=2$, which agrees with established nonperturbative results. This work provides a concrete, low-order cross-check of the Seiberg–Witten framework and underscores the consistency between instanton calculus and the holomorphic prepotential in the weak-coupling regime.

Abstract

Using instanton calculus we check, in the weak coupling region, the nonperturbative relation $$ <\Trφ^2>=iπ\left(\cf-{a\over 2} {\partial\cf\over\partial a}\right)$$ obtained for a N=2 globally supersymmetric gauge theory. Our computations are performed for instantons of winding number k, up to k=2 and turn out to agree with previous nonperturbative results.