Abstract
We apply the instanton counting method to study a class of four-dimensional $\mathcal{N}=2$ supersymmetric quiver gauge theories with alternating $\mathrm{SO}$ and $\mathrm{USp}$ gauge groups. We compute the partition function in the $\Omega$-background and express it as functional integrals over density functions. Applying the saddle point method, we derive the limit shape equations which determine the dominant instanton configurations in the flat space limit. The solution to the limit shape equations gives the Seiberg-Witten geometry of the low energy effective theory. As an illustrating example, we work out explicitly the Seiberg-Witten geometry for linear quiver gauge theories. Our result matches the Seiberg-Witten solution obtained previously using the method of brane constructions in string theory.
Highlights
Four-dimensional N 1⁄4 2 supersymmetric gauge theories are an extremely interesting playground for studying nonperturbative dynamics of quantum field theories
In order to introduce a supersymmetric regulator of the infinite volume of spacetime and to simplify the evaluation of the path integral, the four-dimensional N 1⁄4 2 supersymmetric gauge theory is formulated in the Ω-background, which is a particular supergravity background with two deformation parameters ε1, ε2
Based on field theoretical arguments [3,4], the Seiberg-Witten prepotential F of the low energy effective theory can be extracted from the partition function Z in the following limit, F 1⁄4 −ε1l;εim2→0ε1ε2 log Zðε1; ε2Þ: ð1Þ
Summary
Four-dimensional N 1⁄4 2 supersymmetric gauge theories are an extremely interesting playground for studying nonperturbative dynamics of quantum field theories. Based on field theoretical arguments [3,4], the Seiberg-Witten prepotential F of the low energy effective theory can be extracted from the partition function Z in the following limit, This approach has hitherto been used to derive the Seiberg-Witten solution for gauge theories with a simple classical gauge group [4,5,6], and SUðNÞ quiver gauge theories with hypermultiplets in the fundamental, adjoint or bifundamental representations [7]. These are far from all the N 1⁄4 2 supersymmetric gauge theories for which we are able to compute the partition function in the Ω-background. It is convenient to encode the Coulomb moduli ai and masses mi in the characters of two vector space Ni and Mi assigned for each vertex i ∈ Vγ,
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