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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 partitionfunction in the $\Omega$-background and express it as functional integrals overdensity functions. Applying the saddle point method, we derive the limit shapeequations which determine the dominant instanton configurations in the flatspace limit. The solution to the limit shape equations gives the Seiberg-Wittengeometry of the low energy effective theory. As an illustrating example, wework out explicitly the Seiberg-Witten geometry for linear quiver gaugetheories. Our result matches the Seiberg-Witten solution obtained previouslyusing the method of brane constructions in string theory.

Seiberg-Witten geometry of four-dimensional N=2 SO-USp quiver gauge theories, I