We apply the Jeffrey-Kirwan method to compute the multiple integrals for the $BCD$ type Nekrasov partition functions of four dimensional $\mathcal{ N}=2$ supersymmetric gauge theories. We construct a graphical distinction rule to determine which poles are surrounded by their integration cycles. We compute the instanton correction of the "$Sp(0)$" pure super-Yang-Mills theory and find that $Z^{ Sp(0)}_{ k}=(-1)^{ k}(2^{ k}k!\varepsilon_{ 1}^{ k}\varepsilon_{ 2}^{ k})^{ -1}$ for $k\le 8$, which resembles the formula $Z^{ U(1)}_{ k}=(k!\varepsilon_{ 1}^{ k}\varepsilon_{ 2}^{ k})^{ -1}$ for the pure super-Yang-Mills theory with gauge group $U(1)$.
Categories: hep-th math-ph math.MP