On the Jeffrey-Kirwan residue of BCD-instantons

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 $U(1)$.