# How to perform contour integral in Nekrasov's formula

My question is technical. It is about instanton counting calculation (see this paper).

The partition function of SU(N) gauge theory with $N_f$ fundamental multiplets in k instanton background is given by (3.10) $$Z_k(a,\epsilon_1,\epsilon_2)=\frac{\epsilon^k}{k!(2\pi i\epsilon_1 \epsilon_2)^k }\oint \prod_{I=1}^{k} {{\rm d}{\phi}_I \ Q({\phi}_I) \over {P({\phi}_I) P({\phi}_I + {\epsilon})}} \ {\prod}_{1\leq I < J \leq k} {{{\phi}_{IJ}^2 ( {\phi}_{IJ}^2 - {\epsilon}^2)}\over{({\phi}_{IJ}^2 - {\epsilon}_1^2)({\phi}_{IJ}^2- {\epsilon}_2^2)}}$$ where: $Q(x) = \prod_{f=1}^{N_f} ( x + m_f)$ & $P (x) = \prod_{l=1}^{N} ( x - a_l)$, ${\phi}_{IJ}$ denotes ${\phi}_I - {\phi}_J$, and ${\epsilon} = {\epsilon}_1 + {\epsilon}_2$ .

Then the pole of the integral corresponding to $\vec Y$ is at ${\phi}_I$ with $I$ corresponding to the box $({\alpha},{\beta})$ in the $l$'th Young tableau (so that $0 \leq {\alpha} \leq {\nu}^{l,{\beta}}, \ 0 \leq {\beta} \leq k_{l,{\alpha}}$) equal to: ${\vec Y} \longrightarrow {\phi}_{I} = a_l + {\epsilon}_1 ({\alpha}-1) + {\epsilon}_2 ( {\beta}-1)$.

After the path intagral, the result is (3.20)

\eqalign{&R_{\vec Y} = {1\over{({\epsilon}_1 {\epsilon}_2)^k}} \prod_{l} \prod_{{\alpha}=1}^{{\nu}^{l,1}} \prod_{{\beta}=1}^{k_{l,{\alpha}}} {{\mathcal{S}_{l}({\epsilon}_1 ({\alpha}-1) + {\epsilon}_2 ({\beta}-1))}\over{({\epsilon} ({\ell}(s)+1) - {\epsilon}_2 h(s))({\epsilon}_2 h(s) - {\epsilon} {\ell} (s))}} \times \prod_{l < m} \prod_{{\alpha}=1}^{{\nu}^{l,1}} \prod_{{\beta}=1}^{k_{m,1}} \left( {{\left( a_{lm} + {\epsilon}_1 ({\alpha} - {\nu}^{m,{\beta}})+{\epsilon}_2 (1-{\beta}) \right) \left( a_{lm} + {\epsilon}_1 {\alpha} + {\epsilon}_2 ( k_{l,\alpha} + 1 - {\beta}) \right)}\over{\left( a_{lm} + {\epsilon}_1 {\alpha} + {\epsilon}_2 ( 1 - {\beta}) \right) \left( a_{lm} + {\epsilon}_1 ({\alpha} - {\nu}^{m,{\beta}}) + {\epsilon}_2 ( k_{l,{\alpha}} + 1 - {\beta}) \right)}}\right)^2 } where we have used the following notations: $a_{lm} = a_l - a_m$, $\mathcal{S}_{l}(x) = {Q(a_l + x) \over { \prod_{m \neq l} ( x + a_{lm} ) (x + {\epsilon} + a_{lm} )} }$, and ${\ell}(s) = {k}_{l,{\alpha}} - {\beta}$, $\qquad h(s) = {k}_{l,{\alpha}} + {\nu}^{l,{\beta}} - {\alpha} - {\beta} +1$