# The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral $$\frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge \ldots\wedge d\phi_N \prod_{i<N} (-\phi_i) \prod_{i}(\phi_i+E_1+E_2)$$ $$\times \frac{\prod_{i\neq j}\phi_{ij} }{\prod_{i}(-\phi_i)(\phi_i+E_1+E_2) }\ {\prod}_{i,j} {{(\phi_{ij} +E_1+E_2)}\over{({\phi}_{ij}+ E_1)(\phi_{ij}+ E_2)}}.\tag{6.4}$$

1. The residues are given by $$\phi_i \rightarrow \phi_{(\alpha,\beta)}=(\alpha-1)E_1+(\beta-1)E_2.\tag{6.9}$$ MNS claims that the evalution of the above integral is equivalent to use of fixed point techniques. They also give some references. I can not understand their trick here after skimming over the references. Can this integral be solved by residue theorem?

2. In order to calculate the sum of critical points, they use an equality from Nakajima's Lecture (Ref. 18): $$\Sigma_{i,j \in D}[ e^{\phi_{ij}} +e^{\phi_{ij} +E_1+E_2}-e^{\phi_{ij}-E_1}-e^{\phi_{ij}-E_2}] - \Sigma_{i\in D} [ e^{-\phi_{i}} + e^{\phi_{i}+E_1+E_2}] \tag{6.10}$$ $$~=~-\Sigma_{(\alpha,\beta \in D)}e^{(\nu_\beta-\alpha+1 )E_1+(\beta-\nu_\alpha^{'})E_2 }+e^{(-\nu_\beta+\alpha )E_1+(-\beta+\nu_\alpha^{'}+1)E_2 }. \tag{6.11}$$ Since I am not familiar with Nakajima's topic, I did not find the equality in Nakajima's lecture. Which chapter does the equality (6.10+11) come from? Thanks for pointing me out.