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

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The residues are given by $\phi_i \rightarrow \phi_{(\alpha,\beta)}=(\alpha-1)E_1+(\beta-1)E_2$. 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?

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

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the equality is not explicit in Nakajima's lecture notes (which you can download from here); proposition 5.7 on page 60 comes closest.

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  • $\begingroup$ Thanks,they are really different for me who majors in physics. $\endgroup$
    – user9527
    Jul 26 '13 at 13:44

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