In the celebrated paper Seiberg-Witten prepotential from instanton counting by N. Nekrasov I can't quite understand some parts of section (2.3). The Nekrasov partition function is defined via \begin{equation}\tag{NekPF} Z(a, \epsilon) = \left\langle\exp\left\{\frac{1}{(2\pi i)^2}\int_{\mathbb{R}^4}\left[\omega\wedge\operatorname{Tr}\left(\phi F+\frac{1}{2}\psi \psi \right)-H \operatorname{Tr}(F\wedge F)\right]\right\}\right\rangle_a \end{equation} Here

  • $\omega$ is a symplectic form that breaks the rotational invariance of $\mathbb{R}^4$ and gives it the complex structure $(z_1, z_2)$,
  • $\psi$ is a fermion in the twisted superspace defined in terms of the two Weyl spinors of the $\mathcal{N} = 2$ vector multiplet,
  • $F$ is the curvature of the gauge connection in the $\mathcal{N} = 2$ gauge boson and
  • $\phi$ is the scalar of the $\mathcal{N} = 2$ vector multiplet,
  • $H$ is basically equal to $H(z_1, z_2) = \epsilon_1 |z_1|^2 + \epsilon_2 |z_2|^2$ in the complex coordinates $(z_1, z_2)$.

Also the subscript $a$ means the expectation value is taken in the vacuum in which the expectation value of the adjoint scalar in the $\mathcal{N} = 2$ vector multiplet is $a$ (which is thus an element of the Cartan subalgebra of the gauge algebra).

Let's move to section (2.3). First Nekrasov says that if $H$ were a constant that it would simply renormalize the dynamical scale by $\Lambda \mapsto \Lambda e^{-H}$. I can sort of see this from the fact that in the second term of (NekPF) $H$ would basically be a theta angle.

He then proceeds to say that $H$ is not constant and it is the bosonic part of a superfield $\mathcal{H}(x, \theta) = H(x) + \frac{1}{2}\omega_{\mu\nu} \theta^\mu\theta^\nu = H + \omega$ (here $\theta^\mu$ are coordinates on the twisted superspace). I suppose (NekPF) should be rewritten in terms of the $\mathcal{N}=2$ vector superfield \begin{equation} \Phi = \phi + \theta^\mu \psi_\mu + \frac{1}{2}\theta^\mu\theta^\nu F_{\mu\nu} + \dots = \phi + \psi + F + \dots \end{equation} and this superfield $\mathcal{H}$ as \begin{equation}\tag{NekPFinSuperspace} Z(a, \epsilon) = \left\langle\exp\left[\frac{1}{(2\pi i)^2}\int_{\mathbb{R}^4}\frac{1}{2}\mathcal{H} \wedge \operatorname{Tr}\left\{\Phi^2\right\}\right]\right\rangle_a \end{equation} (here I mean that the top form part of the expression has to be integrated of course) Nekrasov writes this in the remarks after equation (2.10) (which is (NekPF)) to show the operator is cohomologous to the identity. However using the expression for $\Phi$ given above and in the paper I can't show that the two expressions for $Z$ coincide. In particular \begin{equation} \begin{aligned} \Phi^2 &= \Phi \wedge \Phi = \phi^2 + 2\phi \psi + 2 \phi F + \psi \wedge \psi + \psi \wedge F + F \wedge \psi + F \wedge F + \dots\\ &= \phi^2 + 2\phi \psi + 2 \phi F + \psi \wedge \psi + F \wedge F + \dots \end{aligned} \end{equation} then \begin{equation} \frac{1}{2}\mathcal{H} \wedge \operatorname{Tr}\left\{\Phi^2\right\} = \frac{1}{2}\mathcal{H} \operatorname{Tr}\left\{\phi^2\right\} + \mathcal{H} \wedge \operatorname{Tr}\left\{ \phi \psi\right\} + \mathcal{H} \wedge \operatorname{Tr}\left\{ \phi F \right\} + \frac{1}{2}\mathcal{H} \wedge \operatorname{Tr}\left\{ \psi \wedge \psi \right\} + \frac{1}{2}\mathcal{H} \wedge \operatorname{Tr}\left\{ F \wedge F \right\} + \dots \end{equation} and the 4-form part of this is (I actually have not verified exactly what the dots are, I have the full expression of $\Phi$ in the regular superspace but getting all the terms would require quite many calculations and then it would have to be rewritten in the twisted superspace) \begin{equation} \omega \wedge \operatorname{Tr}\left\{ \phi F + \frac{1}{2}\psi \wedge \psi\right\} + \frac{1}{2}H\operatorname{Tr}\left\{ F \wedge F \right\} \end{equation} which is almost the integrand in (NekPF) except for the sign and factor in front of the second term. $\mathcal{H}$ from here clearly has the role that usually the complexified coupling would have in the $\mathcal{N}=2$ action. I can then understand the rest of the section if (NekPFinSuperspace) is correct, but I am not sure of this since the calculation does not work out for me.

I do not understand whether the formula given for $\Phi$ in the paper is just schematic and the actual formula would have different factors (this is also supported by the fact than in another work by Nekrasov The ABCD of instantons the formula is different, but the computation still does not seem to work out)? Or maybe the part that has been ommitted in $\Phi$ actually contributes the the 4-form part of $\frac{1}{2}\mathcal{H} \wedge \operatorname{Tr}\left\{\Phi^2\right\}$ and fixes the sign? Or maybe my calculation is wrong? Or, hopefully not, my understanding of what Nekrasov is doing is wrong on a conceptual level?



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