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I've seen a few papers [1,2,3,4] which defined Nekrasov Partition Function as (in particular [2,3,4]) \begin{equation} Z(\mathbf{a}, \epsilon_1,\epsilon_2,\Lambda) := \sum_{n = 0}^\infty \Lambda^n\int_{M(r,n)}1 \end{equation} but I'm having a difficult time understanding the (physical) motivation behind this definition, so I would like ask a few questions here.

  • Most background explanation I could find was from [1] and [2], from what I understood this Partition Function came from the path integration ((3.2) of [1]) \begin{equation} Z(\mathbf{a},\epsilon_1,\epsilon_2, \Lambda) = \int \mathcal{D}A\mathcal{D}[\text{other fields...}]e^{-\int_{\mathbb{R}^4} \mathcal{L}\sqrt{g}d^4x} \end{equation} where $\mathcal{L}$ is the Lagrangian of $4D$, $\mathcal{N} = 2$, $SU(r)$ SYM in the $(\epsilon_1,\epsilon_2)$-deformed curved spacetime (called $\Omega$-background) which is derived from the dimensional reduction of $6D, \mathcal{N} = 1, SU(r)$ SYM in $\Omega$-spacetime. The field $A$ is the gauge field.

    In normal non-abelian gauge theory this path integral would be (after Wick rotation): \begin{equation} Z[A] = \int \mathcal{D}A e^{\frac{1}{4}\int d^4x F^a_{\mu\nu}F^{a\mu\nu}}. \end{equation} Therefore, after gauge-fixing, the integral $\int\mathcal{D}A$ is the same as integrating over all possible non-equivalent gauges $A$. But the main contribution came from $A$ that satisfies the BPS bound (because they minimise energy), in this case the bound is the anti-self-duality condition. So we can turn $\int \mathcal{D}A$ into the integral over the moduli spaces of $S^4$ anti-instantons $M(r,n)$ (or rather its compactification) for arbitrary $n$, $\sum_{n = 0}^\infty \Lambda^n\int_{M(r,n)}$. The scale parameter $\Lambda$ I guess came from the UV cut-off of $\int \mathcal{D}A$, now it appears in the sum to make the sum converges.

    Back to Nekrasov's, I think the situation should be similar, except we have to add other fields in the supermultiplet of $A$. Lastly, all the fields in $\mathcal{L}$ are re-scaled to zero so the integrand becomes $1$ (I thought this is what [2] suggested on page 6 before (6)).

    My understanding is very imprecise, but is the concept more or less on track? Please correct me or please suggest how I can have a more precise understanding.

  • The point of Nekrasov conjecture is that the Seiberg-Witten prepotential $\mathcal{F}$ can be found via \begin{equation} \mathcal{F}(\mathbf{a}, \Lambda) = \lim_{\epsilon_1,\epsilon_2 \rightarrow 0} \epsilon_1\epsilon_2\log Z(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda). \end{equation} From my understanding, the prepotential was originally define to be the holomorphic function relating the periods $a_i = \oint_{A_i}dS, a^D_i = \oint_{B_i}dS$ on the Seiberg-Witten curve via $a^D_i = \partial \mathcal{F}/\partial a_i$. This conjecture is proven in [1,2,3,4] but what is the physical interpretation of this and what is the physical (or mathematical) intuition that lead to this conjecture in the first place?

  • We have a $(r + 2)$-dimensional torus action $T \curvearrowright M(r,n)$ and $1 \in H^0_T(M(r,n))$. If the moduli space $M(r,n)$ is compact then $\int_{M(r,n)}1 = 0$ because $\dim M(r,n) = 4rn$. Also, from my understanding, because $M(r,n)$ is non-compact the usual Atiyah-Bott localization formula shouldn't apply(?). However it was then taken as a definition that $\int_{M(r,n)} 1$ is to be evaluated via the localization formula ((6) of [2]). I always wonder if there is a better explanation for this? How do we know if we are still getting something meaningful physically after this redefinition? Is there a way to apply the localization theorem directly to something like $\int_{M(r,n)} e^{\frac{1}{4}\int d^4x F^a_{\mu\nu}F^{a\mu\nu}}$ and get similar result?

References:

  1. https://arxiv.org/abs/hep-th/0306238
  2. https://arxiv.org/abs/math-ph/0601062
  3. https://arxiv.org/abs/math/0311058
  4. https://arxiv.org/abs/math/0306198
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