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edited Apr 14, 2018 at 0:23

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.

In (4.25) the author expressed the partition function corresponding to a vector of Young's Diagrams $\mathbf{k}$ (described by a functiion $f_{\mathbf{a},\mathbf{k}}(x)$ as \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) = \exp\left(-\frac{1}{4}\int_{x\neq y}dxdy f''_{\mathbf{a},\mathbf{k}}(x|\epsilon_1,\epsilon_2)f''_{\mathbf{a},\mathbf{k}}(y|\epsilon_1,\epsilon_2)\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)\right). \end{equation} where I suppose the double integral is over a portion of $\mathbb{R}^2$. The function $\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)$ is defined in Appendix A. The $x\neq y$ part isn't really there in the paper, but it appeared in (4.24) and the integral agrees with the formula (3.6) with $x\neq y$, so I suppose that it should be there. Then the author considered the limit $\epsilon_1,\epsilon_2 \rightarrow 0$, $f_{\mathbf{a},\mathbf{k}}(x)$ becomes a continuous function $f(x)$ and claimed that the partition function to the leading term of its exponent in this limit should be (4.29) \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) \approx Z_f(\mathbf{a},\epsilon_1,\epsilon_2, \Lambda) \approx \exp\left(\frac{1}{\epsilon_1\epsilon_2}\mathcal{E}_\Lambda(f)\right) \end{equation} where \begin{equation} \mathcal{E}_\Lambda(f) = -\frac{1}{2}\int_{y < x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right). \end{equation} My problem with this is that if I just substitute the leading term of $\gamma_{\epsilon_1, \epsilon_2}$ as presented in equation (A.3) into the formula for $Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda)$ I would get the leading exponent term withto be \begin{equation} \bar{\mathcal{E}}_\Lambda(f) = -\frac{1}{4}\int_{y \neq x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right) \end{equation} instead (the integral limit is $x\neq y$ instead of $y < x$). When $x < y$, I would naively believe that $\log((x-y)/\Lambda) = \log(|x - y|/\Lambda) + i\pi$ and so $\bar{\mathcal{E}}_\Lambda(f) - \mathcal{E}_\Lambda(f) = 2\pi iN\sum_{l = 1}^N a^2_l$. Since the main result of this paper is that the leading exponent term $\mathcal{E}_\Lambda(f_*)$ is the Seiberg-Witten prepotential and it is well-known (among other things) that the $\partial/\partial a_l$ derivatives of SW prepotential must relate $A$ and $B$-periods of SW curves so I'm quite worried that I keep getting this extra ($\{a_l\}$ dependent term) $2\pi iN\sum_{l = 1}^N a^2_l$ term.

So what did I do wrong in going from (4.25) to (4.28)?

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.

In (4.25) the author expressed the partition function corresponding to a vector of Young's Diagrams $\mathbf{k}$ (described by a functiion $f_{\mathbf{a},\mathbf{k}}(x)$ as \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) = \exp\left(-\frac{1}{4}\int_{x\neq y}dxdy f''_{\mathbf{a},\mathbf{k}}(x|\epsilon_1,\epsilon_2)f''_{\mathbf{a},\mathbf{k}}(y|\epsilon_1,\epsilon_2)\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)\right). \end{equation} where I suppose the double integral is over a portion of $\mathbb{R}^2$. The function $\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)$ is defined in Appendix A. The $x\neq y$ part isn't really there in the paper, but it appeared in (4.24) and the integral agrees with the formula (3.6) with $x\neq y$, so I suppose that it should be there. Then the author considered the limit $\epsilon_1,\epsilon_2 \rightarrow 0$, $f_{\mathbf{a},\mathbf{k}}(x)$ becomes a continuous function $f(x)$ and claimed that the partition function to the leading term of its exponent in this limit should be (4.29) \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) \approx Z_f(\mathbf{a},\epsilon_1,\epsilon_2, \Lambda) \approx \exp\left(\frac{1}{\epsilon_1\epsilon_2}\mathcal{E}_\Lambda(f)\right) \end{equation} where \begin{equation} \mathcal{E}_\Lambda(f) = -\frac{1}{2}\int_{y < x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right). \end{equation} My problem with this is that if I just substitute the leading term of $\gamma_{\epsilon_1, \epsilon_2}$ as presented in equation (A.3) into the formula for $Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda)$ I would get the leading exponent term with \begin{equation} \bar{\mathcal{E}}_\Lambda(f) = -\frac{1}{4}\int_{y \neq x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right) \end{equation} instead (the integral limit is $x\neq y$ instead of $y < x$). When $x < y$, I would naively believe that $\log((x-y)/\Lambda) = \log(|x - y|/\Lambda) + i\pi$ and so $\bar{\mathcal{E}}_\Lambda(f) - \mathcal{E}_\Lambda(f) = 2\pi iN\sum_{l = 1}^N a^2_l$. Since the main result of this paper is that the leading exponent term $\mathcal{E}_\Lambda(f_*)$ is the Seiberg-Witten prepotential and it is well-known (among other things) that the $\partial/\partial a_l$ derivatives of SW prepotential must relate $A$ and $B$-periods of SW curves so I'm quite worried that I keep getting this extra ($\{a_l\}$ dependent term) $2\pi iN\sum_{l = 1}^N a^2_l$ term.

So what did I do wrong in going from (4.25) to (4.28)?

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.

In (4.25) the author expressed the partition function corresponding to a vector of Young's Diagrams $\mathbf{k}$ (described by a functiion $f_{\mathbf{a},\mathbf{k}}(x)$ as \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) = \exp\left(-\frac{1}{4}\int_{x\neq y}dxdy f''_{\mathbf{a},\mathbf{k}}(x|\epsilon_1,\epsilon_2)f''_{\mathbf{a},\mathbf{k}}(y|\epsilon_1,\epsilon_2)\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)\right). \end{equation} where I suppose the double integral is over a portion of $\mathbb{R}^2$. The function $\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)$ is defined in Appendix A. The $x\neq y$ part isn't really there in the paper, but it appeared in (4.24) and the integral agrees with the formula (3.6) with $x\neq y$, so I suppose that it should be there. Then the author considered the limit $\epsilon_1,\epsilon_2 \rightarrow 0$, $f_{\mathbf{a},\mathbf{k}}(x)$ becomes a continuous function $f(x)$ and claimed that the partition function to the leading term of its exponent in this limit should be (4.29) \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) \approx Z_f(\mathbf{a},\epsilon_1,\epsilon_2, \Lambda) \approx \exp\left(\frac{1}{\epsilon_1\epsilon_2}\mathcal{E}_\Lambda(f)\right) \end{equation} where \begin{equation} \mathcal{E}_\Lambda(f) = -\frac{1}{2}\int_{y < x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right). \end{equation} My problem with this is that if I just substitute the leading term of $\gamma_{\epsilon_1, \epsilon_2}$ as presented in equation (A.3) into the formula for $Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda)$ I would get the leading exponent term to be \begin{equation} \bar{\mathcal{E}}_\Lambda(f) = -\frac{1}{4}\int_{y \neq x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right) \end{equation} instead (the integral limit is $x\neq y$ instead of $y < x$). When $x < y$, I would naively believe that $\log((x-y)/\Lambda) = \log(|x - y|/\Lambda) + i\pi$ and so $\bar{\mathcal{E}}_\Lambda(f) - \mathcal{E}_\Lambda(f) = 2\pi iN\sum_{l = 1}^N a^2_l$. Since the main result of this paper is that the leading exponent term $\mathcal{E}_\Lambda(f_*)$ is the Seiberg-Witten prepotential and it is well-known (among other things) that the $\partial/\partial a_l$ derivatives of SW prepotential must relate $A$ and $B$-periods of SW curves so I'm quite worried that I keep getting this extra ($\{a_l\}$ dependent term) $2\pi iN\sum_{l = 1}^N a^2_l$ term.

So what did I do wrong in going from (4.25) to (4.28)?

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asked Apr 13, 2018 at 15:28

user113988

Nekrasov Partition function and the leading term of Prepotential

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.

In (4.25) the author expressed the partition function corresponding to a vector of Young's Diagrams $\mathbf{k}$ (described by a functiion $f_{\mathbf{a},\mathbf{k}}(x)$ as \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) = \exp\left(-\frac{1}{4}\int_{x\neq y}dxdy f''_{\mathbf{a},\mathbf{k}}(x|\epsilon_1,\epsilon_2)f''_{\mathbf{a},\mathbf{k}}(y|\epsilon_1,\epsilon_2)\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)\right). \end{equation} where I suppose the double integral is over a portion of $\mathbb{R}^2$. The function $\gamma_{\epsilon_1,\epsilon_2}(x-y|\Lambda)$ is defined in Appendix A. The $x\neq y$ part isn't really there in the paper, but it appeared in (4.24) and the integral agrees with the formula (3.6) with $x\neq y$, so I suppose that it should be there. Then the author considered the limit $\epsilon_1,\epsilon_2 \rightarrow 0$, $f_{\mathbf{a},\mathbf{k}}(x)$ becomes a continuous function $f(x)$ and claimed that the partition function to the leading term of its exponent in this limit should be (4.29) \begin{equation} Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda) \approx Z_f(\mathbf{a},\epsilon_1,\epsilon_2, \Lambda) \approx \exp\left(\frac{1}{\epsilon_1\epsilon_2}\mathcal{E}_\Lambda(f)\right) \end{equation} where \begin{equation} \mathcal{E}_\Lambda(f) = -\frac{1}{2}\int_{y < x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right). \end{equation} My problem with this is that if I just substitute the leading term of $\gamma_{\epsilon_1, \epsilon_2}$ as presented in equation (A.3) into the formula for $Z_{\mathbf{k}}(\mathbf{a},\epsilon_1,\epsilon_2,\Lambda)$ I would get the leading exponent term with \begin{equation} \bar{\mathcal{E}}_\Lambda(f) = -\frac{1}{4}\int_{y \neq x}dxdy f''(x)f''(y)(x - y)^2\left(\frac{1}{2}\log\left(\frac{x - y}{\Lambda}\right) - \frac{3}{4}\right) \end{equation} instead (the integral limit is $x\neq y$ instead of $y < x$). When $x < y$, I would naively believe that $\log((x-y)/\Lambda) = \log(|x - y|/\Lambda) + i\pi$ and so $\bar{\mathcal{E}}_\Lambda(f) - \mathcal{E}_\Lambda(f) = 2\pi iN\sum_{l = 1}^N a^2_l$. Since the main result of this paper is that the leading exponent term $\mathcal{E}_\Lambda(f_*)$ is the Seiberg-Witten prepotential and it is well-known (among other things) that the $\partial/\partial a_l$ derivatives of SW prepotential must relate $A$ and $B$-periods of SW curves so I'm quite worried that I keep getting this extra ($\{a_l\}$ dependent term) $2\pi iN\sum_{l = 1}^N a^2_l$ term.

So what did I do wrong in going from (4.25) to (4.28)?