Seiberg–Witten theory as a Fermi gas

Abstract

We explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang–Mills theory in a self-dual $\Omega$ background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painlevé $\mathrm{III}_3$ $\tau$ function. In addition, we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a mathematical proof of the conjecture for the local ${\mathbb P}^1 \times {\mathbb P}^1$ geometry.

Appendices

Appendix A: Quantum A-period

The notion of quantum A-period was studied in the context of AGT correspondence [86, 93–96] and topological strings [97, 98]. It is the integral of a quantum differential over the A-cycle of a given curve. In the particular case of local ${\mathbb P}^1 \times {\mathbb P}^1$, the quantum A-period has been computed in [98] and reads

$$\begin{aligned} {t(\mu , \hbar ) \over 2}= & {} \Pi _A(\mu , \hbar )=\mu +(-m_{{\mathbb F}_0}-1) z\nonumber \\&+\,z^2 \left( -\frac{3 m_{{\mathbb F}_0}^2}{2}-m_{{\mathbb F}_0} q-\frac{m_{{\mathbb F}_0}}{q}-4 m_{{\mathbb F}_0}-\frac{3}{2}\right) +O\left( z^3\right) , \end{aligned}$$

(A.1)

where $z=\mathrm{{e}}^{-2\mu }$ and $q=\mathrm{{e}}^{ \mathrm{{i}}\hbar }$. This relation can also be inverted using an ansatz of type

$$\begin{aligned} \mu =\Pi _A+\sum _{n\ge 1} a_n(m_{{\mathbb F}_0})\mathrm{{e}}^{-2n\Pi _A}. \end{aligned}$$

(A.2)

We find

$$\begin{aligned} \mu =\Pi _A+\sum _{n\ge 0} \Pi _n(z_2,q)z_1^n \end{aligned}$$

(A.3)

where

$$\begin{aligned} z_1=\mathrm{{e}}^{- 2\Pi _A}, \quad z_2=m_{{{\mathbb F}_0}}\mathrm{{e}}^{- 2\Pi _A}. \end{aligned}$$

(A.4)

Notice that in the 4d limit (3.2), we have

$$\begin{aligned} z_1 \rightarrow 0, \quad z_2 \rightarrow \mathrm{{e}}^{4 \pi \mathrm{{i}}\sigma }. \end{aligned}$$

(A.5)

Therefore, it is important to resum the $z_2$ expansion. For the first few coefficients, we find

$$\begin{aligned} \Pi _0(z_2,q)= & {} \log (1+z_2) , \nonumber \\ \Pi _1(z_2,q)= & {} 1+\frac{z_2 \left( -q^2+2 q z_2-1\right) }{(z_2-1) (z_2-q) (q z_2-1)}. \end{aligned}$$

(A.6)

In the four-dimensional limit, we have

$$\begin{aligned} \Pi _0(z_2,q)&\xrightarrow {4D}&\log (1+\mathrm{{e}}^{4 \pi \mathrm{{i}}\sigma }), \nonumber \\ \Pi _1(z_2,q)&\quad \xrightarrow {4D}&\quad 1+ \frac{1}{\cos (4 \pi \sigma )-1}. \end{aligned}$$

(A.7)

Notice that in our construction, the mass parameter of local ${\mathbb P}^1 \times {\mathbb P}^1$ and $\hbar$ are both positive. Hence, $\sigma$ is purely imaginary and $\sigma \ne 0$; therefore, $\Pi _1(z_2,q)$ is perfectly well-defined. Similarly, for the other $\Pi _n(z_2,q)$. It follows that

$$\begin{aligned} \mu - {t(\mu , \hbar ) \over 2} \xrightarrow {4D} \log (1+\mathrm{{e}}^{4 \pi \mathrm{{i}}\sigma }) . \end{aligned}$$

(A.8)

Appendix B: Standard and NS free energies

The free energy of the standard topological string at large radius was computed in [51, 99] and it reads

$$\begin{aligned} F^\mathrm{GV}(\mathbf{{t}}, g_s)=\sum _{g\ge 0} \sum _\mathbf{d} \sum _{w=1}^\infty {1\over w} n_g^{ \mathbf{d}} \left( 2 \sin { w g_s \over 2} \right) ^{2g-2} \mathrm{{e}}^{-w \mathbf{d} \cdot \mathbf{t}} . \end{aligned}$$

(B.1)

The variable $\mathbf{{t}}$ denotes the Kähler parameters of the geometry, $g_s$ the string coupling, and $n_g^\mathbf{d}$ the Gopakumar–Vafa invariants. These can be easily computed with the topological vertex [99] formalism or the holomorphic anomaly equation [50]. For the local ${\mathbb P}^1 \times {\mathbb P}^1$ geometry, we have, for instance

$$\begin{aligned} F^\mathrm{GV}(\mathbf{{t}}, g_s)= & {} \frac{2 q( \mathrm{{e}}^{-t_1} + \mathrm{{e}}^{-t_2})}{(q-1)^2}+\frac{4 q \mathrm{{e}}^{-t_1} \mathrm{{e}}^{-t_2}}{(q-1)^2}+\frac{q^2 \mathrm{{e}}^{-2t_1}}{(q-1)^2 (q+1)^2}\nonumber \\&+\frac{q^2 \mathrm{{e}}^{-2t_2}}{(q-1)^2 (q+1)^2}+ \mathcal {O}(\mathrm{{e}}^{-3 t_i}) , \end{aligned}$$

(B.2)

where $q=\mathrm{{e}}^{\mathrm{{i}}g_s}$.

Similarly the free energy of refined topological string in the NS limit reads [59]

$$\begin{aligned} F^\mathrm{NS}(\mathbf{{t}}, \hbar )={1\over 6 \hbar } a_{ijk} t_i t_j t_k +b^\mathrm{NS}_i t_i \hbar + F^\mathrm{NS}_\mathrm{inst}(\mathbf{{t}}, \hbar ) -\sum _{n\ge 1}\frac{1}{n^2}{\mathrm{e}^{-n t_2 } \cot \left( \frac{n \hbar }{2}\right) }\qquad \end{aligned}$$

(B.3)

where

$$\begin{aligned} \begin{array}{l} \displaystyle F^\mathrm{NS}_\mathrm{inst}(\mathbf{{t}}, \hbar )= \sum _{j_L, j_R} \sum _{w, \mathbf{d} } N^{\mathbf{d}}_{j_L, j_R} \frac{\sin \frac{\hbar w}{2}(2j_L+1)\sin \frac{\hbar w}{2}(2j_R+1)}{2 w^2 \sin ^3\frac{\hbar w}{2}} \mathrm{{e}}^{-w \mathbf{d}\cdot \mathbf{t}},\\ \displaystyle \quad \mathbf{d}=\{d_1, d_2\}, \quad d_1>0, \quad d_2 \ge 0 \end{array} \end{aligned}$$

(B.4)

and $N_{j_L,j_R}^\mathbf{d}$ denote the refined BPS invariants [52, 53]. These can be computed using the refined topological vertex [59] or the refined holomorphic anomaly [100, 101]. The last term in (B.3) is often called the one-loop contribution to the NS free energy.

For the local ${\mathbb P}^1 \times {\mathbb P}^1$ geometry, we have, for instance

$$\begin{aligned} F^\mathrm{NS}(\mathbf{{t}}, \hbar )=\ \frac{t_1^3}{6 \hbar }-\frac{t_1^2 (t_1-t_2)}{4 \hbar }-\frac{t_1 \hbar }{12}- \cot \left( \frac{\hbar }{2}\right) \left( \mathrm{{e}}^{-t_1}+\mathrm{{e}}^{-t_2}\right) + \mathcal {O}(\mathrm{{e}}^{-2 t_i}).\nonumber \\ \end{aligned}$$

(B.5)

The expressions (B.1) and (B.3) are valid at the large radius point of the moduli space, where

$$\begin{aligned} t_2, t_1 \rightarrow \infty . \end{aligned}$$

(B.6)

However, thanks to the refined topological vertex formalism [26, 59], it is possible to perform a partial resummation in $t_2$ to obtain an expression which is valid around

$$\begin{aligned} t_1 \rightarrow \infty , \quad t_2 \rightarrow 0. \end{aligned}$$

(B.7)

As an example, we consider the standard free energy of local ${\mathbb P}^1 \times {\mathbb P}^1$. Using the refined topological vertex, we obtain

$$\begin{aligned}&F^\mathrm{GV}(\mathbf{{t}}, g_s)=F_\mathrm{ol}(\mathrm{{e}}^{-t_2})+\frac{2 q \mathrm{{e}}^{-t_1}}{(q-1)^2 (\mathrm{{e}}^{-t_2}-1)^2}\nonumber \\&\quad +\,\frac{q^2 \mathrm{{e}}^{-2t_1}\left( q^2 \mathrm{{e}}^{-4t_1}+q^2+4 q (q+1)^2 \mathrm{{e}}^{-3t_2}-2 (q (q+1) (q (q+3)+4)+1) \mathrm{{e}}^{-2 t_2}+4 q (q+1)^2 \mathrm{{e}}^{-t_2}\right) }{(q-1)^2 (q+1)^2 (\mathrm{{e}}^{-t_2}-1)^4 (q-\mathrm{{e}}^{-t_2})^2 (q \mathrm{{e}}^{-t_2}-1)^2}\nonumber \\&\quad +\,\mathcal {O}(\mathrm{{e}}^{-3t_1}) , \quad q=\mathrm{{e}}^{\mathrm{{i}}g_s} , \end{aligned}$$

(B.8)

where $F_\mathrm{ol}(\mathrm{{e}}^{-t_2})$ is what we call the one-loop contribution of the standard topological strings. In Appendix C, we show that, when this is appropriately combined with the one-loop contribution of the NS free energy, one can resum it using the methods of [54] .

Similarly, using the refined topological vertex, one has

$$\begin{aligned} F^\mathrm{NS}_\mathrm{inst}(\mathbf{{t}}, \hbar )=\frac{\mathrm{{i}}\mathrm{e}^{\mathrm{{i}}\hbar } \left( 1+\mathrm{e}^{\mathrm{{i}}\hbar }\right) \mathrm{{e}}^{-t_1} }{\left( -1+\mathrm{e}^{\mathrm{{i}}\hbar }\right) \left( -\mathrm{{e}}^{-t_2}+\mathrm{e}^{i \hbar }\right) \left( -1+\mathrm{{e}}^{-t_2} \mathrm{e}^{\mathrm{{i}}\hbar }\right) }+\mathcal {O}(\mathrm{{e}}^{-2 t_1}). \end{aligned}$$

(B.9)

Appendix C: Integral representation for the one-loop contribution

In this appendix, we use the results of [54] to compute the one-loop part (3.8) of the spectral determinant (2.5).

It was shown in [54] that

$$\begin{aligned}&\sum _{m=1}^\infty \frac{(-1)^m}{ 2 m}\left( \sin \frac{m g_s}{2}\right) ^{-2} \mathrm{{e}}^{-m t} -\sum _{\ell =1}^\infty \frac{1}{4\pi \ell ^2} 2\csc \left( \frac{2\pi ^2 \ell }{g_s}\right) \nonumber \\&\qquad \times \,\left[ \frac{2\pi \ell }{g_s}\left( t\right) +\frac{2\pi ^2 \ell }{g_s} \cot \left( \frac{2\pi ^2 \ell }{g_s}\right) +1 \right] \mathrm{{e}}^{-\frac{2\pi \ell t}{g_s} } \nonumber \\&\quad = 2\frac{\text {Li}_3(-\mathrm{{e}}^{-t})}{g_s^2} -2\int _0^{\infty } \mathrm{{d}}x\frac{x}{\mathrm{{e}}^{2\pi x}-1}\log (1+\mathrm{{e}}^{-2t}+2\mathrm{{e}}^{-t}\cosh g_sx). \end{aligned}$$

(C.1)

With some algebraic manipulations and by following [54], we can write it as

$$\begin{aligned}&\sum _{m=1}^\infty \frac{1}{ 2 m}\left( \sin \frac{m g_s}{2}\right) ^{-2} \mathrm{{e}}^{-m t} -\sum _{\ell =1}^\infty \frac{1}{4\pi \ell ^2} 2\cot \left( \frac{2\pi ^2 \ell }{g_s}\right) \left[ \frac{2\pi \ell }{g_s}t+1 \right] \mathrm{{e}}^{-\frac{2\pi \ell t}{g_s} }\nonumber \\&\quad -{\pi \over g_s}\sum _{\ell =1}^\infty {1\over \ell } \cot \left( \frac{2\pi ^2 \ell }{g_s}\right) ^2 \mathrm{{e}}^{-\frac{2\pi \ell t}{g_s} } =-{\pi \over g_s} \log \left( 1-\mathrm{{e}}^{-\frac{2\pi t}{g_s} }\right) +2 \frac{\text {Li}_3(\mathrm{{e}}^{-t})}{g_s^2} \nonumber \\&\quad - 2 \mathrm{Re}\int _0^{\infty \mathrm{{e}}^{\mathrm{{i}}0}} \mathrm{{d}}x\frac{x}{\mathrm{{e}}^{2\pi x}-1}\log (1+\mathrm{{e}}^{-2t}-2\mathrm{{e}}^{-t}\cosh g_sx), \end{aligned}$$

(C.2)

which reproduces precisely (3.8). This means that the one-loop part of the standard topological string plus the one-loop part of the NS limit of topological string sum up to give the non-perturbative free energy of topological string on the resolved conifold, as given in [54].

Appendix D: Spectral determinant and Painlevé III equation

In this section, we briefly review the results of [28]. These results will be relevant in Sect. 3.

We define the Zamolodchikov spectral determinant $\Xi _\mathrm{Z}(\kappa , t)$ as

$$\begin{aligned} \Xi _\mathrm{Z}(\kappa , t)=\sum _{N\ge 0} \kappa ^N{D_N(t)\over N!}, \end{aligned}$$

(D.1)

where

$$\begin{aligned} D_N(t)= \int \prod _{i=1}^N {\mathrm{d}z_i\over z_i}\mathrm{{e}}^{- 2 \sum _{i=1}^N u(z_i, t)} \frac{\prod _{i< j} (z_i-z_j)^2}{\prod _{i <j} (z_i+z_j)^2}, \end{aligned}$$

(D.2)

with

$$\begin{aligned} u(z,t)={t\over 4} z+{t\over 4} z^{-1}, \quad t>0. \end{aligned}$$

(D.3)

From [28, 102, 103], it follows that $\Xi _\mathrm{Z}(\kappa , t)$ satisfies

$$\begin{aligned} 4 \left( \left( {\mathrm{d}\over \mathrm{d}t}\right) ^2+{1\over t}{\mathrm{d}\over \mathrm{d}t}\right) (-\log \Xi _\mathrm{Z}(\kappa ,t))=\left( { \Xi _\mathrm{Z}(-\kappa ,t)\over \Xi _\mathrm{Z}(\kappa ,t)}\right) ^2-1. \end{aligned}$$

(D.4)

Similarly

$$\begin{aligned} U(\kappa ,t)=2 \mathrm{{i}}\log \left( \mathrm{{i}}\Xi _\mathrm{Z}(-\kappa ,t)/ \Xi _\mathrm{Z}(\kappa ,t)\right) \end{aligned}$$

(D.5)

satisfies

$$\begin{aligned} \left( \left( {\mathrm{d}\over \mathrm{d}t}\right) ^2+{1\over t}{\mathrm{d}\over \mathrm{d}t}\right) U(\kappa ,t) =-\sin ( U(\kappa ,t) ). \end{aligned}$$

(D.6)

In the context of Painlevé equations, it useful to introduce the so-called $\tau$ function which is related to the solution of (D.6) as

$$\begin{aligned} \mathrm{{e}}^{-\mathrm{{i}}U(\kappa ,t) }={4 \over t }{\mathrm{d} \over \mathrm{d}t} t {\mathrm{d} \over \mathrm{d}t}\log \tau (t^4 2^{-12}, \kappa ). \end{aligned}$$

(D.7)

From (D.4), it follows that

$$\begin{aligned} {4\over t}{\mathrm{d}\over \mathrm{d}t}t {\mathrm{d}\over \mathrm{d}t} \log \left[ \Xi _\mathrm{Z}(\kappa ,t)\mathrm{{e}}^{-t^2/16 }\right] =\left( \mathrm{{i}}{ \Xi _\mathrm{Z}(-\kappa ,t)\over \Xi _\mathrm{Z}(\kappa ,t)}\right) ^2. \end{aligned}$$

(D.8)

This means that

$$\begin{aligned} \Xi _\mathrm{Z}(\kappa ,t)\mathrm{{e}}^{-t^2/16 } \end{aligned}$$

(D.9)

is the $\tau$ function corresponding to the solution of (D.6). The small t expansion of $\Xi _Z(\kappa ,t)$ was also computed in [28], where the author shows that

$$\begin{aligned} \Xi _\mathrm{Z}(\kappa ,t)\approx & {} \left( {t \over 8}\right) ^{4 \sigma ^2-\frac{1}{4}} \exp \left[ 3 \zeta '(-1)+{5\over 6}\log 2 -4 \sigma ^2 (\log (8)-1)+2 \sigma \text {log}\Gamma (-2 \sigma ) \right] \nonumber \\&\times \exp \left[ -2 \sigma \text {log}\Gamma (2 \sigma )+\psi ^{(-2)}(-2 \sigma )+\psi ^{(-2)}(2 \sigma )\right] . \end{aligned}$$

(D.10)

The variable $\sigma$ in (D.10) is related to $\kappa$ through

$$\begin{aligned} 2 \pi \kappa =\cos {2\pi \sigma } \end{aligned}$$

(D.11)

and we can assume without loss of generalities $0 \le \mathrm{Re}(\sigma ) \le 1/2$.

Moreover, for small values of t, it was shown in [28] that

$$\begin{aligned} \mathrm{{e}}^{\mathrm{{i}}U(\kappa ,t)} \approx - \left( {t\over 8}\right) ^{8\sigma -2} {\Gamma ^2\left( {1}-{2\sigma }\right) \over \Gamma ^2\left( {2\sigma }\right) }. \end{aligned}$$

(D.12)

In particular, the monodromy data of the related Fuchsian system for this solution are

$$\begin{aligned} (\sigma , \eta =0). \end{aligned}$$

(D.13)