Large N instantons, BPS states, and the replica limit

Abstract

We study the relation between large N instantons and conventional instantons, focusing on matrix models and topological strings. We show that the resurgent properties of the perturbative series at fixed but arbitrary N, including the replica limit N = 0, can be obtained from large N instantons. In the case of topological strings, it has been conjectured that the resurgent structure encoded by large N instantons is closely related to the spectrum of BPS states. We give direct evidence for this connection in the case of Seiberg-Witten theory and other topological string models, and we show in detail how the resurgent properties at fixed N follow from the large N theory, and therefore can be used to obtain information on BPS invariants.

Figures (5)

• Figure 1: The dots represent the umerical results for $\mu_{0,1}$ (left and right, respectively) for the values of $N$ of the form $i/50$, $i=1, \cdots, 49$, while the lines are the theoretical predictions (\ref{['preds']}).
• Figure 2: The curve of marginal stability in the $u$-plane, defined by the equation (\ref{['cms-eq']}).
• Figure 3: The Borel singularities of ${\cal F}^{\rm reg}(t)$ in SW theory, inside the curve of marginal stability with $u=1/2$ (left), and outside the curve with $u=5/2$ (right). The black dot in the figure in the left signals the value $2 \pi (a + a_D)$, and the black dots in the figure in the right signal the value $2 \pi a$ (on the real axis) and $2 \pi (a+a_D)$ (above the axis).
• Figure 4: In these figures, the continuous line represents the real (left) and imaginary (right) parts of the function $\mu(N)$ defined in (\ref{['fn-ex']}), as a function of $0<N<1$. The dots represent the numerical results obtained from the large order behavior of the series (\ref{['fNSW']}).
• Figure 5: The Borel singularities of ${\cal F}^{\rm reg} (t_c)$ in local ${\mathbb F}_0$, for $z=3/32>1/16$ (left), and $z=1/32<1/16$ (right). These regions are the analogues of the strong coupling (respectively, weak coupling) region in SW theory. The black dots in the figure in the left signal the value $2 \pi (t +{\rm i} t_c)$ on the real axis, $4 \pi^2 {\rm i} + 2\pi {\rm i} t_c$ on the imaginary axis, and $2 \pi (t + {\rm i} t_c) +4 \pi^2 {\rm i} +2\pi {\rm i} t_c$ in the first quadrant. The black dots in the figure in the right signal the value $2 \pi t$ on the real axis, $4 \pi^2 {\rm i} + 2\pi {\rm i} t_c$ on the imaginary axis, and $2 \pi t \pm 2 \pi {\rm i} t_c$ above (respectively, below) the real axis. There is also a point $2 \pi t + 4 \pi^2 {\rm i}$ in the first quadrant.

Theorems & Definitions (1)

• Remark 2.1