Resurgent Wilson loops in refined topological string

TL;DR

The paper investigates the resurgence of refined Wilson loops in topological string theory, showing that their Borel singularities are integral periods and that the associated Stokes constants coincide with refined Donaldson-Thomas invariants. By solving the non-perturbative trans-series through refined holomorphic anomaly equations, it provides a closed-form construction of non-perturbative corrections and demonstrates that Stokes data are frame-independent and DT-predicted. The authors also clarify the behavior in limiting regimes (unrefined and NS) and verify the framework with explicit checks on local ${\mathbb P}^2$ and local ${\mathbb P}^1\times {\mathbb P}^1$, including detailed Borel-plane analyses and large-order consistency with trans-series. These results strengthen the connection between non-perturbative topological string physics, D-brane bound states, and refined DT invariants, and they offer a practical route to compute non-perturbative Wilson loop effects.

Abstract

We study the resurgent structures of Wilson loops in refined topological string theory. We argue that the Borel singularities should be integral periods, and that the associated Stokes constants are refined Donaldson-Thomas invariants, just like the free energies, except that the Borel singularities cannot be local flat coordinates. We also solve the non-perturbative series in closed form from the holomorphic anomaly equations for the refined Wilson loops. We illustrate these results with the examples of local P^2 and local P^1 x P^1.

Figures (26)

• Figure 5.1: Borel singularities of refined Wilson loop BPS sectors $\mathcal{F}[1](\mathsf{b};g_s)$ for local $\mathbb{P}^2$ with $\mathsf{b}=2$ up to $g=50$ in the large radius frame, respectively (a) near the large radius point $z=0$ and (b) near the conifold point $z=-1/27$. The red dots are approximate singularities from numerical calculations, which would accumulate to branch cuts. The branch points (black dots) on the imaginary axis are $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,0,0)_\text{LR}}$, and those in the four quadrants are $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,1,0)_\text{LR}}$ and $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,-1,-1)_\text{LR}}$.
• Figure 5.2: Borel singularities of refined Wilson loop BPS sectors $\mathcal{F}[2](\mathsf{b};g_s)$ for local $\mathbb{P}^2$ with $\mathsf{b}=2$ up to $g=50$ in the large radius frame, respectively (a) near the large radius point $z=0$ and (b) near the conifold point $z=-1/27$. The red dots are approximate singularities from numerical calculations, which would accumulate to branch cuts. The branch points (black dots) are the same as in Fig. \ref{['fig:P2C1b2LR-brl']}.
• Figure 5.3: Borel singularities of refined free energies $\mathcal{F}[0](\mathsf{b};g_s)$ for local $\mathbb{P}^2$ with $\mathsf{b}=2$ up to $g=50$ in the large radius frame, respectively (a) near the large radius point $z=0$ and (b) near the conifold point $z=-1/27$. The red dots are approximate singularities from numerical calculations. The branch points (black dots) on the imaginary axis are $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,0,0)_{\text{LR}}}$, and those away from the imaginary axis are $\mathsf{b}^{-1}\mathcal{A}_{\pm(0,1,n)_\text{LR}} (n=-1,0,1,2)$.
• Figure 5.4: Borel singularities of refined Wilson loop BPS sectors $\mathcal{F}[1](\mathsf{b};g_s)$ for local $\mathbb{P}^2$ with $\mathsf{b}=2$ up to $g=50$ in the conifold frame, respectively (a) near the large radius point with $z<0$, (c) near the conifold point, and (c) with $z>0$. The red dots are approximate singularities from numerical calculations. The branch points (black dots) are (a) $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,0,n)_\text{c}} (n=-1,0,1,2)$, (b) $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,1,0)_{\text{c}}}$, $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,-1,1)_{\text{c}}}$ and (c) $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,0,0)_{\text{c}}}$ (horizontal), $\mathsf{b}^{-1}\mathcal{A}_{\pm(-3,-1,0)_{\text{c}}}$ respectively.
• Figure 5.5: Borel singularities of refined Wilson loop BPS sectors $\mathcal{F}[2](\mathsf{b};g_s)$ for local $\mathbb{P}^2$ with $\mathsf{b}=2$ up to $g=50$ in the conifold frame, respectively (a) near the large radius point with $z<0$, (b) near the conifold point $z=-1/27$, and (c) with $z>0$. The red dots are approximate singularities from numerical calculations. The branch points (black dots) are the same as in Fig. \ref{['fig:P2C1b2Con-brl']}.