Finite N from Resurgent Large N

TL;DR

This work addresses how to extract finite $N$ gauge-theory observables from the inherently asymptotic large $N$ expansion by constructing a resurgent transseries that combines perturbative $1/N$ terms with nonperturbative instanton sectors. Using the quartic matrix model as a testing ground, the authors derive a transseries for the free energy $F(N,t)$ and recurrence data, featuring an instanton action $A(t)$ and a two-step Borel--Écalle resummation (often via Padé approximants) to yield finite $N$ results that agree with exact calculations to high precision. They show how Stokes phenomena control which sectors contribute as one moves in the complex $t$-plane, and how analytic continuation to complex $N$ is made possible by the resurgent structure. The framework demonstrates that instantons are essential for finite-$N$ physics and provides a nonperturbative definition of gauge theories as functions of complex $N$, with potential extensions to other matrix models and topological strings.

Abstract

Due to instanton effects, gauge-theoretic large N expansions yield asymptotic series, in powers of 1/N^2. The present work shows how to generically make such expansions meaningful via their completion into resurgent transseries, encoding both perturbative and nonperturbative data. Large N resurgent transseries compute gauge-theoretic finite N results nonperturbatively (no matter how small N is). Explicit calculations are carried out within the gauge theory prototypical example of the quartic matrix model. Due to integrability in the matrix model, it is possible to analytically compute (fixed integer) finite N results. At the same time, the large N resurgent transseries for the free energy of this model was recently constructed. Together, it is shown how the resummation of the large N resurgent transseries matches the analytical finite N results up to remarkable numerical accuracy. Due to lack of Borel summability, Stokes phenomena has to be carefully taken into account, implying that instantons play a dominant role in describing the finite N physics. The final resurgence results can be analytically continued, defining gauge theory for any complex value of N.

Figures (15)

• Figure 1: Illustration of the (double) monodromy of the confluent hypergeometric function of the second kind, for fixed values of $a=\frac{3}{4}$ and $b=\frac{1}{2}$, and different values of $|z|=0.1$ (lower right), $|z|=1.1$ (upper right), and $|z|=10.1$ (left). The figures plot $U \left( \left. a,b\, \right| z \right)$ over the complex $z$-plane as $\arg z \in (-\pi,3\pi)$, with the solid line representing the first turn, $\arg z \in (-\pi,\pi)$, and the dotted line the second, $\arg z \in (\pi,3\pi)$. Note the difference in scales between principal and secondary sheets, increasingly significant as $|z|$ grows. In the left plot we have inclosed a zoom-in close to the origin, in order to show the trajectory along the first sheet in more detail.
• Figure 2: Monodromy of the recursion coefficient $r_3$, for different values of $|t|=0.7$ (left), $|t|=1.2$ (upper right), and $|t|=2$ (lower right). The solid line corresponds to the first sheet, $\arg(t) \in (-\pi, \pi)$, and the dotted line to the second, $\arg(t) \in (\pi, 3\pi)$. In the left plot we inclose a zoom-in close to the origin, to see the trajectory along the first sheet in more detail.
• Figure 3: Monodromy of the free energy ${\cal F}(3)$, when $N=3$, for $|t|=0.6$ (left); and of the free energy ${\cal F}(2)$, when $N=2$, for different values $|t|=0.8$ (upper right) and $|t|=0.6$ (lower right). As usual, the solid line corresponds to $\arg(t) \in (-\pi, \pi)$ while the dotted line now corresponds to $\arg(t) \in \pm (\pi, \pi+ \delta\theta)$, where we set $\delta\theta=\pi/4$ (left), $\delta\theta=7\pi/8$ (upper right), and $\delta\theta=\pi$ (lower right). In some plots we have not included the full dotted curves as they show no more relevant features beyond what is displayed (their range gets naturally enlarged by force of the logarithm).
• Figure 4: Approximate complex Borel $s$-planes for the perturbative (left, ${\cal F}^{(0)}$) and one-instanton (right, ${\cal F}^{(1)}$) free energies obtained by plotting poles of the Padé approximant when $t=2$. Due to limited computational resources we have less points for higher instanton sectors, as compared to the perturbative sector. Still, the instanton action singularities are very clear, with the accumulation of poles signaling their associated logarithmic branch cuts.
• Figure 5: Number of decimal places up to which the resummation of the recursion coefficients (left) and the free energy (right) match their exact counterparts, with $N=1,\dots,5$ and $t=6$.