A Primer on Resurgent Transseries and Their Asymptotics

Abstract

The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries---usually described mathematically via alien calculus---are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.

Figures (41)

• Figure 1: The quartic potential \ref{['quarticpotential']} in the complex $z$-plane, with ${\mathbb{R}}{\mathrm{e}}\, \lambda > 0$ ($\lambda=1$ in the plot). The brown (darker) region is where ${\mathbb{R}}{\mathrm{e}}\, V(z) > 0$, while the blue (lighter) region is where ${\mathbb{R}}{\mathrm{e}}\, V(z) < 0$. The four contours are the admissible contours, leading to three homologically independent integration paths. The ${\mathbb Z}_2$ symmetry $V(z) = V(-z)$ of the quartic potential \ref{['quarticpotential']} further narrows down the number of independent results one may compute over homologically distinct contours.
• Figure 2: Steepest descent (ascent) contours for the quartic integral, through the several saddles (black for the "perturbative" saddle and red for the "instanton" saddle), along with the positivity of the potential, for different values of $\theta = \arg x$ ($|x|=1$ in the plot). This makes clear which are Stokes and anti-Stokes lines: Stokes lines at $\arg x = 0, \pi$ when the steepest descent contours hit the three saddles; anti-Stokes lines at $\arg x = \frac{\pi}{2}, \frac{3\pi}{2}$. See the main text for a full discussion of the plot sequence.
• Figure 3: The first image shows the coupling constant $x$-plane, representing the phase diagram of the quartic partition function. Stokes lines are in red (solid), at $\arg x = 0,\pi$, while anti-Stokes lines are in blue (dashed), at $\arg x = \frac{\pi}{2}, \frac{3\pi}{2}$. The second image shows the Lee--Yang accumulation of zeros of the modified Bessel function describing the quartic partition-function, signaling a phase transition. One can clearly see that indeed the anti-Stokes lines correspond to phase boundaries.
• Figure 4: The monodromy of the transseries solution to the quartic integral. We start at $\arg x = 0^-$ with ${\cal Z} (0.21\, {\mathrm{e}}^{{\mathrm{i}} \theta}, 1, 0)$, and rotate $\theta$ four times around the origin according to \ref{['quarticmonodromy']}. This four-fold rotation is abstractly represented in the upper plot, where the different colors/types are signaling different regions (on the Borel plane) in-between Stokes transitions. For instance, at $2\pi n$, the Stokes transition \ref{['crossingthetazero']} takes place, while at $\pi + 2\pi n$ it is \ref{['crossingthetapi']} instead ($n=0,1,2,3$). In this way, each color/type occurs in an interval of length $\pi$ and the complete sequence yields the full $8\pi$ rotation as shown. As we represent this four-fold rotation back into the $x$-plane (illustrated in the lower plots, with matching colors/types as in the upper plot), it is clear how the transseries trajectory closes upon itself, fulfilling the predicted monodromy and showing how indeed the transseries encodes the full nonperturbative solution. The axes in the lower plots correspond to real and imaginary parts of the partition function. A word on numerics: we have resummed the asymptotic series with Padé approximants, and chosen relatively small $|x|$ for improved accuracy. This, however, results in the different scales appearing in the lower plots due to the exponential factor. The central figure is plotted with a logarithmic scale and also gives an (almost complete) overview of the trajectory. The exponential factor makes it hard to understand what happens closer to the origin and, hence, the lateral images zoom-in onto the small "loops" which are already visible in the central plot.
• Figure 5: The left Borel resummation ${\cal S}_{\theta^+}$ crosses the singular Stokes line along $\theta$, to yield the right Borel resummation ${\cal S}_{\theta^-}$ alongside the Stokes discontinuity ${\mathrm{Disc}}_\theta$. This discontinuity has a geometrical representation as sum over the (red) Hankel contours associated to each (multi-instanton) singularity.