Contents

# Contents

## Idea

What is called resurgence theory is a mathematical formalization of what in quantum field theory is known as reading off non-perturbative effects from the Feynman perturbation series (S-matrix) in perturbative quantum field theory. It tries to regard this non-converging formal power series in the coupling constant $g$ as an asymptotic series of a function involving non-analytic contributions such as $e^{-1/g^2}$.

In quantum mechanics and quantum field theory, renormalized perturbation theory produces expansions in powers of the coupling constant $g$. By appropriate resummation using the renormalization group one can also capture terms involving logarithmic contributions of the form $\log (g/g_0)$. However, it is well-known that, e.g., instantons contribute terms of order $e^{-c/g}$ or $e^{-c/g^2}$, whose Taylor expansion is identically zero at $g = 0$, so that they do not contribute at all to the power series expansion. These contributions are therefore intrinsically nonperturbative.

A transseries is an expansion of a function $f(x)$ as a power series in a vector $z$ whose components are fixed functions of $x$. In quantum mechanics and quantum field theory, the relevant case is $x=g$ or $x=g^2$ and $z_1=x$, $z_2=e^{-c/x}$, and in QFT usually also $z_3=\log(x/x_0)$. Clearly, transseries are more flexible than ordinary power series.

Resurgent functions are functions arising in the analysis of singular points of germs of analytic functions. Resurgent transseries are transseries that arise in methods for resummation that generalize Borel summation by looking at the obstructions for Borel summability such as renormalons.

It turns out that using resurgent transseries, one can obtain under certain conditions information from the nonperturbative sector, starting from the Feynman perturbation series alone. The basic idea is to construct from the perturbative series the terms appearing in the renormalization group equation (RGE) by Callan and Symanzik, to substitute into it the leading terms of an appropriate transseries, and to determine the coefficients so that they match the RGE to the order specified.

To check that the method really works one can apply it to simple examples form quantum mechanics where other methods can be used to compute nonperturbative information. For example, for the double well potential, this was done by Jentschura & Zinn-Justin 04 , and for some other cases by (Dunne-Unsal 14). For applications to gauge theories and to string theory, see (Marino 12); of course, in the latter cases there is no known alternative way of checking the results. A more mathematically oriented overview is presented in (Dorigoni 14)

Thus it seems possible (and sometimes is claimed) that the perturbative series already contains all nonperturbative information. The future will tell.

## References

### General

Original articles include

A clear account is in

• M. V. Berry, C. J. Howls, Hyperasymptotics for Integrals with Saddles, Proceedings: Mathematical and Physical Sciences Vol. 434, No. 1892 (Sep. 9, 1991), pp. 657-675 (jstor:51890)

See p. 9 of https://arxiv.org/abs/1206.1890

Review includes

• Marcos Marino, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings (arXiv:1206.6272)

• Daniele Dorigoni, An Introduction to Resurgence, Trans-Series (arXiv:1411.3585)

• Ovidiu Costin, The Mathematical Theory of Resurgence and Resummation, and Their Applications in Mathematics and Physics, 2017 (web)

• Gerald Dunne, Introduction to Resurgence, Trans-series and Non-perturbative Physics, 2018 (pdf)

Part of the above text is taken from