There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In this thesis we consider certain supersymmetric string theories on curved backgrounds containing the three-dimensional anti-de Sitter space, AdS3. On the one hand, these are gravity theories in three dimensions, which makes them simpler than real-world gravity but still rich in interesting phenomena. On the other hand, according to the holographic duality, each of these gravitational theories should be equivalent to a suitable quantum field theory with conformal symmetry in two dimensions, a CFT2. In that low dimension, such conformal field theories enjoy an infinite-dimensional Virasoro symmetry which strongly constrains them, but remain non-trivial. Our study is confined to a particular regime, the 't Hooft limit, in which the strings propagate freely. Free string theory can be described as a non-linear sigma model from the string worldsheet (geometrically, a two-dimensional cylinder) into a target space of the form AdS3xM, where M is a manifold without boundary. For us, M is a product of circles and three-dimensional spheres, namely M=S3xT4 or M=S3xS3xS1, both of which enjoy 16 supersymmetries. For the sake of brevity, our presentation is mostly focused on the massive sector of the former geometry, while more general cases are discussed in the concluding chapter. Our main result is to describe how these theories are integrable at the quantum level, i.e. how their worldsheet scattering matrix factorises and their spectrum can be efficiently computed by Bethe ansatz techniques. This is done both from the point of view of the worldsheet theory and of a dual spin chain picture. We also discuss how this procedure is supported by overwhelming evidence from independent (perturbative or semiclassical) calculations, up to two loops in the worldsheet theory expansion.

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Towards integrability for AdS3/CFT2

Consistent formulations of relativistic viscous hydrodynamics involve short-lived modes, leading to asymptotic rather than convergent gradient expansions. In this Letter we consider the Muller-Israel-Stewart theory applied to a longitudinally expanding quark-gluon plasma system and identify hydrodynamics as a universal attractor without invoking the gradient expansion. We give strong evidence for the existence of this attractor and then show that it can be recovered from the divergent gradient expansion by Borel summation. This requires careful accounting for the short-lived modes which leads to an intricate mathematical structure known from the theory of resurgence.

Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Theta angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model

Ten years ago, relativistic viscous fluid dynamics was formulated from first principles in an effective field theory framework, based entirely on the knowledge of symmetries and long-lived degrees of freedom. In the same year, numerical simulations for the matter created in relativistic heavy-ion collision experiments became first available, providing constraints on the shear viscosity in QCD. The field has come a long way since then. We present the current status of the theory of non-equilibrium fluid dynamics in 2017, including the divergence of the fluid dynamic gradient expansion, resurgence, non-equilibrium attractor solutions, the inclusion of thermal fluctuations as well as their relation to microscopic theories. Furthermore, we review the theory basis for numerical fluid dynamics simulations of relativistic nuclear collisions, and comparison of modern simulations to experimental data for nucleus-nucleus, nucleus-proton and proton-proton collisions.

Relativistic Fluid Dynamics In and Out of Equilibrium -- Ten Years of Progress in Theory and Numerical Simulations of Nuclear Collisions

A primer on resurgent transseries and their asymptotics

Nonperturbative eects in string theory are usually associated to D{branes. In many cases it can be explicitly shown that D{brane instantons control the large{order behavior of string perturbation theory, leading to the well{known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large{order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non{linear systems, which are developed at length herein as they apply to string theory. The cases of topological strings, the Painlev e I equation describing 2d quantum gravity, and the quartic matrix model, are explicitly addressed. These results generalize to minimal strings and general matrix models. It is shown that, in order to completely understand string theory at a fully nonperturbative level, new sectors are required beyond the standard D{brane sector.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2012/0006/0002/CNTP-2012-0006-0002-a003.pdf

The Resurgence of Instantons in String Theory

In a wide range of quantum theoretical settings—from quantum mechanics to quantum field theory, from gauge theory to string theory—singularities in the complex Borel plane, usually associated to instantons or renormalons, render perturbation theory ill-defined as they give rise to nonperturbative ambiguities. These ambiguities are associated to choices of an integration contour in the resummation of perturbation theory, along (singular) Stokes directions in the complex Borel plane (rendering perturbative expansions non-Borel summable along any Stokes line). More recently, it has been shown that the proper framework to address these issues is that of resurgent analysis and transseries. In this context, the cancelation of all nonperturbative ambiguities is shown to be a consequence of choosing the transseries median resummation as the appropriate family of unambiguous real solutions along the coupling-constant real axis. While the median resummation is easily implemented for one-parameter transseries, once one considers more general multi-parameter transseries the procedure becomes highly dependent upon properly understanding Stokes transitions in the complex Borel plane. In particular, all Stokes coefficients must now be known in order to explicitly implement multi-parameter median resummations. In the cases where quantum-theoretical physical observables are described by resurgent functions and transseries, the methods described herein show how one may cancel nonperturbative ambiguities, and define these observables nonperturbatively starting out from perturbation theory. Along the way, structural results concerning resurgent transseries are also obtained.

Nonperturbative Ambiguities and the Reality of Resurgent Transseries

Localization methods have recently led to a plethora of new exact results in supersymmetric gauge theories, as certain observables may be computed in terms of matrix integrals. These can then be evaluated by making use of standard large N techniques, or else via perturbative expansions in the gauge coupling. Either approximation often leads to observables given in terms of asymptotic series, which need to be properly defined in order to obtain nonperturbative results. At the same time, resurgent analysis has recently been successfully applied to several problems, e.g., in quantum, field and string theories, precisely to overcome this issue and construct nonperturbative answers out of asymptotic perturbative expansions. The present work uses exact results from supersymmetric localization to address the resurgent structure of the free energy and partition function of Chern-Simons and ABJM gauge theories in three dimensions, and of $$ \mathcal{N}=2 $$
 supersymmetric Yang-Mills theories in four dimensions. For each case, the complete structure of Borel singularities is exactly determined, and the relation of these singularities with the large-order behavior of (multi-instanton) perturbative expansions is made fully precise.

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Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

It has recently been understood that the hydrodynamic series generated by the M\"uller-Israel-Stewart theory is divergent, and that this large order behaviour is consistent with the theory of resurgence. Furthermore, it was observed, that the physical origin of this is the presence of a purely damped nonhydrodynamic mode. It is very interesting to ask whether this picture persists in cases where the spectrum of nonhydrodynamic modes is richer. We take the first step in this direction by considering the simplest hydrodynamic theory which, instead of the purely damped mode, contains a pair of nonhydrodynamic modes of complex conjugate frequencies. This mimics the pattern of black brane quasinormal modes which appear on the gravity side of the AdS/CFT description of \symm\ plasma. We find that the resulting hydrodynamic series is divergent in a way consistent with resurgence and precisely encodes information about the nonhydrodynamic modes of the theory.

Inês Aniceto

Papers

A primer on resurgent transseries and their asymptotics

The Resurgence of Instantons in String Theory

Nonperturbative Ambiguities and the Reality of Resurgent Transseries

Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

Resurgence in extended hydrodynamics