In general interacting theories — quantum mechanical, field, gauge, or string theories — perturbation theory is divergent: perturbative expansions have zero radius of convergence and seemingly cannot be summed. Nonperturbatively well-defined results can still be constructed out of perturbation theory by the uses of resurgence and transseries.
Asymptotic series require the use of resurgence and transseries in order for their associated observables to become nonperturbatively well-defined. Resurgent transseries encode 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 coefficients. By means of two very explicit examples, we plan to introduce the aforementioned resurgent, large- order asymptotics of general perturbative expansions, including discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus.
Time permitting, we will discuss advanced examples in matrix models, minimal and topological strings, and Jackiw-Teitelboim gravity. The discussion will focus both on the construction of the resurgent transseries, their resonance, and the computation of associated Stokes data.
The program of the lectures will include:
In this talk, we will explore the relationship between non critical string theories, SU(2) supersymmetric gauge theories and WKB analysis. In particular, we will examine the role of Seiberg-Witten geometry in the context SU(2) of theories, and how this is related to the concept of string duality. We will examine the role of WKB analysis in solving the equations that come out in this context. We will focus on the physical interpretation of the various quantities that can be computed in WKB analysis, and how they can be related to observables in Seiberg-Witten theory or string theory. We will conclude by presenting the author's work on WKB analysis of finite difference equations, that naturally appear in this context.
We will illustrate the importance of the above concepts in known solutions of the Painlevé equations (tronquée and tritronquée phases). Furthermore, we will expand on the above solutions by introducing some specific characteristics of these equations like resonance, relations between Stokes constants.Then, we will expand on this by explaining a conjecture on the analytic form of these constants that has been checked with our numerical method up to very high precision. Finally, we will talk about possible future directions.
In these talks we will report the findings of our unpublished work on resurgent properties of the Painlevé I and II equations. These equations play a fundamental role in Minimal String Theories as the specific heat of 2D (Super)-Quantum Gravity. We will do a short introduction to the topic of resurgence and explain the role of Stokes constants when constructing solutions to these equations. While reviewing these tools, we will present our new method for numerically calculating these constants in very general setups to reasonably high precision.
I will give a broad overview of techniques in resurgent analysis and transseries, as they apply in string theory and matrix models (and with some focus on Painlevé type equations).
We will motivate and introduce the study of conformal defects in superconformal field theories (SCFTs). We will show how symmetries constrain the anomaly coefficients of BPS defects. In the case of N=(2,2) surface defects in four-dimensional N=2 SCFTs these anomaly coefficients can be computed by studying a protected subsector captured by a vertex operator algebra, or two-dimensional chiral algebra.
I will review recent and on-going work concerning applications of resurgence within the realm of minimal-string and JT quantum gravities in 2d.
Integrability techniques have played a major role in the study the AdS/CFT correspondence, providing an accurate description of different string theoretic observables beyond the weak or strong coupling perturbation theory. However, the case of string on certain AdS_3 backgrounds provided new challenges in the form of massless excitations. Difficulties in incorporating these into the integrable description have led to disagreements concerning the energy of massive physical states.
In general integrable theories, massless and massive sectors can generally be treated separately. We know this cannot be the case in AdS_3, but a full TBA description of the interaction between the sectors is yet to be found. Surprisingly, such a description can found in a family of integrable field theories — homogeneous sine-Gordon models. Here, one can take a double scaling limit of the adjustable parameters and zoom into a regime described by a TBA where the massless sector does not decouple and contributes to the energy of massive particles at the same order as for which the Bethe ansatz would suffice in a massive theory.
Integrability techniques have played a major role in the study the AdS/CFT correspondence, providing an accurate description of different string theoretic observables beyond the weak or strong coupling perturbation theory. However, the case of string on certain AdS3 backgrounds provided new challenges in the form of massless excitations. Difficulties in incorporating these into the integrable description have led to disagreements concerning the energy of massive physical states.
In general integrable theories, massless and massive sectors can generally be treated separately. We know this cannot be the case in AdS3, but a full TBA description of the interaction between the sectors is yet to be found. Surprisingly, such a description can found in a family of integrable field theories — homogeneous sine-Gordon models. Here, one can take a double scaling limit of the adjustable parameters and zoom into a regime described by a TBA where the massless sector does not decouple and contributes to the energy of massive particles at the same order as for which the Bethe ansatz would suffice in a massive theory.
We study BPS surface defects in 4d superconformal field theories and show how symmetries constrain their anomaly coefficients. Focusing on N=(2,2) surface defects we review the protected subsector captured by a two-dimensional chiral algebra. We study the properties of the defect in this subsector and discuss how to compute the aforementioned anomaly coefficients.
We study BPS surface defects in 4d superconformal field theories and show how symmetries constrain their anomaly coefficients. Focusing on N=(2,2) surface defects we review the protected subsector captured by a two-dimensional chiral algebra. We study the properties of the defect in this subsector and discuss how to compute the aforementioned anomaly coefficients.
We will give introductions to the topics of Kahler geometry, geometry on the space of Kahler metrics and geometric quantization. We will then see how the 3 topics interact strongly with each other and will describe some examples. Namely we will describe how quantization in so-called real polarizations can sometimes be related to the (easier to define) quantization in Kahler polarizations. Geodesics for a natural metric structure on the space of Kahler metrics play a central role in this relation.
We study symmetry constraints on BPS surface defects in four-dimensional superconformal field theories, showing how supersymmetry imposes relations on anomaly coefficients. Turning to dynamics, we analyze a protected subsector of N=(2,2) surface defects that is captured by a two-dimensional chiral algebra. We discuss how to compute observables of interacting defects from the chiral algebra, including the aforementioned anomaly coefficients.
One can obtain a two-dimensional chiral algebra, or vertex operator algebra, as a protected subsector of any four-dimensional N>1 SCFT. In these lectures we will review the construction of the chiral algebra, and the basic properties that follow from its four-dimensional origin. We will also explore some of the consequences for four-dimensional physics.
Following a recent revival of interest in 2-dimensional gravity due to the holographic properties of Jackiw-Teitelboim gravity, we investigate the non-perturbative properties of such models using the tools offered by Resurgence. This leads to non-trivial results concerning the asymptotics of Weil-Petersson volumes and the instanton contributions to 2-dimensional topological gravity.
We study symmetry constraints on BPS surface defects in four-dimensional superconformal field theories, showing how supersymmetry imposes relations on anomaly coefficients. Turning to dynamics, we analyze a protected subsector of N=(2,2) surface defects that is captured by a two-dimensional chiral algebra. We discuss how to compute observables of interacting defects from the chiral algebra, including the aforementioned anomaly coefficients.
We study the spectrum of defect conformal field theories (CFTs), and show the existence of universal accumulation points in the defect spectrum. This is achieved by obtaining an inversion formula for the bulk to defect OPE, akin to the Lorentzian inversion formula of Caron-Huot for CFTs without defects. We conclude with an outlook for the bootstrap approach to defect CFTs, in particular for cases where extra symmetries, namely supersymmetry, prove constraining.
The perturbative expansions of many physical quantities are divergent, and defined only as asymptotic series. It is well known that this divergence reflects the existence of nonperturbative, exponentially damped contributions, which are not captured by a perturbative analysis. This connection between perturbative and non-perturbative contributions to a given physical observable can be systematically studied using the theory of resurgence, allowing us to construct a full non-perturbative solution from perturbative asymptotic data.
In this talk I will start by reviewing the essential role of resurgence theory, coupled to exponentially accurate numerical methods, in the description of the analytic solution behind an asymptotic series, and its relation to the so-called Stokes phenomena and phase transitions. I will then exemplify how these techniques can be applied to to the calculation of poles of solutions to the Painlevé I non-linear ODE, a subject of great interest in both mathematics and physics.
The non-equilibrium hydrodynamic phenomena in strongly coupled quantum systems such as the N=4 supersymmetric Yang-Mills plasma, can be addressed with the tools of the AdS/CFT correspondence, where calculations in the gravity side of the duality can be performed through a large proper-time expansion. This expansion is asymptotic and encodes a spectrum of non-hydrodynamic modes reflecting the frequencies of AdS black-brane quasinormal modes. The emerging structure for the description of physical quantities such as the energy density is that of a resurgent transseries.
This connection between perturbative and non-perturbative contributions can be systematically studied using the theory of resurgence, allowing us to construct a full non-perturbative solution from perturbative asymptotic data. In this talk, I will analyse the essential role of resurgence theory, coupled to exponentially accurate numerical methods, in the description of the analytic solution behind the asymptotic late time behaviour of the Yang-Mills plasma.
The perturbative expansions of many physical quantities are divergent, and defined only as asymptotic series. It is well known that this divergence reflects the existence of nonperturbative, exponentially damped contributions, such as instanton effects, which are not captured by a perturbative analysis. This connection between perturbative and non-perturbative contributions of a given physical observable can be systematically studied using the theory of resurgence, allowing us to construct a full non-perturbative solution from perturbative asymptotic data. In this talk I will start by reviewing the essential role of resurgence theory in the description of the analytic solution behind an asymptotic series, and its relation to the so-called Stokes phenomena, phase transitions and ambiguity cancelations. I will then exemplify with some recent applications of resurgence in the context of string and gauge theories.
I will present a broad and pedagogical overview of the resurgence program as it applies to topological string theory. In particular, we’ll have in mind understanding the asymptotics and resurgent structures underlying enumerative Gromov–Witten invariants (i.e., their behavior at large degree and large genus).
Matrix models offer non-perturbative descriptions of quantum gravity in simple settings, allowing us to study important dualities, the so-called large-N dualities, between the seemingly very different gauge and string theories. However, the large-N expansions of matrix models lead divergent series, only defined as asymptotic series. Furthermore, by fine-tuning the couplings of the matrix model we obtain models of pure gravity coupled to minimal conformal field theories. The simplest of these "minimal models" is 2d gravity, and the perturbative expansion of its partition function is asymptotic and formally satisfies the Painlevé I equation.
These asymptotic properties are connected to the existence of exponentially small contributions not captured by a perturbative analysis. The emerging structure can be accurately described by means of a resurgent transseries, capturing this perturbative/non-perturbative connection and its consequences. In this talk, I will analyse essential role of this resurgent transseries for the cases of Painlevé I and the quartic matrix model: together with exponentially accurate numerical and summation methods, I will show how to go beyond the asymptotic results and obtain (analytically and numerically) non-perturbative data.
We study the spectrum of defect conformal field theories (CFTs), and show the existence of universal accumulation points in the defect spectrum. This is achieved by obtaining an inversion formula for the bulk to defect OPE, akin to the Lorentzian inversion formula of Caron-Huot for CFTs without defects. We conclude by applying the result in examples and with an outlook.
Recently, a new approximation scheme for matrix quantum mechanics was proposed. It is a large D limit for models in which D U(N) matrices interact through an O(D) invariant action. For a specific choice of the interaction terms, this limit has been shown to reproduce the physics of the SYK model without the need of random couplings. In this talk, I will show how a generalization of the concept of D-brane probe can be introduced for such an SYK-like matrix model, and how the corresponding probe brane action can be computed exactly. This leads to a test of a non-trivial relation with the free energy of the model. I will also provide new insight on the properties of the new large D limit of matrix quantum mechanics, by addressing several new model building issues.
Understanding the asymptotic properties of solutions of the Painlev ́e I non-linear ODE is ofgreat interest in both mathematics and physics. It is well known that the asymptotic behaviour of these solutions is connected to the existence of exponentially small contributions, directly linked to physical phenomena not captured by a perturbative analysis. The theory of resurgence perfectly captures this perturbative/non-perturbative connection and its consequences. Moreover, it allows us to construct a full non-perturbative solution from perturbative data. In this talk, I will demonstrate the essential role of resurgence theory, coupled to exponentially accurate numerical methods, in going beyond the perturbative results and obtain (analytically and numerically) non-perturbative data. In particular, I will exemplify how these techniques can be applied to to the calculation of poles of Painlevé I solutions.
In this talk, we will illustrate why resurgent analysis is a promising framework for studying non-perturbative aspects of physical theories. We will present examples from mathematics and physics in which resurgence has been successfully applied, and why resurgence is a very good tool for understanding phase diagrams of physical systems. We will then talk about the basic ideas of resurgence, showing how the non-perturbative information is encoded in the perturbative asymptotic series and the concept of the Stokes phenomenon. We will conclude the exposition by presenting ongoing work in the context of Painlevé equations, that are related to the partition function of 2D gravity and supergravity.
I will review older work, and report on work in progress, concerning applications of resurgence and trans series within the realms of matrix models and minimal/topological strings.
Perturbation theory is generically divergent, leading to series with zero radius of convergence. When such asymptotic perturbative-series are resurgent, this problem can be tackled by extending the perturbative series into a non-perturbative trans-series, in a specified fashion. Resurgent trans-series may then be used to go beyond perturbation theory in generic problems across theoretical physics, and address diverse non-perturbative phenomena. This colloquium will cover a brief introduction to resurgence and trans-series, with some illustrative applications within (zero-dimensional) gauge theories (i.e., matrix models) and (topological) string-theoretic settings
I will briefly review applications of resurgence within topological string theory, with the goal of uncovering the resurgent properties of Gromov-Witten invariants.
Lectures based on https://arxiv.org/abs/1802.10441
The geodesics for the Mabuchi metric on the space H of Kahler metrics on a compact symplectic manifold M correspond to solutions of a homogeneous complex Monge-Ampere (HCMA) equation. The space H is an infinite dimensional analogue of the symmetric spaces of noncompact type G_C/G for compact Lie groups G. In H the role of G is being played by the group of Hamiltonian symplectomorphisms. I will describe a method for reducing the relevant Cauchy problem for the HCMA eq with analytic initial data to finding a related Hamiltonian flow followed by a "complexification". For Hamiltonian G-spaces, with G-invariant Kahler structure, the geodesic corresponding to the norm square of the moment map or its Hamiltonian flow in imaginary time (= gradient flow for the changing metric following the geodesic) leads to the convergence of the holomorphic sections to sections supported on Bohr-Sommerfeld leaves. For M=T*G, starting from the vertical or Schrodinger polarization, one obtains the Segal-Bargman-Hall coherent state transform.
We start by a review of numerical and analytical conformal bootstrap techniques at our disposal. We then use these tool to approach the “simplest” Argyres-Douglas theory, a four-dimensional N=2 strongly coupled superconformal field theory. We “zoom in” to the theory by specifying Coulomb branch data, and obtain strong constraints on its operator spectrum and OPE coefficients. We conclude with an outlook for the prospects of approaching other strongly coupled N=2 conformal theories.
Resurgencia es un método utilizado para resolver equaciones dife-renciales, con un amplio rango de aplicaciones. Se basa principalmen-te en la aplicación del llamado Alien Calculus. Tiene su origen en elanálisis del fenómeno de Stokes, que en análisis complejo consiste enque el comportamiento asintótico de ciertas funciones puede variar endistintas regiones del plano complejo. Resurgencia nos dirá cómo rela-cionar estos dos comportamientos, es decir, cómo cruzar las llamadaslíneas de Stokes, las cuales separan dichas regiones.La principal característica de Resurgencia es que es un procesoconstructivo que nos permite extraer información sobre soluciones for-males dadas en forma de transseries (series de potencias, logaritmos yexponenciales) con radio de convergencia nulo.En esta charla daremos una pequeña noción sobre dónde aplicamosResurgencia e introduciremos los conceptos y procedimientos acom-pañados de un ejemplo ilustrativo.
The goal of this seminar is to introduce the concept of integrability, showing how it can be applied to obtain solutions for dynamical systems. We will then apply the tools that we build in the first part of the talk to attack a problem that is relevant in topological string theory, the problems of the KP and KdV hierarchy. We will introduce those problems in a self-contained way, explore the use of the technique of the Lax pairs to solve them and finally we will see how those solutions are of use in matrix models, that appear in minimal string theories of interest.
We start by a review of numerical and analytical conformal bootstrap techniques at our disposal. We then use these tool to approach the “simplest” Argyres-Douglas theory, a four-dimensional N=2 strongly coupled superconformal field theory. We “zoom in” to the theory by specifying Coulomb branch data, and obtain strong constraints on its operator spectrum and OPE coefficients. We conclude with an outlook for the prospects of approaching other strongly coupled N=2 conformal theories.
Mirror Symmetry is a conjecture that suggests the connection between the structures of two mirror manifolds. In this talk, we will present an introduction to this symmetry, first through the Strominger-Yau-Zaslow conjecture and then in more general terms. Finally, we will mention the origin of Mirror Symmetry in the context of Topological Strings and we will make some comments on the topological A and B models.
The non-equilibrium hydrodynamic phenomena in strongly coupled quantum systems such as the N=4 supersymmetric Yang-Mills plasma, can be addressed with the tools of the AdS/CFT correspondence, where calculations in the gravity side of the duality can be performed through a large proper-time expansion. This expansion is asymptotic and encodes a spectrum of non-hydrodynamic modes reflecting the frequencies of AdS black-brane quasinormal modes. The emerging structure for the description of physical quantities such as the energy density is of a resurgent transseries. The theory of resurgence perfectly captures this perturbative/non-perturbative connection and its consequences. Moreover, it allows us to construct a full non-perturbative solution from perturbative data. In this talk, I will analyse essential role of resurgence theory, coupled to exponentially accurate numerical methods, to go beyond the perturbative results and obtain non-perturbative data. I will use these techniques to thoroughly study the resurgent transseries for the energy density of the N=4 SYM plasma.
We start by a review of numerical and analytical bootstrap techniques at our disposal. We then use these tool to approach the “simplest” Argyres-Douglas theory. Specifically, we attempt to “zoom in” to the theory by specifying Coulomb branch data, and obtain constraints on its operator spectrum and OPE coefficients. We conclude with an outlook for the prospects of approaching other Argyres-Douglas theories.
Resurgence is a method used to solve differential equations with a wide range of applications. It is based on the so called alien calculus. The talk will give a brief insight on what resurgence is used for. Then, via example, a short introduction to alien calculus is given.
This is the third in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.
This is the second in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.
This is the first in a series of talks introducing the subject of resurgence in quantum mechanics, field theory and string theory.