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.