# Séminaire de Physique Théorique

### Schedule

 15th Dec, 2021 Henri Epstein Archeological Remarks on Analyticity Properties in Momentum Space in QFT 6th Dec, 2021 Sylvain Ribault Global Symmetry and Conformal Bootstrap in the Two-Dimensional O(n) Model 15th Nov, 2021 Andrei Smilga Spin(7) and Generalized SO(8) Instantons in Eight Dimensions 27th Oct, 2021 Antoine Tilloy Variational Method in 1+1 Dimensional Relativistic Field Theory 18th Oct, 2021 Ritam Sinha The Bi-conical Vector Model at 1/N 28th Jan, 2020 Emil Akhmedov Ultraviolet Phenomena in AdS Self-interacting Quantum Field Theory 20th Jan, 2020 Alexander M. Polyakov Turbulence and Quantum Field Theory 26th Nov, 2019 Yifei He Geometrical Four-Point Functions of 2d Critical Q-State Potts Model 26th Sep, 2019 Dmitri Bykov Flag Manifold Sigma-Models 16th Sep, 2019 Jens Hoppe U(1)-invariant minimal 3-manifolds 10th Jul, 2019 Guillaume Bossard On Supersymmetric E11 Exceptional Field Theory 9th Jul, 2019 Eliezer Rabinovici On the Time Dependence of Complexities 22nd May, 2019 Anatoly Dymarsky Quantum KdV Hierarchy in 2nd CFTs 13th May, 2019 Pierpaolo Mastrolia Feynman Integrals and Intersection Theory 26th Apr, 2019 Apratim Kaviraj & Emilio Trevisani Supersymmetry and Dimensional Reduction in Random Field Models 25th Apr, 2019 Fidel Schaposnik Massolo On Phases of Melonic Quantum Mechanics 24th Apr, 2019 Stefanos Kousvos Three Dimensional Cubic Symmetric CFTs in the Bootstrap and their Applications 23rd Apr, 2019 Pierre Toledano Introduction to Structural Phase Transitions 19th Apr, 2019 Miguel Paulos A Functional Approach to the Numerical Conformal Bootstrap 18th Apr, 2019 Mikhail Isachenkov Solving Large N Double-Scaled SYK 17th Apr, 2019 Volker Schomerus The Casimir Equation for 4D Superconformal Blocks 20th Mar, 2019 Stefan Hollands Modular Flows in Quantum Field Theory 31st Jan, 2019 Sergiu Klainerman On the Nonlinear Stability of Black Holes 5th Dec, 2018 Pierre Cartier Some remarks on the energy-momentum tensor in general relativity 3rd Dec, 2018 Miguel Paulos A more functional bootstrap 29th Nov, 2018 Georgios Papadopoulos Robinson Structures and the Double Copy 22nd Nov, 2018 Frank Verstraete Tensor networks for describing correlated quantum systems 20th Nov, 2018 Anatoly Dymarsky Timescale of ergodicity: when many-body quantum systems can be described by Random Matrix Theory? 8th Nov, 2018 Junya Yagi Unification of integrability in supersymmetric gauge theories 9th May, 2018 Gordon Slade Critical exponents for long-range O(n) models 8th Mar, 2018 Sandipan Kundu Three Lectures on Causality in Conformal Field Theory (3/3) 7th Mar, 2018 Sandipan Kundu Three Lectures on Causality in Conformal Field Theory (2/3) 22nd Feb, 2018 Sandipan Kundu Three Lectures on Causality in Conformal Field Theory (1/3) 21st Feb, 2018 Marco Meineri Universality at Large Transverse Spin in Defect CFT 30th Jan, 2018 Pavel Saponov Cayley-Hamilton Identity and Drinfeld-Sokolov Reduction in Quantum Algebras 21st Dec, 2017 Mikhail Isachenkov Conformal Blocks and Integrability 19th Dec, 2017 Emil Akhmedov Surprises of quantization in de Sitter space 5th Dec, 2017 Fidel I. Schaposnik Massolo Phase Diagram of Planar Matrix Quantum Mechanics, Tensor and SYK Models (arXiv: 1707.03431) 6th Oct, 2017 Bruno Le Floch Surface defects and instanton-vortex moduli spaces 26th Sep, 2017 Carlangelo Liverani The Lorentz Gas : New Results and Open Problems 28th Jun, 2017 Ali CHAMSEDDINE Resolving Space-Time Singularities in Mimetic Gravity 21st Jun, 2017 Ali CHAMSEDDINE Beyond the Standard Model in Noncommutative Geometry and Mimetic Dark Matter 16th Jun, 2017 Raymond GOLDSTEIN Dynamic Interconversions of Minimal Surfaces 18th Apr, 2017 Nikita SOPENKO Surface defects and instanton-vortex interaction 15th Mar, 2017 Brian WILLIAMS Higher chiral differential operators 1st Mar, 2017 Philsang YOO Physics of Langlands Dualities 22nd Feb, 2017 Yegor ZENKEVICH Ding-Iohara-Miki algebra and gauge theories 20th Feb, 2017 Pavel SAFRONOV A q-deformation of the geometric Langlands correspondence 10th Feb, 2017 Jürg FRÖHLICH The Classical XY Model – Vortex- and Random Walk Representations 17th Jan, 2017 Pavel SAPONOV Braided Yangians 28th Nov, 2016 Vyacheslav RYCHKOV State of the art of conformal bootstrap (2/2) 21st Nov, 2016 Vyacheslav RYCHKOV State of the art of conformal bootstrap (1/2) 15th Nov, 2016 Bethan CROPP Hints of quantum gravity from the horizon fluid 15th Nov, 2016 Taro KIMURA Elliptic deformation of W-algebras from 6d quiver gauge theory 20th Oct, 2016 Michele CIRAFICI Framed BPS states from framed BPS quivers 14th Sep, 2016 Peter KOROTEEV Elliptic algebras and large-N supersymmetric gauge theories 12th Jul, 2016 Artan SHESHMANI Donaldson-Thomas theories and modular forms and S-duality conjecture 18th May, 2016 Dmytro VOLIN All unitary representations of su(p,qIm) 13th Apr, 2016 Sergei ALEXANDROV D-instantons, mock modular forms and BPS partition functions 6th Apr, 2016 Nicolas BOULANGER Spin two duality in linearised gravity around ads 1st Apr, 2016 Anne TAORMINA Mathieu Moonshine 12th Jan, 2016 Dmitri GUREVICH Quantum matrix algebras and their applications 8th Jan, 2016 Tin SULEJMANPASIC An inextricable link : semi-classics and complex saddles 2nd Dec, 2015 Vasily SAZONOV Convergent series : from lattice models to QCD 12th Nov, 2015 Seth HOPPER Finding self-force quantities in a post-Newtonian expansion: Eccentric orbits on a Schwarzschild background 27th Oct, 2015 Hovhannes M. KHUDAVERDIAN The modular class of an odd Poisson supermanifold and second order operators on half-densities 13th Oct, 2015 Semyon KLEVTSOV Geometry of Quantum Hall states 30th Sep, 2015 Michele CIRAFICI Theories of class S and line defects 20th May, 2015 Jnan MAHARANA Scattering of Stringy States and T-duality 21st Apr, 2015 Bertrand EYNARD CFTs, and the (quantum) geometry of integrable systems 13th Apr, 2015 Ehsan HATEFI String amplitudes of type IIA,IIB string theory with their α’ corrections 7th Apr, 2015 Alexander Alexandrov Matrix models for intersection numbers 10th Feb, 2015 Tomas PROCHAZKA W infinity and triality 19th Dec, 2014 Marco BAGGIO tt∗ equations, localization and exact chiral rings in 4d N=2 SCFTs 20th Jun, 2014 Misha SHIFMAN N=(0, 2) Deformation of (2, 2) Sigma Models: Geometric Structure, Holomorphic Anomaly and Exact Beta Functions 22nd May, 2014 Piotr SULKOWSKI 3d gauge theories from homological knot invariants 8th Aug, 2013 Nikita NEKRASOV BPS/CFT Correspondence 1st Feb, 2013 Eliezer RABINOVICI BPS/CFT (In)Stabilities and complementarity in AdS/CFT

### Archeological Remarks on Analyticity Properties in Momentum Space in QFT

I will describe the foundations of the program of studying the analyticity properties of the n-point functions in momentum space : the primitive domain of analyticity and methods to enlarge it. If time permits, some of the results for the 4-point function will be described.

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IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 70 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 70 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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### Global Symmetry and Conformal Bootstrap in the Two-Dimensional O(n) Model

The torus partition function of the critical O(n) model, which is known since 1987, does not fully characterize the space of states. For complex n, I will conjecture a determination of that space in terms of irreducible O(n) representations and indecomposable Virasoro representations. I will then describe the interplay between O(n) symmetry and crossing symmetry in four-point correlation functions, and explain how the solutions of crossing symmetry can be counted numerically. This leads to the determination of some of the model's fusion rules.

This talk is based on the preprint https://arxiv.org/abs/2111.01106 with Linnea Grans-Samuelsson, Rongvoram Nivesvivat, Jesper Lykke Jacobsen, and Hubert Saleur.

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IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: "subscribe seminaire_physique PRENOM NOM"
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### Spin(7) and Generalized SO(8) Instantons in Eight Dimensions

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IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: "subscribe seminaire_physique PRENOM NOM"
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### Variational Method in 1+1 Dimensional Relativistic Field Theory

The variational method is a powerful approach to solve many-body quantum problems non perturbatively. However, in the context of relativistic quantum field theory (QFT), it needs to meet 3 seemingly incompatible requirements outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In practice, variational methods usually break one of the 3, which translates into the need to have an IR or UV cutoff. I will explain how a relativistic modification of continuous matrix product states allows us to satisfy the 3 requirements jointly in 1+1 dimensions. Optimizing over this class of states, one can solve scalar QFT without UV cutoff and directly in the thermodynamic limit, and numerics are promising. I will try to cover both the general philosophy of the method, the basics of the computations, and mention the many open problems.

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IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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### The Bi-conical Vector Model at 1/N

We study finite N aspects of the O(m) × O(N-m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d=4-ε dimensions. The large rank limit of the bi-conical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N. In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.

==================================================================

IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: "subscribe seminaire_physique PRENOM NOM"
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### Ultraviolet Phenomena in AdS Self-interacting Quantum Field Theory

We study the one-loop corrections to correlation functions in quantum field theories in the Anti de Sitter space-time. Our calculation shows the existence of non-local counterterms which however respect the AdS isometry. Our arguments are general and applicable to general non-conformal AdS field theories. We also explain why calculations in Euclidean and Lorentzian signatures should differ even at the leading order in non globaly hyperbolic manifolds.

### Geometrical Four-Point Functions of 2d Critical Q-State Potts Model

In this talk I will describe the geometrical four-point functions (cluster connectivities) of 2d critical Q-state Potts model and its relation to minimal models four-point functions. By studying on the RSOS lattice of the ADE type, we provide a geometrical formulation of minimal models four-point functions of operators related to the Potts model and they involve the same types of cluster/loop expansions as that of the Potts model albeit with different weights. I will discuss how our results can be used to extract useful information about the Potts model from minimal models.

### Flag Manifold Sigma-Models

The talk is dedicated to flag manifold sigma-models, which are theories of generalized harmonic maps from a Riemann surface to a manifold of flags in $C^N$. These theories feature interesting geometric properties and are in certain cases examples of the so-called ‘integrable’ models. I will review some of these facts.

### On Supersymmetric E11 Exceptional Field Theory

We shall review how supergravity theories can emerge from an exceptional field theory based on the Kac-Moody group E11 (i.e. E8+++) with gauge symmetry a set of generalised diffeomorphisms' acting on the fundamental module while preserving E11. The construction relies on a super-algebra T  that extends E11 and provides a differential complex for the exceptional fields. A twisted self-duality equation underlying the dynamics can be shown to be invariant under generalised diffeomorphisms provided a certain algebraic identity holds for structure coefficients of the super-algebra T. The fermions of the theory belong to an unfaithful representation of the double cover of a maximal Lorentzian subgroup K(E11). We conjecture that certain tensor products of unfaithful representations are homomorphic to the quotient of specific indecomposable modules of E11. Using these conjectures, we can write a linearised Rarita-Schwinger equation and show that the E11 twisted self-duality equations are supercovariant. The conjectures are checked through computations in level decompositions with respect to maximal parabolic subgroups.

### On the Time Dependence of Complexities

The emergence of very long time scales on the gravity side of the AdS/CFT correspondence has led to the introduction of the notion of complexity in the research of quantum gravity. Various notions of complexity have been studied in the area of Quantum Information as well as in Quantum Field Theory. I will discuss, in this context, complexities and the time scales of their evolution following in the various definitions. The physics involved will be stressed where it is understood.

### Quantum KdV Hierarchy in 2nd CFTs

Infinite-dimensional conformal symmetry in two dimensions renders conformal field theories integrable with an infinite hierarchy of quantum KdV charges being in involution. These charges govern the structure of Virasoro descendant states and provide correct formulation for the Eigenstate Thermalization in 2d theories. After covering recent results on Eigenstate Thermalization, I will talk about an ongoing progress of calculating the spectrum of quantum KdV charges and generalized partition function of two dimensional theories in the limit of large central charge. The talk is based on https://arxiv.org/abs/1903.03559 as well as https://arxiv.org/abs/1812.05108 and https://arxiv.org/abs/1810.11025

### Feynman Integrals and Intersection Theory

I will show that Intersection Theory (for twisted de Rham cohomology) rules the algebra of Feynman integrals. In particular I will address the problem of the direct decomposition of Feynman integrals into a basis of master integrals, showing that it can by achieved by projection, using intersection numbers for differential forms. After introducing a few basic concepts of intersection theory, I will show the application of this novel method, first, to special mathematical functions, and, later, to Feynman integrals on the maximal cuts, also explaining how differential equations and dimensional recurrence relations for master Feynman integrals can be directly built by means of intersection numbers. The presented method exposes the geometric structure beneath Feynman integrals, and offers the computational advantage of bypassing the system-solving strategy characterizing the standard reduction algorithms, which are based on integration-by-parts identities. Examples of applications to multi-loop graphs contributing to multiparticle scattering, involving both massless and massive particles are presented.

### Supersymmetry and Dimensional Reduction in Random Field Models

In this talk we will discuss an ongoing work on random field models. First we will review a work by Parisi and Sourlas. They conjectured that the infrared fixed point of such random field models should be described by a supersymmetric conformal field theory (CFT). From this they argued that the disordered CFT admits a description in terms of a CFT in two less spacetime dimensions but without the disorder. We will explain how the dimensional reduction is realized. Finally we will discuss when and how the RG flow of the random field theory reaches the SUSY fixed point.

### On Phases of Melonic Quantum Mechanics

We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large D limit, or as disordered models. Both models have a mass parameter m and the transition from the perturbative large m region to the strongly coupled "black-hole" small m region is associated with several interesting phenomena. One model, with U(n)^2 symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m*. For 0<m<m*, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions Δ+ and Δ- for the Euclidean two-point function when t→ +∞ and t→ -∞

### Three Dimensional Cubic Symmetric CFTs in the Bootstrap and their Applications

I will discuss results of recent numerical bootstrap work performed for systems with cubic symmetry. Under certain assumptions, we find an isolated region in parameter space, which given prior intuition with the numerical bootstrap, indicates the existence of a CFT in this region. We find critical exponents for the conjectured CFT which are in discrepancy with the epsilon expansion (but in good agreement with experiments for structural phase transitions). The disagreement of critical exponents for structural phase transitions calculated in the epsilon expansion with those measured in experiments is something that was noticed since the 70s. I will briefly discuss some resolutions proposed at the time.

### A Functional Approach to the Numerical Conformal Bootstrap

We apply recently constructed functional bases to the numerical conformal bootstrap for 1D CFTs. We argue and show that numerical results in this basis converge much faster than the traditional derivative basis. In particular, truncations of the crossing equation with even a handful of components can lead to extremely accurate results, in opposition to hundreds of components in the usual approach. We explain how this is a consequence of the functional basis correctly capturing the asymptotics of bound-saturating extremal solutions to crossing. We discuss how these methods can and should be implemented in higher dimensional applications.

### Solving Large N Double-Scaled SYK

I will review a method to evaluate correlation functions of certain statistical systems via chord diagrams and apply it to compute correlators in the double-scaled version of SYK model, in particular in its large N limit. The results are exact at all energies and allow to extract corrections to the maximal Lyapunov exponent. Time permitting, I will comment on the suggested relation of this model to a Hamiltonian reduction of quantum particle moving on the non-compact quantum group SU_q(1,1).

### The Casimir Equation for 4D Superconformal Blocks

Applications of the bootstrap to superconformal field theories require the construction of superconformal blocks for four-point functions of arbitrary supermultiplets. Up until recently, only sporadic results had been obtained. In my talk I explain the key ingredients of a new systematic construction that apply to a large class of superconformal field theories, including 4-dimensional models with any number N of supersymmetries. It hinges on a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact.

### Modular Flows in Quantum Field Theory

The reduced density matrix of a subsystem induces an intrinsic internal dynamics called the modular flow''. The flow depends on the subsystem and the given state of the total system. It has been subject to much attention in theoretical physics in recent times because it is closely related to information theoretic aspects of quantum field theory. In mathematics, the flow has played an important role in the study of operator algebras through the work of Connes and others.

It is known that the flow has a geometric nature (boosts resp. special conformal transformations) in case the subsystem is defined by a spacetime region with a simple shape. For more complicated regions, important progress was recently made by Casini et al. who were able to determine the flow for multi-component regions for free massless fermions or bosons in 1+1 dimensions.

In this introductory lecture, I describe the physical and mathematical backgrounds underlying this research area. Then I describe a new approach which is not limited to free theories, based in an essential way on two principles: The so-called KMS-condition'' and the exchange relations between primaries (braid relations) in rational CFTs in 1+1 dimensions. A combination of these ideas and methods from operator algebras establish that finding the modular flow of a multi-component region is equivalent to a certain matrix Riemann-Hilbert problem. One can therefore apply known methods for this classic problem to find or at least characterize the modular flow.

### Some remarks on the energy-momentum tensor in general relativity

Following the method outlined b Emmy Noether in her famous 1918 paper, we propose a version of the momentum-energy tensor in general relativity which is geometric and suitably covariant, giving conservation laws .

This conssuction appears to be novel in a field well explored.

### A more functional bootstrap

The conformal bootstrap aims to systematically constraint CFTs based on crossing symmetry and unitarity.

In this talk I will introduce a new approach to extract information from the crossing symmetry sum rules, based on the construction of linear functionals with certain positivity properties. I show these functionals allow us to derive optimal bounds on CFT data. Furthemore I will argue that special extremal solutions to crossing form a basis for the crossing equation, with the functionals living in the dual space. As an application we reconstruct physics of QFTs in AdS2 from the properties of 1d CFTs.

### Tensor networks for describing correlated quantum systems

Quantum tensor networks provide a new language for describing many body systems. They model the entanglement structure of many body wavefunctions, and give a precise description of symmetries such as arising in systems exhibiting topological quantum order. In this talk, an overview will be given of the challenges, prospects and limitations of this approach.

### Timescale of ergodicity: when many-body quantum systems can be described by Random Matrix Theory?

In this talk I will argue that after certain timescale (which scales with the system size as L^{d+2}) dynamics of a local observable becomes universal and it can be described by a random matrix.

This talks is based on https://arxiv.org/abs/1804.08626 and other recent works.

### Unification of integrability in supersymmetric gauge theories

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory. I will explain how string theory unifies these phenomena.

This is based on my work with Kevin Costello [arXiv:1810.01970].

### Critical exponents for long-range O(n) models

Séminaire de Probabilités et de Physique Théorique

We present results on the critical behaviour of long-range models of multi-component ferromagnetic spins and weakly self-avoiding walk in dimensions 1, 2, and 3. The range of the interaction is adjusted so that the models are below their upper critical dimension.  Critical exponents are computed for the susceptibility, specific heat, and critical two-point function, using a renormalisation group method to perturb around a non-Gaussian fixed point.  This provides a mathematically rigorous version of the epsilon expansion.

### Three Lectures on Causality in Conformal Field Theory (3/3)

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014arXiv:1601.07904arXiv:1610.05308.

### Three Lectures on Causality in Conformal Field Theory (2/3)

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014arXiv:1601.07904arXiv:1610.05308.

### Three Lectures on Causality in Conformal Field Theory (1/3)

Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014arXiv:1601.07904arXiv:1610.05308.

### Universality at Large Transverse Spin in Defect CFT

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators.  We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE and is analytic in s, analogous to the Caron-Huot formula for the four-point function. Some important assumptions are made in deriving this result: we comment on them.

### Cayley-Hamilton Identity and Drinfeld-Sokolov Reduction in Quantum Algebras

Different forms of the matrix Cayley-Hamilton identity in some quantum algebras will be presented. In particular, I plan consider the so-called braided Yangian -- some generalization of Drinfeld Yangian -- recently introduced in my joint work with D.Gurevich. A quantum counterpart of the Drinfeld-Sokolov reduction based on the Cayley-Hamilton identity will be discussed as well.

### Conformal Blocks and Integrability

I will discuss a relation between conformal blocks, describing kinematics of a CFT, and integrable models of quantum-mechanical particles. I will show how the dependence of blocks on cross-ratios is encoded in equations of motion of a Calogero-Sutherland model and their dependence on conformal dimension and spin of the exchanged operator - in those of a relativistic Calogero-Sutherland model. Both are simultaneously controlled by an integrable connection generalizing 2d Knizhnik-Zamolodchikov equations. I will review how this connection, associated to representations of degenerate double affine Hecke algebra, comes from a q-deformed bispectrally symmetric setting.

### Surprises of quantization in de Sitter space

I will talk about loop infrared effects in de Sitter QFT. Namely about their types, physical meaning and origin and also about their resumation and physical consequences. The talk is based on arXiv:1701.07226.

### Phase Diagram of Planar Matrix Quantum Mechanics, Tensor and SYK Models (arXiv: 1707.03431)

In this talk I present the phase diagram of a U(N)^2 x O(D) invariant fermionic planar matrix quantum mechanics (equivalently tensor or complex SYK models) in the new large D limit dominated by melonic graphs. The Schwinger-Dyson equations can have two solutions describing either a "large" black hole phase a la SYK or a "small" black hole with trivial IR behavior. In the strongly coupled region of the mass-temperature plane, there is a line of first order phase transitions between the small and large black hole phases. This line terminates at a new critical point which can be studied numerically in detail. The critical exponents are non-mean-field and different on the two sides of the transition. If time allows, I will compare this to purely bosonic unstable and stable melonic models.

### Surface defects and instanton-vortex moduli spaces

Instantons on R4, namely anti-self-dual Yang-Mills connections, are in bijection with framed locally free sheaves on CP2. Ramified instantons have an imposed singularity along R2 in R4 that translates to a parabolic structure along a CP1 divisor, or equivalently to a cyclic orbifold.  Such a singularity (Gukov-Witten defect) can be obtained in 4d N=2 supersymmetric Yang-Mills theory by adding 2d N=(2,2) degrees of freedom on R2, and gauging a global symmetry of the 2d theory using the R2 restriction of the 4d gauge connection.  The moduli space of ramified instantons should thus be related to a moduli space of instanton-vortex configurations of the 4d-2d pair of gauge theories.  I propose an incomplete definition of the latter moduli space by fibering (over the instanton moduli space) a recent description of the vortex moduli space as based maps to the Higgs branch stack.  As evidence I compare Nekrasov partition functions, namely equivariant integrals over these moduli spaces.  The equality relies on Jeffrey-Kirwan technology, applicable thanks to the ADHM construction of the moduli spaces as Kähler quotients.

### The Lorentz Gas : New Results and Open Problems

I will make a quick review of old and new results concerning the Lorentz gas; discuss new directions in which I’d like to proceed (e.g. non periodic obstacles, interacting particles, …) and some (very) partial results toward such directions.

### Dynamic Interconversions of Minimal Surfaces

A classical problem in mathematics is the determination of the minimal surface that spans a given contour, which can be realized in the laboratory as a soap film supported by a wire frame. In the early 1940s Richard Courant pointed out nontrivial situations in which a small deformation of certain frames can render unstable the supported surface, leading by a rapid dynamical process to a new minimal surface. For example, a soap film Möbius strip can transition to a disc. Despite the enormous body of work on the mathematics of minimal surfaces themselves, the understanding of these dynamical problems is at a very early stage. In this talk I will summarize our recent experimental and theoretical work on problems of this type, in which a combination of high-speed imaging and stability theory has revealed new insights. (Work done in collaboration with A.I. Pesci, H.K. Moffatt, T. Machon and G.P. Alexander)

### Surface defects and instanton-vortex interaction

I’ll present a general prescription for the 4d-2d partition function of half-BPS surface defects in d = 4, N = 2 gauge theories in Omega-background which is applicable for any surface defect obtained by gauging a 2d flavour symmetry using a 4d gauge group and reproduces known results obtained via the Higgsing procedure and Kanno-Tachikawa orbifold calculation for Gukov-Witten defects. The role of “negative vortices” which appear in the background of instantons will be emphasized.

### Higher chiral differential operators

The sheaf of chiral differential operators is a sheaf of vertex algebras defined by Gorbounov, Malikov, and Schechtman in the early nineties that exists on any manifold with vanishing second component of its Chern character. Later on it was proposed by Witten to be related to the chiral operators of the (0,2)-supersymmetric sigma-model. Recently, we have proved this using an approach to QFT developed by Costello: the BV-quantization of the holomorphic twist of the (0,2) theory is isomorphic to the sheaf of chiral differential operators. Along with Gorbounov and Gwilliam, we prove this using the language of holomorphic factorization algebras in one complex dimension. In this talk I will sketch the proof of this result while also motivating a family of BV theories that produce sheaves of higher dimensional holomorphic factorization algebras that deserve to be called “higher” CDOs. We discuss the meaning of the OPE for these theories as encoded by the higher dimensional factorization structure.

### Physics of Langlands Dualities

In the first part of the talk, I will discuss a joint project with Chris Elliott on realizing the geometric Langlands correspondence as an instance of S-duality by careful analysis of Kapustin and Witten's work using derived algebraic geometry. In the second part of the talk, I will report on work in progress to produce new instances of Langlands duality in geometric representation theory through the lens of quantum field theory.

### Ding-Iohara-Miki algebra and gauge theories

We study the role of the Ding-Iohara-Miki (DIM) algebra, which is the simplest example of quantum toroidal algebra, in gauge theories, matrix models, q-deformed CFT and refined topological strings. We use DIM algebra to write down the Ward identities for the matrix models and show how it is connected to quiver W-algebras of the A-series. We describe the integrable structure of refined topological strings arising from DIM algebra: the R-matrix, T-operators and RTT relations. Finally, we write down the q-KZ equation for the DIM algebra intertwiners and interpret its solutions as refined topological string amplitudes.

### A q-deformation of the geometric Langlands correspondence

The geometric Langlands correspondence was introduced by Beilinson and Drinfeld as a tool to solve quantum Hitchin systems such as the Gaudin model. The correspondence can be understood following Kapustin and Witten as arising from S-duality in a topologically twisted 4-dimensional super Yang-Mills theory. In this talk I will describe a multiplicative deformation of the Hitchin system and explain a conjectural statement of the corresponding q-deformed correspondence. I will also give some motivations for the statement from the theory of higher deformation quantization and a deformed setup of Kapustin and Witten.

### The Classical XY Model – Vortex- and Random Walk Representations

A review of results concerning the classical XY model in various dimensions is presented.

I start by showing that the XY model does not exhibit any phase transitions in a non-vanishing external magnetic field, and that connected spin-correlations have exponential decay. These results can be derived from the Lee-Yang theorem.

Subsequently, I study the XY model in zero magnetic field: The McBryan-Spencer upper bound on spin-spin correlations in two dimensions is derived. The XY model is then reformulated as a gas of vortices of integer vorticity (Kramers-Wannier duality). This representation is used to explain some essential ideas underlying the proof of existence of the Kosterlitz-Thouless transition in the two-dimensional XY model. Remarks on the existence of phase transitions accompanied by continuous symmetry breaking and the appearance of Goldstone modes in dimension three or higher come next.

Finally, I sketch the random-walk representation of the XY model and explain some consequences thereof – such as convergence to a Gaussian fixed point in the scaling limit, provided the dimension is > 4; and the behaviour of the inverse correlation length as a function of the external magnetic field.




### Braided Yangians

: In my talk I consider a q-deformation of the so-called Yangian Y(gl(m)). The standard Yangian Y(gl(m)) (associated with the Yang R-matrix) was introduced by V.Drinfeld and is rather well known. It possesses a lot of interesting properties and has applications in integrable models of mathematical physics (for example, in the non-linear Schroedinger model), W-algebras and so on. Its q-analog, called the q-Yangian, is usually defined as a "half" of a quantum affine group. D. Gurevich and me suggest a new construction for such a q-analog of the Yangian Y(gl(m)). We call it "braided Yangian". We associate the braided Yangians with rational and trigonometric quantum R-matrices, depending on a formal parameter. These R-matrices arise from constant involutive or Hecke R-matrices by means of the Baxterization procedure. Our braided Yangians admit the evaluation morphism onto quantum matrix algebras and due to this one can construct a rich representation theory for them. In my talk I also plan to define analogs of symmetric polynomials (full, elementary and powers sums) which form a commutative subalgebra in the braided Yangian and exibit some noncommutative matrix identities similar to the Newton-Cayley-Hamilton identities of the classical matrix anlysis.

### State of the art of conformal bootstrap (2/2)

MINI-COURS

Conformal bootstrap is a mathematically well-defined framework for performing non-perturbative computations in strongly coupled conformal field theories, including theories of real physical interest like the critical point of the 3d Ising model.  In these lectures I will describe the recent advances in this field and the challenges it faces.

### State of the art of conformal bootstrap (1/2)

MINI-COURS

Conformal bootstrap is a mathematically well-defined framework for performing non-perturbative computations in strongly coupled conformal field theories, including theories of real physical interest like the critical point of the 3d Ising model.  In these lectures I will describe the recent advances in this field and the challenges it faces.

### Hints of quantum gravity from the horizon fluid

For many years researchers have tried to glean hints about quantum gravity from black hole thermodynamics. We try a different approach using a minimal statistical mechanical model for the horizon fluid based on Damour-Navier-Stokes (DNS) equation. For asymptotically flat black hole spacetimes in General Relativity, we show explicitly that at equilibrium the entropy of the horizon fluid is the Bekenstein-Hawking entropy. Further we show that, for the bulk viscosity of the fluctuations of the horizon fluid to be identical to Damour, a confinement scale exists for these fluctuations, implying quantization of the horizon area.

### Elliptic deformation of W-algebras from 6d quiver gauge theory

In this talk, we show that the elliptic deformation of W-algebra is naturally realized using quiver gauge theory in six dimensions compactified on a torus. This construction is based on the gauge theoretical realization of W-algebra proposed in our previous study [arXiv:1512.08533]. In particular, double quantization of Seiberg-Witten geometry for Γ-quiver gauge theory provides a generating current of W(Γ)-algebra in the free field realization. We also show that the partition function is given by a correlator of the corresponding W(Γ)-algebra, which is equivalent to the AGT relation under the gauge/quiver (base/fibre; spectral) duality. This talk is based on a collaboration with V. Pestun [arXiv:1608.04651]

### Framed BPS states from framed BPS quivers

In this talk I will focus on BPS states in supersymmetric field theories with N=2. In this theories one can consider a certain class of supersymmetric line operators. Such operators support a new class of BPS states, called framed BPS states. I will discuss a formalism based on quivers to understand these objects and their properties. Time permitting I will discuss a relation with the theory of cluster algebras.

### Elliptic algebras and large-N supersymmetric gauge theories

We shall address the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-n limit of the dual gauge theory.

### Donaldson-Thomas theories and modular forms and S-duality conjecture

I will start by an introduction to Donaldson Thomas theory and some of the statements about its modularity properties, as well as its connection to S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin. I will then provide an algebraic geometric approach to prove this conjecture for DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold.

### All unitary representations of su(p,qIm)

Classification of all unitary representations of su(p,q|m) algebra with non-zero p,q,m should have been achieved a while ago, given the current level of the representation theory development. However, to our surprise, the literature on the subject contains some incomplete or incorrect  statements,  save the well-understood su(2,2|N) case. We therefore decided to address the question from scratch and were able to get a complete and concise description of the unitary dual for generic su(p,q|m).

In the current talk:

- The classification statement is presented in full generality, we also mention all the other real forms of gl(p+q|m,C).

- Shortening conditions naturally arise from considering of all possible choices of the Kac-Dynkin-Vogan diagram at once.

- Schwinger oscillators are used to prove unitarity, with a novel option to work with non-integer weights  by representing the oscillator algebra in a generalisation of the Fock module.

- A generalisation of Young diagrams inscribed into a T-hook [almost] bijectively labels the unitary dual. This opens interesting opportunities for new combinatorial identities.

### D-instantons, mock modular forms and BPS partition functions

I'll discuss the modular properties of D3-brane instantons appearing in Calabi-Yau string compactifications. I'll show that the D3-instanton contribution to a certain geometric potential on the hypermultiplet moduli space can be related to the elliptic genus of (0,4) SCFT. The modular properties of the potential imply that the elliptic genus associated with non-primitive divisors of Calabi-Yau is only mock modular. I'll show how to construct its modular completion and prove the modular invariance of the twistorial construction of D-instanton corrected hypermultiplet moduli space.

### Mathieu Moonshine

I shall give a brief introduction to Mathieu Moonshine, an observation made in 2010 in the context of string theory compactified on a K3 surface and whose significance in string theory remains elusive. Attempts to understand the mathematical structure behind this observation have included techniques from Number Theory, Group Theory and Geometry. I will discuss how geometry provides an interesting angle when attempting to explain the presence of the huge Mathieu 24 discrete symmetry in string theories compactified on a K3 surface.

### Quantum matrix algebras and their applications

By quantum matrix algebras I mean these related to braidings (solutions to Quantum Yang-Baxter Equation) and in a sense similar to the classical matrix algebras. In first turn, I am interested in the so-called Reflection Equation algebra. By using it, me (in collaboration with P.Saponov) have introduced the notion of partial derivatives on the enveloping algebra U(gl(m)). This leads to a new type of Noncommutative Geometry (we call it Quantum Geometry), which is deformation of the classical one. In my talk I plan to consider a way of defining some dynamical models on U(u(2)) background.

#### 8th January, 2016 - 11:01:00Tin SULEJMANPASIC (North Carolina State University & IHÉS)

I will discuss the use of semi-classics and instanton calculus and argue that, contrary to common wisdom, complex solutions of the equations of motion are a necessary ingredient of semi-classical expansion. In particular, I will show that without the complex solutions semi-classical expansion of supersymmetric theories cannot be reconciled with the constraints of supersymmetry. This has a natural interpretation in the Picard-Lefschetz theory.

### Convergent series : from lattice models to QCD

The standard perturbation theory leads to the asymptotic series because of the illegal interchange of the summation and integration. However, changing the initial approximation of the perturbation theory, one can generate the convergent series. We study the lattice \phi4-model and compare observables calculated using the convergent series and Monte Carlo simulations. Then, we discuss the generalization of the same ideas for the continuum \phi4-model and QCD.

### Finding self-force quantities in a post-Newtonian expansion: Eccentric orbits on a Schwarzschild background

Small compact objects orbiting supermassive back holes are an important potential source of gravitational radiation. Detection of such waves and the parameter estimation of their sources will require accurate waveform templates. To this eventual end, I present work on bound eccentric motion around a static black hole. In two separate approaches, I examine solutions to the first order (in mass-ratio) field equations. First, I consider solving the field equations entirely analytically in a double post-Newtonian/small-eccentricity expansion. Then I show numeric work wherein we use the MST formalism to solve the field equations to 200 digits. We use this extreme accuracy to fit for previously unknown PN energy flux parameters, extending the previous state of the art from 3PN to 7PN.

### The modular class of an odd Poisson supermanifold and second order operators on half-densities

 Second order operator $\Delta$ on half-densities can be uniquely defined by its principal symbol  $E$ up to a potential' $U$. If $\Delta$ is an odd operator such that order of  operator $\Delta^2$ is less than $3$ then principal symbol $E$ of this operator defines an odd Poisson bracket. We define the modular class of an odd Poisson supermanifold in terms of $\Delta$ operator defining the odd Poisson structure. In the case of non-degenerate odd Poisson structure (odd symplectic case) the modular class vanishes, and we come to canonical odd Laplacian on half-densities, the main ingridient of Batalin-Vilkovisky
formalism.  Then we consider examples of odd Poisson supermanifolds with non-trivial modular classes related with the Nijenhuis bracket.

The talk is based on the joint paper with M. Peddie: arXive: 1509.05686

### Geometry of Quantum Hall states

I will talk about recent progress in understanding quantum Hall states on curved backgrounds and in inhomogeneous magnetic fields and their large N limits, N being the number of particles. The large N limit of the free energy of the Laughlin states in the integer Quantum Hall is controlled by the Bergman kernel expansion, and, in a sense, is exactly solvable to all orders in 1/N. For the fractional Laughlin states, the large N limit can be determined from free field representation. The terms in the large N expansion are given by various geometric functionals. In particular, the Liouville action shows up at the order O(1) in the expansion, and signifies the effect gravitational anomaly. The appearance of this term leads us to argue for the existence of a third quantized kinetic coefficient, precise on the Hall plateaus, in addition to Hall conductance and anomalous viscosity. Based on: 1309.7333, 1410.6802, 1504.07198 and upcoming work.

### Theories of class S and line defects

The goal of this talk is to discuss some properties of line defects in certain supersymmetric QFTs, the so-called theories of class S. I will spend some time reviewing some old work of Gaiotto Moore and Neitzke on the emergence of Hitchin systems in these theories, and on certain coordinates on the associated moduli spaces. Line defects can then be understood as certain functions on these moduli spaces. If there is time I will present new results. The tone of the discussion will be informal.

### Scattering of Stringy States and T-duality

I shall briefly review the salient features of target space duality (T-duality) and recall some essential results. I shall outline a prescription to derive the S-matrix for the scattering of massless stringy states that arise in the compactification of closed bosonic strings on a torus at the tree level. It will be shown that the S-matrix elements can be expressed in a T-duality invariant form. The Kawai-Lewellen-Tye formalism plays an important role in our approach.

### CFTs, and the (quantum) geometry of integrable systems

It has been realized recently that the c=1 conformal block of 4 point function in Liouville CFT is related to the Tau function of the Painlevé 6 integrable system. Here we propose a general construction: starting from a very general integrable system (a Hitchin system: the moduli space of flat G-connections over a Riemann surface, with G an arbitrary semi-simple Lie group), we define some "amplitudes", and we show that these amplitudes satisfy all the axioms of a CFT: they satisfy OPEs, Ward identities and crossing symmetry. The construction is very geometrical, by defining a notion of "quantum spectral curve" attached to a flat connection, defining homology and cohomology on it, and showing that amplitudes satisfy Seiberg-Witten like relations, and behave well under modular transformations. So this link between CFTs and integrable systems unearths a new and beautiful quantum geometry.

### String amplitudes of type IIA,IIB string theory with their α’ corrections

We would like to make various remarks on string amplitudes in type II string theory in which getting the exact and final form of the world sheet integrals up to five point mixed closed-open amplitudes to all orders are given. We are also going to to talk about all kinds of effective actions involving DBI, Chern-Simons and more importantly new Wess Zumino actions. Indeed we try to provide a comprehensive explanation even for D-brane-anti D-brane systems. We also introduce various new techniques to be able to derive all order α’ corrections to all type II super string effective actions. If time allows, we then mention several issues related to those effective actions as well.

### Matrix models for intersection numbers

In my talk I will discuss a family of matrix models, which describes the generating functions of intersection numbers on moduli spaces both for open and closed Riemann surfaces. Linear (Virasoro\W-constraints) and bilinear (KP\MKP integrable hierarchies) equations follow from the matrix model representation.

### N=(0, 2) Deformation of (2, 2) Sigma Models: Geometric Structure, Holomorphic Anomaly and Exact Beta Functions

We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. We prove that despite the chiral nature of the model anomalies in the isometry currents do not appear for CP(N-1) at any N. This is in contradistinction with the minimal heterotic model (with no right-moving fermions) which is anomaly-free only for N=2, i.e. in CP(1). We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the "Konishi anomaly." This gives us another method for finding exact β functions. A clear–cut parallel between N=1 4D Yang-Mills and N=(0,2) 2D sigma models is revealed.

### 3d gauge theories from homological knot invariants

Compactifications of M5-branes on non-trivial 3-manifolds lead to a N=2 supersymmetric theories in 3 dimensions. If the 3-manifold in question is a knot complement, various Chern-Simons amplitudes - or corresponding knot invariants - for this knot determine the properties of the resulting 3d, N=2 theory. This relation between N=2 theories and Chern-Simons theory is referred to as the 3d-3d correspondence. M-theory realization of this correspondence implies that N=2 theories obtained in this way possess one special flavor symmetry, which is related to certain deformation of Chern-Simons theory and homological knot invariants. In this talk I will discuss properties of these refined/homological invariants, and their role in the 3d-3d correspondence.