Below you can find names and abstracts of the previous talks given in our seminar series
March 2, 2023 Speaker: Nikita Nikolaev (Birmingham U.) Title: Borel Resummation in Exact Perturbation Theory and the Exact WKB Method Abstract: I will present recent progress in the amazing newly emerging subject of exact perturbation theory. This theory provides methods to upgrade divergent perturbative expansions to analytic information. One such method is called Borel resummation: it involves a detailed analysis (via a Borel transform) of a given divergent perturbative expansion in order to recover the nonperturbative corrections (i.e., exponentially small terms) that otherwise cannot be captured by the ordinary perturbation theory. The astounding big-picture upshot of this theory is that — contrary to Freeman Dyson’s conclusion that perturbation theory is incomplete — the divergent sector of perturbation theory actually already encodes all the necessary nonperturbative information, and it is therefore only a matter of applying suitable methods to extract it.
One of the most classical settings for exact perturbation theory is the so-called exact WKB method for solving singularly perturbed linear ODEs such as the Schrödinger equation. Such ODEs can be easily solved in (exponential) power series (the so-called WKB ansatz), but they are almost always divergent. Attempting to apply Borel resummation to get true analytic solutions turns out to be the correct approach, but a difficult mathematical problem. I will describe a solution I have developed through a series of recent works.
March 16, 2023
Speaker: Saebyeok Jeong (CERN)
Title: Correlation function of surface defects and Gaudin spectral problem
Abstract: I'll explain how half-BPS surface defects in N=2 supersymmetric gauge theories provide solutions to the spectral problem of the associated quantum Hitchin integrable system. We consider two types of surface defects, the "canonical" surface defect and the "regular monodromy" surface defect, inserted on top of each other. With the former the chiral ring equations give the oper encoding the quantum spectra, while the latter constructs the common eigenfunctions. The quantum Hamiltonians can also be constructed by studying the chiral ring relations in the presence of both (which turn out to give the universal oper). There is a string duality that connects our N=2 gauge theory configuration to the twisted M-theory and further to the topologically twisted (GL-twist) N=4 gauge theory, where we can give a dual interpretation of the aligned surface defects as the 't Hooft line attached to Dirichlet boundaries. Compactifying to the two-dimensional sigma model, this is the Hecke operator acting on the Hecke eigensheaf.
March 30, 2023
Speaker: Claudia Rella (UniGe)
Title: Resurgence, Stokes constants, and arithmetic functions in topological string theory
Abstract: The quantization of the mirror curve to a toric Calabi-Yau threefold gives rise to quantum-mechanical operators. Their fermionic spectral traces produce factorially divergent power series in the Planck constant, which are conjecturally captured by the refined topological string in the Nekrasov-Shatashvili limit via the Topological Strings/Spectral Theory correspondence. In this talk, I will discuss how the machinery of resurgence can be applied to study the non-perturbative sectors associated with these asymptotic expansions, producing infinite towers of periodic singularities in the Borel plane and infinitely-many rational Stokes constants, which are encoded in generating functions given in closed form by q-series. I will then present an exact solution to the resurgent structure of the semiclassical limit of the first fermionic spectral trace of the local P^2 geometry, which unveils a remarkable arithmetic construction. The same analytic approach is applied to the dual weakly-coupled limit of the conventional topological string on the same background. The Stokes constants are explicit divisor sum functions, the perturbative coefficients are particular values of known L-functions, and the duality between the two scaling regimes appears in number-theoretic form. This talk is based on arXiv:2212.10606.
April 27, 2023
Speaker: Tom Bridgeland (Sheffield U.)
Title:Geometric structures defined by Donaldson-Thomas invariants
Abstract: This will be a report on an ongoing programme whose aim is to use the Donaldson-Thomas invariants of a three-dimensional Calabi-Yau category to define a geometric structure on its space of stability conditions. There are close connections to questions about resurgence of partition functions in topological string theory. The relevant geometry has been christened a Joyce structure: it is a complex hyperkahler structure on the total space of the tangent bundle with certain extra symmetries. At this point we only understand how to construct the Joyce structure in a few special cases, e.g. derived categories of coherent sheaves on non-compact CY3s without compact 4-cycles, or Fukaya categories of non-compact CY3s related to theories of class S[A1]. In this talk I will introduce Joyce structures and describe the construction in the case of the CY3 category associated to the A2 quiver.
May 11, 2023 Speaker: Bruno le Floch (Sorbonne Université and CNRS) Title:BPS Dendroscopy on Local P2 Abstract: The spectrum of BPS states in D=4 supersymmetric field theories famously jumps across codimension-one walls in vector multiplet moduli space. The Split Attractor Flow Conjecture posits that the BPS index Ω_z(γ) for given charge γ and moduli z is a finite sum, over attractor flow trees rooted at z and ending at “attractor points”, of combinations of “attractor indices” Ω_*(γ_i) that count BPS states at these points. When it applies, this conjecture provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. The conjecture is known to hold for quiver quantum mechanics descriptions valid near orbifold-like points in moduli space. Combining this with large-volume results, we proved the conjecture for the whole moduli space of IIA strings compactified on KP2 (the canonical bundle over the projective plane P2). I will present this proof and the resulting dendroscopy, with numerous figures. This is joint work with Pierrick Bousseau, Pierre Descombes, and Boris Pioline.
October 7, 2022 Speaker: Gabriele Rembado (Bonn U.) Title: An intrinsic approach to irregular isomonodromic deformations Abstract: Meromorphic connections on Riemann surfaces are central objects in 2d gauge theory. The quantisation of their moduli spaces provide an entry point for mathematical constructions of QFTs with conformal symmetries, namely the Wess--Zumino--Novikov--Witten model (WZNW). The quantisation is properly defined for families of such moduli spaces, as soon as some deformation parameter is varied: this talk is about giving an intrinsic (topological) description of the spaces of such deformations in the case where the connections have wild/irregular singularities at some marked points on the surface, i.e. high-order poles. In particular one can deform certain local moduli at each marked point, controlling the principal part of the meromorphic connections, in addition to the standard variations of the underlying pointed Riemann surface.
The archetype is the work of Schlesinger. He studied deformations of Fuchsian systems on the Riemann sphere such that the monodromy representation (of the punctured sphere) is kept locally constant as the position of the simple poles are moved: whence isomonodromic. The quantisation of the resulting Schlesinger system yields the connection of Knizhnik--Zamolodchikov in the genus-zero WZNW model. But much more general situations can be considered and quantised, provided one has a description of the isomonodromy `times' in general, which does not rely on choices of local coordinates/trivialisations.
This is joint work with P. Boalch, J. Douçot and M. Tamiozzo. If possible we will also discuss work with G. Felder about the construction of `irregular' conformal blocks in the WZNW model, also related to the quantisation of moduli spaces of wild meromorphic connections.
October 20, 2022 Speaker: Séverin Charbonnier (UniGe) Title: Solutions of integrable hierarchies via combinatorial maps and topological recursion Abstract: Solutions of some integrable hierarchies such as KdV, KP or 2-Toda can be proved to satisfy Topological Recursion, a procedure developed by Chekhov, Eynard and Orantin and which appears in various branches of mathematical physics, geometry and combinatorics.
One way to tackle this result is to encode the tau-functions of the hierarchies as generating functions of combinatorial maps. Those combinatorial objects are particularly well-suited to prove that the generating functions satisfy topological recursion. I will show two instances of such combinatorial treatment of the solutions of integrable hierarchies: ciliated maps for the r-KdV hierarchy (j.w. Belliard, Eynard and Garcia-Failde), and constellations for the Orlov-Sherbin tau-functions of the 2-Toda hierarchy (j.w. Bonzom, Chapuy and Garcia-Failde).
November 11,2022 Speaker: Miroslav Rapcak (CERN) Title: Knot Homologies from Landau Ginsburg Models Abstract: In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.
November 25, 2022 Speaker: Rita Teixeira da Costa (Princeton U.) Title: Mode stability for Kerr black holes Abstract: Kerr black holes are a family of solutions to the Einstein vacuum equations represented by a mass, M, and angular momentum |a|≤M. Understanding their stability is a longstanding open problem in General Relativity. As a first essential step towards stability, Whiting showed in 1989 that the Teukolsky equation, which describes the behavior of gauge-invariant curvature components under linearized gravity, admits no exponentially growing modes for any |a|≤M black hole.
In this talk, we review Whiting’s classical proof and a recent adaptation thereof to the extremal Kerr case. We also present a new approach to mode stability, which is inspired by the connection of the Teukolsky equation to Seiberg-Witten theory drawn by Aminov, Grassi and Hatsuda in 2020. Part of this talk is based on joint work with Marc Casals (Leipzig/CBPF/UCD).
December 9, 2022 Speaker: Yegor Zenkevich (SISSA) Title: R-matrix formalism for qq-characters in gauge theories Abstract: qq-characters can be viewed in two ways. From physics point of view they are certain observables in supersymmetric gauge theories with eight supercharges having special regularity properties. Two q letters indicate two deformation parameters of the Omega-backgraound in which the gauge theory lives. This approach has been studied a lot in the works of N. Nekrasov.
From a representation theoretic perspective, pioneered by E. Frenkel and N. Reshetikhin, qq-characters are deformations of q-characters. The latter are traces of R-matrices of quantum affine Lie algebras U_q(g) in finite-dimensional representations. However, after the deformation the nice R-matrix interpretation seems to be lost.
In our work we propose a very different R-matrix formalism for qq-characters involving a quantum toroidal (double affine) algebra. The qq-character is no longer a trace, but instead a matrix element of the corresponding R-matrix. The type of the character depends both on the representation in which the R-matrix is evaluated and the external states of the representation defining the matrix element.
We provide several examples of qq-characters corresponding to classical series of root systems. We also relate our calculations to refined topological strings and 5-brane webs in Type IIB string theory.
This talk is based on a joint work with Mehmet Batu Bayindirli, Dilan Demirtas and Can Kozcaz.
February 24, 2022 Speaker: Kento Osuga (Warsaw University) Title: Supersymmetric matrix models and their recursive structures Abstract: Hermitian matrix models are simple yet rich models of quantum gauge field theories, and they admit two interesting properties: Virasoro constraints and recursive structure. Then, one may ask: can we generalise Hermitian matrix models by upgrading Virasoro constraints to super-Virasoro constraints? In this talk I will discuss one such super-generalisation called supereigenvalue models, and explore their recursive structure as well as its relation to 2d supergravity. If time permits, I will present recent progress in a more mathematical framework called super topological recursion and its potential physics applications.
March 10, 2022 Speaker: Maxim Zabzine (Uppsala University) Title: The index of M-theory and equivariant volumes Abstract: Motivated by M-theory, I will review rank n nn K-theoretic Donaldson-Thomas theory on a toric threefold and its factorisation properties in the context of 5d/7d correspondence. In the context of this discussion I will revise the use of the Duistermaat-Heckman formula for non-compact toric Kahler manifolds, pointing out mathematical and physical puzzles.
March 24, 2022 Speaker: Ricardo Schiappa (Universidade de Lisboa) Title: Resurgent Strings and Their Stokes Data Abstract: I will review how minimal and multicritical string models may be computable beyond perturbation theory, via resurgence and transseries methods. This also includes their large central-charge limits, where they relate to Jackiw-Teitelboim gravity. Whereas these constructions are recursive, hence can be made very explicit, they still lack Stokes data to be fully defined. I will then review how to compute non-trivial Stokes data in the simplest models.
April 28, 2022 Speaker: Nicolas Orantin (Geneva University) Title: Topological recursion, quantisation and Painleve equations Abstract: The topological recursion is a universal formalism allowing to solve many problems of enumerative geometry, including the enumeration of discrete surfaces or the computation of Gromov–Witten invariants. From its origin in matrix model theory, it was believed that it should also give rise to solutions of differential equations appearing in classical integrable systems as well as some associated isomonodromic tau functions. In this talk, I will briefly recall some of the applications of the topological recursion to enumerative geometry before explaining how it can be used to quantise algebraic equations. As an example, I will show how this formalism gives a solution to the Painleve equations as well as some associated Lax systems.
May 12, 2022 Speaker: Harini Desiraju (The University of Sydney) Title: The semiclassical limit of probabilistic conformal blocks on a torus Abstract: The form of the semiclassical limit of Liouville conformal blocks was conjectured by Zamolodchikov in 1986. In this talk, I will outline the proof of the conjecture for the case of the one point torus starting from the recently formulated probabilistic conformal blocks by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun. A key role is played by the information coming from isomonodromic equations on the one point torus and the accessory parameter of the Lamé equation. This talk is based on an upcoming paper with Promit Ghosal and Andrei Prokhorov.
May 25, 2022 Speaker: Sara Pasquetti (Universita de Milano-Bicocca) Title: Rethinking mirror symmetry as a local duality on fields Abstract: We introduce an algorithm to piecewise dualise linear quivers into their mirror dual. The algorithm uses two basic duality moves and the properties of the S-wall which can all be derived by iterative applications of Seiberg-like dualities.
June 9, 2022 Speaker: Albrecht Klemm (Bonn University) Title: Feynman Integrals, Calabi-Yau motives and Integrable Systems Abstract: Recently it has been realized that the parameter dependence of Feynman integrals in dimensional regularisation can be calculated explicitly using period-- and chain integrals of suitably chosen Calabi-Yau motives, where the transcendality weight of the motive is proportional to the dimension of the Calabi Yau geometry and the loop order of the Feynman graphs. We exemplify this for the Banana graphs, the Ice Cone graphs and the Train Track graphs in two dimensions. In the latter case there is a calculational very useful relation between the differential realisation of the Yangian symmetries and the Picard-Fuchs system of compact Calabi-Yau spaces M as well as between the physical correlations functions and the quantum volume of the manifolds W that are the mirrors to M.