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School and Workshop on Topological Recursion: TRSalento2021 (September) http://trsalento2020.eu (deadline April 30)
Abstract: The main theme of the lectures will be the reconstruction of linear ODEs from their monodromy. It leads to two interesting classes of special functions : accessory parameters (for the simplest 2nd order scalar ODEs) and Painlevé functions (for the simplest linear systems). We will start by discussing a few methods of computation of these functions (Hill’s determinant, continued fractions, Widom’s determinant) and explain their connection to $c=\infty$ and $c=1$ conformal blocks of the Virasoro algebra. A classical relation between accessory parameters and Painlevé functions leads to identities involving conformal blocks of both types which allow to express each of them in terms of the other.
Date: 13.04.2021 Time: EST 10am / GMT 2pm / London 3pm / Paris 4pm / Moscow 5pm
Zoom: Link Password: 900582
Abstract: Within the approach of the bootstrap program, the physically pertinent observables in a massive integrable quantum field theory in 1+1 dimensions are expressed by means of the so-called form factor series expansion. This corresponds to a series of multiple integrals in which the n th summand is given by a n-fold integral. While being formally effective for various physical applications, so far, the question of convergence of such form factor series expansions was essentially left open. Still, convergence results are necessary so as to reach the mathematical well-definiteness of such construction and appear as necessary ingredients for the justification of numerous handlings that are carried out on such series and which play the role in the analysis of the physics at the roots of such models.
In this series of two talks, I will first go through the detailed construction of the Sinh-Gordon quantum field theory in 1+1 dimensions within the bootstrap program approach. Once this is settled, I will discuss the physical origin as well as the various motivations for studying the convergence of form factor series expansions in massive quantum integrable field theories. Finally, I will describe the main features of the technique that allows one to prove this convergence. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-n behaviour of the n-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-n behaviour of integrals over the spectrum of n × n random Hermitian matrices.
Date: 16.03.2021 Time: 3pm GMT
Zoom: Link Password: 753053
Abstract: Last time we have recalled the Givental group action on semi-simple CohFTs and its DOSS identification with topological recursion. We will quickly go through several application of the Chiodo classes Cohft, which plays a role in so many different enumerative problems, and the fact that the partition function of its integrals is a KP tau-function. It is interesting to notice how is the spectral curve that transfers the KP integrability from Hurwitz numbers to the Chiodo class. Then, we will see some new applications to Hurwitz numbers of type BKP and their ELSV formula involving an “odd” version of the Chiodo CohFT.
Date: 09.03.2021 Time: 3pm GMT
Zoom: Link Password: 086962
Abstract: We will recall the definition of Cohomological field theories (CohFTs) and their relation with topological recursion. We will continue Gaetan’s review of examples involving intersection theory and ELSV-type formulae, in which Chiodo classes play a central role. We will then address the integrability of Chiodo classes integrals (as opposed to the integrability of the enumerative geometric problems they realise intersection theoretically). This is studied in a work in progress with Giacchetto and Norbury, which extends the proof of Kazarian for the Hodge class. It can be interesting to observe how, in both proofs, the change of variable defined by the spectral curve transfers the integrability from the enumerative problem to the integrals over the moduli spaces of curves.
Date: 02.03.2021 Time: 3pm GMT
Zoom: Link Password: 671483
Abstract: I will review the interpretations (when known or at least conjectured) of the partition function of the basic Airy structures associated to admissible plane curve singularities in terms of intersection theory on moduli space of curves. We will see how, using the symmetries that correspond to deformations of spectral curves, the topological recursion amplitudes can be expressed terms of intersection theory (involving kappa classes and sums over boundary divisors/stable graphs). For regular spectral curves with simple ramification points, this is a result of Eynard. For instance, it gives triple Hodge integrals when applied to the mirror curve of C^3, reproducing the ELSV formula (simple Hodge integrals) in a special case. We will also discuss in the same vein Chiodo classes (for the spectral curve describing r-spin, q-orbifold Hurwitz theory), their inverse (relevant for Masur-Veech volumes). For spectral curves with dx meromorphic, this takes a simpler form to a factorization property that we will explain and explain the relation to the R-action in Givental group.
Date: 23.02.2021 Time: 3pm GMT
Zoom: Link Password: 709935
Abstract: We will see how to reformulate the computation of partition function of Airy structures as residue formula on spectral curves, hence make the connection to the original spirit of Chekhov-Eynard-Orantin formula. In particular, this perspective allows extending the class of admissible spectral curves, to admit some singular covers (i.e. perhaps reducible with components intersecting at branchpoints), by considering twisting by arbitrary permutations instead of just full cycles. Quite mysteriously, we find the type of singularities of the spectral curves is constrained.
Date: 16.02.2021 Time: 3pm GMT
Zoom: Link Password: 544947
Abstract: I finish the description of the W(gl_r) VOA, of some graded Lie ideals inside them, and construction of free field representation from twisted representation of the Heisenberg VOA. I then show how to obtain Airy structures from them by a procedure of diltaon shift, which has to be adjusted together with the choice of ideal so as to match the axioms of Airy structures.
Date: 09.02.2021 Time: 3pm GMT
Zoom: Link Password: 980218
Abstract: I will introduce the formalism of Airy structure, initially proposed by Kontsevich and Soibelman as an algebraic setup to reformulate Chekhov-Eynard-Orantin topological recursion (TR). It subsequently led to the generalisation of the original TR definition to allow a larger class of initial data, via a strategy finding its roots in the work of Milanov, and systematized in joint work with Bouchard, Chidambaram, Creutzig, Noshchenko, and Kramer, Schüler. It consists in starting from a VOA with a free field representation, and constructing modules with certain properties. I will explain how to implement it for the orbifold quotients of W(gl_r) at critical central charge, and present results (believed to be generically optimal) for the classification of Airy structures that can be obtained in this way.
Date: 02.02.2021 Time: 3pm GMT
Zoom: Link Password: 926423