Prof. Dr. Gaetan Borot

Profil

• Generalized Quatum Batalin-Vilkovisky Formalism and Graphical Calculus

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  Förderer: Horizon Europe: Postdoctoral Fellowship EU (PF-EU) Zeitraum: 09/2026 - 08/2028 Projektleitung: Prof. Dr. Gaetan Borot

• GRK 2965/1: „Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte“

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  Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Gavril Farkas

• GRK 2965: Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Stefan Schreieder

• Wissenschaftliche Kommunikation

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 03/2026 - 12/2026 Projektleitung: Prof. Dr. Gaetan Borot

• Europäische Kommission

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• A matrix model for simple Hurwitz numbers, and topological recursion

  2010

  Journal of Geometry and Physics · 125 Zitationen · DOI

• Abstract loop equations, topological recursion and new applications

  2015

  Communications in Number Theory and Physics · 84 Zitationen · DOI

  We formulate a notion of "abstract loop equations," and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one-and two-Hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N ) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

• Asymptotic expansion of beta matrix models in the multi-cut regime

  2013

  arXiv (Cornell University) · 64 Zitationen · DOI

  We establish the asymptotic expansion in $β$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling fractions of these segments are fixed, and show the existence of a $1/N$ expansion. We then study the asymptotics of the sum over the filling fractions, to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($β= 2$) as well as orthogonal ($β= 1$) and skew-orthogonal ($β= 4$) polynomials outside the bulk.

• Geometry of Spectral Curves and All Order Dispersive Integrable System

  2012

  Symmetry Integrability and Geometry Methods and Applications · 46 Zitationen · DOI

  We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N ) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N , where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

• Large-<i>N</i>Asymptotic Expansion for Mean Field Models with Coulomb Gas Interaction

  2015

  International Mathematics Research Notices · 40 Zitationen · DOI

  We derive the large-&lt;f&gt;$N$&lt;/f&gt;, all order asymptotic expansion for a system of &lt;f&gt;$N$&lt;/f&gt; particles with mean field interactions on top of a Coulomb repulsion at temperature &lt;f&gt;$1/\\beta$&lt;/f&gt;, under the assumptions that the interactions are analytic, off-critical, and satisfy a local strict convexity assumption.

• More on the<i>O</i>(<i>n</i>) model on random maps via nested loops: loops with bending energy

  2012

  Journal of Physics A Mathematical and Theoretical · 35 Zitationen · DOI

  We continue our investigation of the nested loop approach to the O(n) model on random maps, by extending it to the case where loops may visit faces of arbitrary degree. This allows to express the partition function of the O(n) loop model as a specialization of the multivariate generating function of maps with controlled face degrees, where the face weights are determined by a fixed point condition. We deduce a functional equation for the resolvent of the model, involving some ring generating function describing the immediate vicinity of the loops. When the ring generating function has a single pole, the model is amenable to a full solution. Physically, such situation is realized upon considering loops visiting triangles only and further weighting these loops by some local bending energy. Our model interpolates between the two previously solved cases of triangulations without bending energy and quadrangulations with rigid loops. We analyze the phase diagram of our model in details and derive in particular the location of its non-generic critical points, which are in the universality classes of the dense and dilute O(n) model coupled to 2D quantum gravity. Similar techniques are also used to solve a twisting loop model on quadrangulations where loops are forced to make turns within each visited square. Along the way, we revisit the problem of maps with controlled, possibly unbounded, face degrees and give combinatorial derivations of the one-cut lemma and of the functional equation for the resolvent.

• The ABCD of topological recursion

  2024

  Advances in Mathematics · 31 Zitationen · DOI

• Topological recursion for Masur–Veech volumes

  2022

  Journal of the London Mathematical Society · 24 Zitationen · DOI

  We study the Masur–Veech volumes MVg,n of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g with n punctures. We show that the volumes MVg,n are the constant terms of a family of polynomials in n variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g and n, which leads us to propose conjectural formulae for low g but all n. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

• Abstract loop equations, topological recursion, and applications

  2013

  arXiv (Cornell University) · 23 Zitationen · DOI

  We formulate a notion of abstract loop equations, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.

• All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials

  2015

  Quantum Topology · 21 Zitationen · DOI

  We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = e^{\frac{2u}{N}}) when N \rightarrow \infty . Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a formal wave function of an integrable system whose semiclassical spectral curve \mathcal{C} would be the \mathrm{SL}_2(\mathbb{C}) character variety of the knot (the A-polynomial), and is formulated in the framework of the topological recursion. It takes as starting point the proposal made recently by Dijkgraaf, Fuji and Manabe (who kept only the perturbative part of the wave function, and found some discrepancies), but it also contains the non-perturbative parts, and solves the discrepancy problem. These non-perturbative corrections are derivatives of Theta functions associated to \mathcal{C} . For a large class of knots, this expansion is still in powers of 1/N due to the special properties of A-polynomials. We provide a detailed check of our proposal for the figure-eight knot and the once-punctured torus bundle L^2R . We also present a heuristic argument inspired from the case of torus knots, for which knot invariants can be computed by a matrix model.

• Higher Airy structures, W algebras and topological recursion

  2018

  arXiv (Cornell University) · 20 Zitationen · DOI

  We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{g})$ algebras at self-dual level, with $\mathfrak{g}= \mathfrak{gl}_{N+1}$, $\mathfrak{so}_{2 N }$ or $\mathfrak{e}_N$. We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these $\mathcal{W}$ constraints have already appeared in the literature, but we find many new ones. For $\mathfrak{gl}_{N+1}$ our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard-Eynard topological recursion gives symmetric $ω_{g,n}$s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to $\mathcal{W}(\mathfrak{gl})$ constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

• Higher Airy structures and topological recursion for singular spectral curves

  2023

  Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions · 17 Zitationen · DOI

  We give elements towards the classification of quantum Airy structures based on the W(\mathfrak{gl}_r) -algebras at self-dual level based on twisted modules of the Heisenberg VOA of \mathfrak{gl}_r for twists by arbitrary elements of the Weyl group \mathfrak{S}_{r} . In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion à la Chekhov–Eynard–Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard–Eynard topological recursion (valid for smooth curves) to a large class of singular curves and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open r -spin intersection theory.

• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

  2022

  Mathematische Annalen · 16 Zitationen · DOI

  Abstract Double Hurwitz numbers enumerate branched covers of $${{{\mathbb {C}}}}{{{\mathbb {P}}}}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers. We prove an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson–Pandharipande–Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> -classes.

• Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

  2016

  Selecta Mathematica · 16 Zitationen · DOI

  We study in detail the large N expansion of SU(N ) and SO(N )/Sp(2N ) Chern-Simons partition function Z N (M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of PSL 2 (C). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern-Simons theory on Seifert spaces. When 1 (M) is finite-i.e., for manifolds M that are quotients of S 3 by a finite isometry group of type ADE-we find that the Weyl group associated with the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large N expansion of Z N (M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in M.

• Asymptotic expansion of matrix models in the multi-cut regime

  2024

  Forum of Mathematics Sigma · 13 Zitationen · DOI

  Abstract We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ( $\beta = 2$ ) as well as orthogonal ( $\beta = 1$ ) and skew-orthogonal ( $\beta = 4$ ) polynomials outside the bulk.

• Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

  2018

  Advances in Theoretical and Mathematical Physics · 13 Zitationen · DOI

  We consider the Gopakumar-Ooguri-Vafa correspondence, relating U(N ) Chern-Simons theory at large N to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients S = \S 3 of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large N dual description in terms of both A-and B-twisted topological strings on (in general non-toric) local Calabi-Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of . Its mirror Amodel theory is realized as the local Gromov-Witten theory of suitable ALE fibrations on P 1 , generalizing the results known for lens spaces. We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large N analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern-Simons partition function. Mathematically, our results put forward an identification between the 1/N expansion of the sl N +1 LMO invariant of S and a suitably restricted Gromov-Witten/Donaldson-Thomas partition function on the A-model dual Calabi-Yau. This 1/N expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in S , is computed by the Eynard-Orantin topological recursion.

• Higher Airy Structures, 𝒲 Algebras and Topological Recursion

  2024

  Memoirs of the American Mathematical Society · 12 Zitationen · DOI

  We define higher quantum Airy structures as generalizations of the Kontsevich–Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W left-parenthesis German g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {W}(\mathfrak {g})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> algebras at self-dual level, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g equals German g German l Subscript upper N plus 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}= \mathfrak {gl}_{N+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German o Subscript 2 upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {so}_{2 N }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German e Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">e</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {e}_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {W}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constraints have already appeared in the literature, but we find many new ones. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l Subscript upper N plus 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {gl}_{N+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard–Eynard topological recursion gives symmetric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper W left-parenthesis German g German l right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">W</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {W}(\mathfrak {gl})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

• Lecture notes on topological recursion and geometry

  2017

  arXiv (Cornell University) · 12 Zitationen · DOI

  These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of topological recursion: in two dimensional quantum field theories, in cohomological field theories, in the computation of Weil-Petersson volumes of the moduli space of curves; (b) relate them more specifically to Eynard-Orantin topological recursion (revisited from Kontsevich-Soibelman point of view based on quantum Airy structures).

• Nesting Statistics in the O(n) Loop Model on Random Planar Maps

  2023

  Communications in Mathematical Physics · 11 Zitationen · DOI

  Abstract In the O ( n ) loop model on random planar maps, we study the depth—in terms of the number of levels of nesting—of the loop configuration, by means of analytic combinatorics. We focus on the ‘refined’ generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O ( n ) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O ( n ) model where loops cross only triangles and have a bending energy, we explicitly compute the refined generating series. We analyse their non generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble ( $$\textrm{CLE}_{\kappa }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>CLE</mml:mtext> <mml:mi>κ</mml:mi> </mml:msub> </mml:math> ) in the Euclidean unit disk, as obtained by Miller et al. (Ann Probab 44(2):1013–1052, 2016, arXiv:1401.0217 ), or to its natural generalisation to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O ( n ) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs.

• Blobbed topological recursion

  2015

  Theoretical and Mathematical Physics · 11 Zitationen · DOI

• Modular functors, cohomological field theories, and topological recursion

  2018

  Proceedings of symposia in pure mathematics · 10 Zitationen · DOI

  Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $\psi$-classes in $\bar{\mathcal{M}}_{g,n}$ is computed by the topological recursion of \cite{EOFg}, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the dimensions $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n}) = \dim \mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is retrieved from the topological recursion. We analyze the consequences of our result on two examples: modular functors associated to a finite group $G$ (for which $D_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $\vec{\lambda}$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $\mathcal{V}_{\vec{\lambda}}(\mathbf{\Sigma}_{g,n})$ is the Verlinde bundle).

• Topological recursion and geometry

  2020

  Reviews in Mathematical Physics · 9 Zitationen · DOI

  This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the computation of volumes of the moduli space of curves. It gives an introduction to the formalism of quantum Airy structures on which the topological recursion is based, which is seen at work in the above topics.

• Asymptotic Expansion of a Partition Function Related to the Sinh-model

  2016

  Mathematical physics studies · 9 Zitationen · DOI

• Large-N asymptotic expansion for mean field models with Coulomb gas interaction

  2013

  arXiv (Cornell University) · 9 Zitationen · DOI

  We derive the large-N, all order asymptotic expansion for a system of N particles with mean-field interactions on top of a Coulomb repulsion at temperature 1/β, under the assumptions that the interactions are analytic, off-critical, and satisfy a local strict convexity assumption.

• Whittaker vectors for $$\mathcal {W}$$-algebras from topological recursion

  2024

  Selecta Mathematica · 8 Zitationen · DOI

  Abstract We identify Whittaker vectors for $$\mathcal {W}^{\textsf{k}}(\mathfrak {g})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of G -bundles over $$\mathbb {P}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> for G a complex simple Lie group, can be computed by a non-commutative version of the Chekhov–Eynard–Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure $$\mathcal {N} = 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

• Deutsche Forschungsgemeinschaft

  Wissenschaftliche Kommunikation

  other

• Europäische Kommission

  Generalized Quatum Batalin-Vilkovisky Formalism and Graphical Calculus

  other

• Gottfried Wilhelm Leibniz Universität Hannover

  GRK 2965: Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte

  university

Name

Prof. Dr. Gaetan Borot

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Physik

Arbeitsgruppe

Mathematische Physik: Mathematische Aspekte der Quantenfeld- und Stringtheorie

Telefon

+49 30 2093-66413

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26.4.2026, 01:02:59