# Gaëtan Borot

Since April 2014, I am W2-Gruppenleiter at the Max Planck Institute for Mathematics, Bonn. Before, I have been a visiting scholar at MIT and a postdoc at University of Geneva (both maths departments). I spend my PhD years in theoretical physics at CEA Saclay, under the supervision of Bertrand Eynard. Earlier, I was a student at ENS Paris.

You can find here information on my research, a CV, events I am co-organizing, upcoming/former talks and slides, information related to my teaching. Here is also some outreach activities I am involved in, popular science texts written earlier, and a big picture.

## Interests

I am interested in the combinatorial and algebraic structures underpinning quantum field theories: Frobenius manifolds, integrable systems, conformal field theories, topological recursions, geometric quantization, operads ... This has strong links to problems in enumerative geometry of surfaces, knot theory, and random matrices. I am also interested in the probabilistic aspects of random matrix theory and interacting particle systems.

An introductory masterclass on topological recursion followed by a conference was organized in QGM, Aarhus in January 2013 (videos). As a follow-up of discussions initiated in Aarhus, I organized in November 2014 a workshop on Topological recursion and geometric quantization in Bonn (videos). Here is a overview list of mathematical results that came out of topological recursion and its applications, which can also serve a guide through bibliography.

A trimester at IHP on Combinatorics and Interactions was held January-March 2017 at IHP, Paris, in coorganization with Marie Albenque, Guillaume Chapuy and Valentin Féray. This represents well my interests related to combinatorics and geometry.

Alessandro Giacchetto and Danilo Lewanski run a reading seminar on the geometric recursion which we developed with Andersen and Orantin. The lectures are recorded.

## Publications

With Eynard and Orantin, we have developed an abstract version of the topological recursion, which takes weaker assumptions than the initial one, and enlarges the set of possible applications:

*Abstract loop equations, topological recursion, and applications*, with B. Eynard and N. Orantin.

math-ph/1303.5808, Communications in Number Theory and Physics **9** 1, 51-187 (2015)

*Formal multidimensional integrals, stuffed maps, and topological recursion*.

math-ph/1307.4957, Annales Henri Poincaré: Comb. Phys. Interact. Volume 1, Issue 2 (2014) 225–264.

* Blobbed topological recursion: properties and applications*.

with S. Shadrin

math-ph/1502.00981, Math. Proc. Cam. Phil. Soc. **162** 1 39--87 (2017)

Complementing a work of Kontsevich and Soibelman, we studied a Lie-algebraic/symplectic approach to topological recursion, based on consistent systems of quadratic differential operators, called Airy structures.

* The ABCD of topological recursion*, with J.E. Andersen, L.O. Chekhov and N. Orantin.

math-ph/1703.03307, submitted.

*Higher Airy structures, W algebras and topological recursion.*

with V. Bouchard, T. Creutzig, N.K. Chidambaram and D. Noshchenko.

math-ph/1812.08738, submitted.

* Geometric recursion*, with J.E. Andersen and N. Orantin.

math.GT/1711.04729, preprint.

With a group in Bonn (Séverin Charbonnier, Alessandro Giacchetto, Danilo Lewanski, Campbell Wheeler) we pursue several applications of the geometric recursion. First, together with Delecroix, we show that the Masur-Veech volumes for the principal stratum of the moduli space of quadratic differentials can be computed by topological recursion.

* Topological recursion for Masur-Veech volumes. *.

with J.E. Andersen, S. Charbonnier, V. Delecroix, A. Giacchetto, D. Lewanski and C. Wheeler.

math.GT/1905.10352, preprint.

* A second topological recursion for Masur-Veech volumes. *

appendix to "Masur-Veech volumes and intersection theory: the principal stratum of quadratic differentials"

by D. Chen, M. Möller, A. Sauvaget.

with A. Giacchetto and D. Lewanski.

math.AG/1912.02267, preprint.

Second, if we run the geometric recursion using the combinatorial Teichmuller space (instead of its hyperbolic model), we can also obtain statistics of combinatorial lengths of simple multicurves. The interplay between the hyperbolic and the combinatorial aspects give via geometric recursion a completely geometric proof of Witten's conjecture/Kontsevich conjecture, which was known before, and generalizations thereof. We also describe

* On the Kontsevich geometry of the combinatorial Teichmüller space*.

with J.E. Andersen, S. Charbonnier, V. Delecroix, A. Giacchetto, D. Lewanski and C. Wheeler.

in preparation.

With Andersen, Eynard and Orantin, we have a long-term project to establish that the topological recursion provides an (asymptotic) construction of states in geometric quantization. The main example we have in view is the moduli space of Higgs bundles on surfaces. Our motivation is to justify the conjecture (see below) that the topological recursion for the A-polynomial curve computes -- in some sense -- the asymptotics of the colored Jones polynomial in the context of the generalized volume conjecture. The second (related) example we are working on is the description of the asymptotic expansion of Wess-Zumino-Witten conformal blocks via the topological recursion.

As a spinoff of this project, we showed that for any modular functor (an axiomatisation of rational conformal field theories), one can construct a vector bundle over the moduli space of curves, whose Chern character factorizes nicely on the Deligne-Mumford boundary (more precisely, it defines a cohomological field theory). It follows that its integrals against psi-classes over the moduli space of curves are computed by the topological recursion. This applies in particular to the character of the Verlinde bundle in Wess-Zumino-Witten theory.

* Modular functors, cohomological field theories, and topological recursion*, with J.E. Andersen and N. Orantin.

math-ph/1509.01387, Proceedings of Symposia in Pure Mathematics, Volume 100 (2018).

### Enumerative geometry

In my first paper, we showed that an appropriate generating series for the number of simple coverings of the sphere by a Riemann surface of genus $g$, obeys the topological recursion of the Lambert curve $ye^{-y} = e^{-x}$. A subtle caveat about convergence of generating series has been observed by Zvonkine, so this should currently only be considered as a "physical proof" -- the statement itself has received a different, complete proof by Eynard-Mulase-Safnuk in math.AG/0907.5224 .

*A matrix model for simple Hurwitz numbers, and topological recursion*, with B. Eynard, M. Mulase and B. Safnuk.

math-ph/0906.1206 , J. Geom. Phys. ** 61 ** (2011).

I am since then interested in Gromov-Witten theory, the BKMP conjecture (now a theorem, stating that GW theory of toric Calabi-Yau 3-folds is computed by topological recursion), Landau-Ginzburg models, etc.

With the group of Sergey Shadrin, we proved that topological recursion computes $r$-spin $q$-orbifold Hurwitz numbers, in the case $r = 2$ for any $q$, and for any $q,r$ in genus $0$. Our approach combines cut-and-join equation with the abstract description of topological recursion from math-ph/1502.00981.

*Special cases of the orbifold version of Zvonkine's r-ELSV formula*

with D. Lewanski, R. Kramer, A. Popolitov and S. Shadrin

math.AG/1705.10811, to appear in Mich. J. Math.

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

with N. Do, M. Karev, D. Lewanski and E. Moskowsky

math-ph/2002.00900, preprint.

With Paul Norbury, we revisit the topological recursion for the Gromov-Witten invariants of $\mathbb{P}^1$. We prove that the non-stationary invariants are also stored in the topological recursion correlators, and that the abstract loop equations (which are of local nature) imply the Virasoro constraints of Okounkov and Pandharipande for ${\rm GW}(\mathbb{P}^1)$.

*Loop equations for Gromov-Witten invariants of $\mathbb{P}^1$*.

with P. Norbury.

to appear in the special issue of SIGMA: Integrability, Geometry, Moduli (2019)

in honor of Motohico Mulase on his 65th birthday.

math.AG/1905.01890

### Combinatorics of maps

I study statistical physics models, like the $O(n)$ model or the Potts model , on random lattices of all topologies. A first problem is to enumerate (decorated) maps, i.e. how many inequivalent surfaces do can we obtain by gluing (with special rules) polygons along their edges ? The final goal is to understand the geometry of large random lattices , in relation with conformal field theory and SLE processes.

In the $O(n)$ model, maps carry self-avoiding loops with a weight $n$ for each (see the picture). In my second paper, we showed the enumeration problem in the trivalent $O(n)$ model is solved by a twisted version of the topological recursion.

*Enumeration of maps with self-avoiding loops and the $O(n)$ model on random lattices of all topologies*

with B. Eynard.

math-ph/0910.5896, J. Stat. Mech. ** P01010 ** (2011).

*A recursive approach to the $O(n)$ model on random maps via nested loops*, with J. Bouttier and E. Guitter.

math-ph/1106.0153, J. Phys. A: Math. Theor. **45** 045002 (2012).

*More on the $O(n)$ model on random maps via nested loops: loops with bending energy*,

with J. Bouttier and E. Guitter.

math-ph/1202.5521, J. Phys. A: Math. Theor. **45** 275206 (2012).

*Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model*, with J. Bouttier and E. Guitter.

math-ph/1207.4878, J. Phys. A, special issue "Lattice models and integrability" in honor of F.Wu (2013).

With J. Bouttier and B. Duplantier, we studied the distribution of nesting in the $O(n)$ model on random planar maps - i.e. number of loops separating two points - and showed that it agrees with the distribution of nesting of the ${\rm CLE}_{\kappa}$ processes with $n = 2\cos\pi|1 - 4/\kappa|$ measured in terms of Liouville quantum gravity. This matches a continuum set of critical exponents for all values of $\kappa$ in the conjectured (in mathematics) correspondence between random maps and Liouville gravity coupled to conformal processes.

* Nesting statistics in the O(n) loop model on random planar maps*, with J. Bouttier and B. Duplantier.

math-ph/1605.02239, submitted.

* Nesting statistics in the O(n) loop model on random maps of arbitrary topologies*, with E. Garcia-Failde.

math-ph/1609.02074, submitted.

* Simple maps, Hurwitz numbers and topological recursion*, with E. Garcia-Failde.

math-ph/1710.07851, submitted.

* Relating ordinary and fully simple maps via monotone Hurwitz numbers*

with S. Charbonnier, N. Do and E. Garcia-Failde.

accepted in Electronic Journal of Combinatorics - A, 2019.

math.CO/1904.02267,

### Random matrix theory

I am interested in random matrix theory, especially in $\beta$ ensembles, i.e. probability measures on $\mathbb{R}^N$ of the form $\mathrm{d}\mu(\lambda_1,\ldots,\lambda_N) = \prod_{i = 1}^N \mathrm{d}\lambda_i\,g(\lambda_i)\,\prod_{1 \leq i \neq j \leq N} |\lambda_i - \lambda_j|^{\beta}$. The $\lambda_i$ can be thought as eigenvalues of certain random operators.

In particular, the $\beta$ Tracy-Widom distribution appears as the limit distribution of $\mathrm{max}_i\,\lambda_i$. Apart from $\beta = 1/2,1$ and $2$ where $\mathsf{TW}_{\beta}$ can be computed in relation with integrable systems and Painlevé equations (graph courtesy of J.M. Stéphan), little is known on $\mathsf{TW}_{\beta}$.

We showed how to compute (without a rigorous proof) the all order asymptotic expansion of the left and right tails of $\mathsf{TW}_{\beta}$.

*Large deviations of the maximal eigenvalue of random matrices*, with B. Eynard, S.N. Majumdar and C. Nadal.

math-ph/1009.1945, J. Stat. Mech ** P11024 ** (2011).

*Right tail expansion of Tracy-Widom beta laws*, with C. Nadal.

math-ph/1111.2761, Random Matrices: Theory and Applications (2012).

For $\beta = 1$ (hermitian random matrices), we proved that our result at the left tail matches the expression found by Tracy and Widom.

*The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants*, with B. Eynard.

math-ph/1012.2752, preprint.

With Alice Guionnet, we proved a technical result on $\beta$ ensembles, namely the existence of a $1/N$ expansion of the partition function under natural assumptions (the one-cut regime). Our result justifies one of the steps in the derivation of asymptotics of $\mathsf{TW}_{\beta}$

*Asymptotic expansion of beta matrix models in the one-cut regime*, with A. Guionnet.

math.PR/1107.1167, Commun. Math. Phys. **317** 2 (2013), pp 447-483.

*Asymptotic expansion of beta matrix models in the multi-cut regime*, with A. Guionnet.

math.PR/1303.1045, preprint.

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

with A. Guionnet and K.K. Kozlowski

math-ph/1312.6664, International Mathematics Research Notices (2015).

We have an ongoing project with Vadim Gorin and Alice Guionnet to extend these results to the case of discrete matrix models, which appear in the study of random tilings. A long-term motivation here is to prove the conjecture of Kenyon-Okounkov that the fluctuations of the random surface described by the tilings behave as the Gaussian Free Field.

We are interested on refinements of the results with weaker assumptions, which would be needed to study critical points of the model. It would have applications e.g. in the physics of quantum integrable systems. We started to develop these techniques in the study of the sinh-model with real weights, that is a prototype integral for correlation functions of SL(2,C) XXZ spin chains, and requires a two-scale analysis. This developed into a book presenting the general strategy of asymptotic analysis from Schwinger-Dyson equation, and establishing our results for this prototype integral.

*Asymptotic expansion of a partition function related to the sinh-model*,

with A. Guionnet and K.K. Kozlowski

math-ph/1412.7721, Math. Phys. Studies, Springer (2016).

### Integrable systems and applications

I am studying the interplay between integrability and loop equations (also called Virasoro constraints). In a joint work with Bertrand Eynard, we conjectured that a certain solution of the loop equations, built from the topological recursion of an algebraic curve, and derivatives of theta function, is the tau function of an integrable system depending on a dispersive parameter.

* Geometry of spectral curves and all order dispersive integrable system*, with B. Eynard.

math-ph/1110.4936, SIGMA **8** (2012), 100.

* Rational differential systems, loop equations, and application to $q$-th reductions of KP*,

with M. Bergère and B. Eynard.

math-ph/1312.4237, Annales Henri Poincaré (2015).

### Knot theory and Chern-Simons theory

There exist knot invariants indexed by a compact group G and a representation R, and their asymptotic properties are interesting for several reasons (arithmetic properties, modular properties, relations with symplectic geometry, etc.). In particular, several independent methods have been proposed to compute those asymptotics, and it is fascinating to understand (or even show) they give the same result. I have been especially interested in two asymptotic regimes: G of large rank (e.g. G = SU(N) for large N) but finite size representation, or fixed G but large representations.

For G = SU(2) and large symmetric representation, the knot invariant is the colored Jones polynomial, and the asymptotic problem is related to the generalized volume conjecture. Inspired by our work on integrable systems, we conjectured that this formalism can be applied to compute the all-order asymptotics of the Jones polynomial of a hyperbolic knot. The theta functions then represent non-perturbative effects. We checked this conjecture to first order for the figure-eight knot depicted here. This improves a conjecture of Dijkgraaf, Fuji, and Manabe (who considered only the "perturbative part" and found some mismatch).

* All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials*, with B. Eynard.

math-ph/1205.2261, Quantum Topology **6** 1 (2015) 40-138.

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

with B. Eynard + a numerical section by A. Weisse

math-ph/1407.4500, Sel. Math. New Ser. 1-111 (2016).

* Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems*, with A. Brini

hep-th/1506.06887, Adv. Theor. Math. Phys. **22** 2 (2018).

### Statistical physics

Linear singular integral equations intervenes in many areas of statistical physics, often in relation with thermodynamical Bethe Ansatz. The techniques developed for the $O(n)$ model turned out to have applications to other problems in physics, like:

- Quantum entanglement. Collaboration with Céline Nadal at Rudolf Peierls Center for Theoretical Physics in Oxford.
*Purity distribution for generalized random Bures mixed states*, with C. Nadal.

cond-mat.stat-mech/1110.3838, J. Phys. A: Math. Theor.**45**(2012).

(Best paper Prize in J. Phys. A 2012)

- Combustion-related problems. Collaboration with Bruno Denet from IRPHE in Marseille, and Guy Joulin from Institut P' in Poitiers.
*Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation*, with B. Denet and G. Joulin.

physics.flu-dyn/1207.5416 , J. Stat. Mech. (2012),**P10023**.

## List of collaborators

Last modified: February 4th 2020.