# 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, and information related to my teaching. Here is also some 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).

A trimester at IHP on Combinatorics and Interactions in January-March 2017 will be held at IHP, Paris, in coorganization with Marie Albenque, Guillaume Chapuy and Valentin Féray.

## Applications

I welcome serious applications for PhD students and post-docs, desiring to work in relation with those thematics.

## 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)

I then showed how the most general (multi-trace) 1 hermitian matrix models are solved by the same topological recursion up to the addition of initial conditions (that I propose to call "blobbed" topological recursion).

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.

In combinatorics, a consequence of this work is that, roughly said, counting maps (build by gluing any type of 2-cells along edges) in all topologies is always done by the same topological recursion, taking as input the counting of disks and of cylinders. We studied the general properties of the blobbed topological recursion, including a general expression in terms of intersection numbers on Deligne-Mumford moduli spaces:

Blobbed topological recursion: properties and applications, with S. Shadrin
math-ph/1502.00981, to appear in Math. Proc. Cam. Phil. Soc. (2016)

Complementing a work of Kontsevich and Soibelman, we studied a Lie-algebraic/symplectic approach to topological recursion.

The ABCD of topological recursion , with J.E. Andersen, L.O. Chekhov and N. Orantin.
math-ph/1703.03307, preprint.

This opens the way to a more geometrical understanding of the nature of topological recursion.

With Andersen, Eynard and Orantin, we started 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, preprint.

We are also developing a hyperbolic generalization of the topological recursion with Andersen and Orantin.

### 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.

### 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).

I extended later this result to a larger class of $O(n)$ loop models. In a joint project with Jérémie Bouttier and Emmanuel Guitter from CEA Saclay, we study them by cutting along loops (these papers are written in a combinatorialist-friendly style). By duality between the $n^2$-Potts model and a loop model with $n$ colors, we could also study the Potts model on general random lattices with defects.

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, preprint.

We generalized these results to random maps of any topology, with use of the topological recursion, with my PhD student Elba Garcia-Failde.

Nesting statistics in the O(n) loop model on random maps of arbitrary topologies, with E. Garcia-Failde.
math-ph/1609.02074, preprint.

### 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.

We extended this work to show the existence of the full asymptotic expansion in the (including oscillatory behavior when $N$ is large) in the multi-cut regime, thus justifying a heuristic derivation proposed earlier by Eynard. This allows to study the asymptotic expansion of orthogonal and skew-orthogonal polynomials by probabilistic methods, i.e. without relying on integrability.

Asymptotic expansion of beta matrix models in the multi-cut regime, with A. Guionnet.
math.PR/1303.1045, preprint.

We gave a more general theory establishing similar results for the asymptotic expansion of multidimensional integrals with arbitrary multlinear interactions, such that the only singularity is a logarithmic pairwise interactions in $\beta$ ensembles. This is typically needed for Chern-Simons matrix models.

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.

We can also go going the other way round: starting with a rational Lax systems (=an integrable system), we gave a result which, under some assumptions, allows to compute the all order WKB expansion from the topological recursion, and we illustrate it on the example of the (p,q) minimal models.

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).

An unjustified point has been found (and fixed by different methods) by Kohei Iwaki and Olivier Marchal for the application to all (p,q) minimal models in this paper.

### 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.

In collaboration with Stavros Garoufalidis and Bertrand Eynard, we are currently trying to extend this proposal to G = SU(k) for any fixed k. From another side, we also try to construct a knot invariant directly from the topological recursion, that should be compared with the expansion of the Kashaev invariant.

The large rank asymptotics is closely related to the theory of LMO invariants. For G = SU(N) and fixed representation, the theory of "geometric transition" allow to compute the asymptotic expansion of knot invariants via topological strings -- this was first put forward by Gopakumar and Vafa for the unknot in the 3-sphere. For very simple knots in finite quotients of the 3-sphere, combining with our work on abstract topological recursion, we could show that the large N expansion of these knot invariants are computed by the topological recursion for an algebraic spectral curve. We also extended this statement to groups of the series SO/Sp, which are related to Kauffman invariants:

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).

Then, with Andrea Brini (Montpellier), we related those spectral curves to the spectral curve of an ADE, relavistic version of the Toda integrable system. We also constructed non-toric target 3-fold, for which suitably defined Gromov-Witten invariants should match large N expansion invariant of the knot invariants computed by topological recursion, hence proposing an extension of Gopakumar-Vafa proposal:

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems, with A. Brini
hep-th/1506.06887, to appear in Advances in Theoretical and Mathematical Physics (2016).

### 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.