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Julien Marché

Publications and source records attributed to Julien Marché.

At least 19 recordsLinked to original sources
arXiv2025

Signatures in TQFT : Asymptotics and Modularity

We study the signature $\sigma_g(\frac q p)$ of $\mathrm{SU}_2$-TQFT vector spaces associated to surfaces of genus $g$, as a function of the defining root of unity $\zeta=e^{i\pi q/p}$. We prove that $\frac{1}{p^2}\sigma_2(\frac{q}{p})$ converges to $\Lambda(\theta)=\frac{16}{\pi^3}\sum\limits_{n\ge 1, \textrm{ odd}}\frac{1}{n^3\sin(n\pi\theta)}$ when $\frac{q}{p}$ goes to an irrational number $\theta\in [0,1]$ under certain conditions. We also observe that the function $\Lambda(\theta)$ is the boundary value of an Eichler integral of a level $2$ modular form of weight $4$, and use this to propose a conjectural transformation law for the signature function in genus 2 similar to the reciprocity formula for classical Dedekind sums.

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arXiv2025

Reidemeister torsion of two-bridge knots and signatures of TQFT

We establish an explicit relation between the adjoint Reidemeister torsion of the two-bridge knot $K(p,q)$ at any parabolic representation and the Frobenius algebra governing the signatures of SU$_2$-TQFT vector spaces at the root $\zeta=\exp(i\pi q/p)$. As applications, (a) we prove that the inverse sum of torsions is constant (i.e., independent of $p$ and $q$); and (b) we show that along sequences of roots of the form $\zeta_n = \exp\left(i\pi\tfrac{a+bn}{c+dn}\right)$, the signatures have the same asymptotic behavior as the Verlinde formula.

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arXiv2023

Signatures of TQFTs and trace fields of two-bridge knots

Let $0<s<r$ be coprime odd integers. We show that the Frobenius algebras governing the signatures of SO$_3$ TQFTs at the root $q=\exp(iπs /r)$ contain (and are often equal to) the trace field of the two-bridge knot of parameters $(r,s)$. This gives an intriguing relationship between these two a priori unrelated objects of low-dimensional topology.

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arXiv2022

Toledo invariants of Topological Quantum Field Theories

We prove that the Fibonacci quantum representations $ρ_{g,n}:\rm{Mod}_{g,n}\to \rm{PU}(p,q)$ for $(g,n)\in\{(0,4),(0,5),(1,2),(1,3),(2,1)\}$ are holonomy representations of complex hyperbolic structures on some compactifications of the corresponding moduli spaces $\mathcal{M}_{g,n}$. As a corollary, the forgetful map between the corresponding compactifications of $\mathcal M_{1,3}$ and $\mathcal M_{1,2}$ is a surjective holomorphic map between compact complex hyperbolic orbifolds of different dimensions higher than one, giving an answer to a problem raised by Siu. The proof consists in computing their Toledo invariants: we put this computation in a broader context, replacing the Fibonacci representations with any Hermitian modular functor and extending the Toledo invariant to a full series of cohomological invariants beginning with the signature $p-q$. We prove that these invariants satisfy the axioms of a Cohomological Field Theory and compute the $R$-matrix at first order (hence the usual Toledo invariants) in the case of the $\rm{SU}_2/\rm{SO}_3$-quantum representations at any level.

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arXiv2021

Valuations on the character variety: Newton polytopes and Residual Poisson Bracket

We study the space of measured laminations ML on a closed surface from the valuative point of view. We introduce and study a notion of Newton polytope for an algebraic function on the character variety. We prove for instance that trace functions have unit coefficients at the extremal points of their Newton polytope. Then we provide a definition of tangent space at a valuation and show how the Goldman Poisson bracket on the character variety induces a symplectic structure on this valuative model for ML. Finally we identify this symplectic space with previous constructions due to Thurston and Bonahon.

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arXiv2020

Overcommuting pairs in groups and 3-manifolds bounding them

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifold and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL_2, related to the Steinberg relation.

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arXiv2019

The Kauffman skein module at first order

For a 3-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+ε$ where $ε^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$ which we prove in particular cases. As a maybe useful tool, we then introduce a notion of twisted self-linking and prove that it satisfies the Kauffman relations at first order. These questions come from considerations on asymptotics of quantum invariants.

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arXiv2019

The parallelogram identity on groups and deformations of the trivial character in SL_2(C)

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth point if dim H_1(G,C)>2.

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arXiv2019

Automorphisms of character varieties

We show that the algebraic automorphism group of the SL(2,C) character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.

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arXiv2018

Introduction to quantum representations of mapping class groups

We provide an (almost) self-contained construction of the Witten-Reshetikhin-Turaev representations of the mapping class group. We describe its properties including its Hermitian structure, irreducibility and integrality (at prime level). The construction of these notes relies only on skein theory (Kauffman Bracket) and does not use surgery techniques. We hope that they will be accessible to non-specialists.

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arXiv2018

Volume function and Mahler measure of exact polynomials

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the 2-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $π$ is greater than the amplitude of the volume function. We also prove a $K$-theoretical criterium for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

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arXiv2017

Distribution of Chern-Simons invariants

Let $M$ be a 3-manifold with a finite set $X(M)$ of conjugacy classes of representations $ρ:π_1(M)\to$SU$_2$. We study here the distribution of the values of the Chern-Simons function CS$:X(M)\to \mathbb{R}/2π\mathbb{Z}$. We observe in some examples that it resembles the distribution of quadratic residues. In particular for specific sequences of $3$-manifolds, the invariants tends to become equidistributed on the circle with white noise fluctuations of order $|X(M)|^{-1/2}$. We prove that for a manifold with toric boundary the Chern-Simons invariants of the Dehn fillings $M_{p/q}$ have the same behaviour when $p$ and $q$ go to infinity and compute fluctuations at first order.

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arXiv2016

Asymptotics of quantum representations of surface groups

For a banded link $L$ in a surface times a circle, the Witten-Reshetikhin-Turaev invariants are topological invariants depending on a sequence of complex $2p$-th roots of unity $(A_p)_{p\in 2\mathbb{N}}$. We show that there exists a polynomial $P_L$ such that these normalized invariants converge to $P_L(u)$ when $A_p$ converges to $u$, for all but a finite number of $u$'s in $S^1$. This is related to the AMU conjecture which predicts that non-simple curves have infinite order under quantum representations (for big enough levels). Estimating the degree of $P_L$, we exhibit particular types of curves which satisfy this conjecture. Along the way we prove the Witten asymptotic conjecture for links in a surface times a circle.

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arXiv2015

Character varieties in SL_2 and Kauffman skein algebras

These lecture notes concern the algebraic geometry of the character variety of a finitely generated group in SL(2,C) from the point of view of skein modules. We focus on the case of surface and 3-manifolds groups and construct the Reidemeister torsion as a rational volume form on the character variety.

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arXiv2015

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $π$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $Σ$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $Σ$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.

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arXiv2014

Singular intersections of subgroups and character varieties

We prove a global local rigidity result for character varieties of 3-manifolds into $\rm{SL}_2$. Given a 3-manifold with toric boundary $M$ satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings $M_{p/q}$ are globally locally rigid in the following sense: every irreducible representation $ρ:π_1(M_{p/q})\to\rm{SL}_2(\mathbb{C})$ is infinitesimally rigid, meaning that $H^1(M_{p/q},\textrm{Ad}_ρ)=0$. This question arose from the study of asymptotics problems in topological quantum field theory developed by L. Charles and the first author. The proof relies heavily on recent progress in diophantine geometry and raises new questions of Zilber-Pink type. The main step is to show that a generic curve lying in a plane multiplicative torus intersects transversally almost all subtori of codimension 1. We prove an effective result of this form, based mainly on a height upper bound of Habegger.

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arXiv2013

The modular action on PSL(2,R)-characters in genus 2

We explore the dynamics of the action of the mapping class group in genus 2 on the PSL(2,R)-character variety. We prove that this action is ergodic on the connected components of Euler class 1 and -1, as it was conjectured by Goldman. In the connected component of Euler class 0 there are two invariant open subsets, on one of them the action is ergodic. In this process we give a partial answer to a question of Bowditch.

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