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Contributions to the theory of KZB associators

Abstract : In this thesis, following the work initiated by V. Drinfeld and pursued by B. Enriquez, then by the latter together with D. Calaque and P. Etingof, we study the universal twisted elliptic (ellipsitomic in short) KZB connection, associated to the moduli space of elliptic curves with n marked points and a (M,N)-level structure. The flatness of this connection allows us to study monodromy relations satisfied by this connection, opening the way to a general theory of ellipsitomic associators and Grothendieck-Teichmüller groups corresponding to them, which is released via the use of the formalism of operads (and some of their variants) basing ourselves on the work of B. Fresse. On the one hand, this formalism allows us to study the structure of associators in higher genus. On the other hand, the ellipsitomic KZB associator allows us to derive a theory of elliptic multiple zeta values at torsion points, from which some of their first associator-like properties are distinguished. We will begin by setting up the operadic machinery necessary to define the ellipsitomic associators starting successively with the genus 0 situation, which is well-known, then the genus 1 situation and their cyclotomic variants. Then, in light of this formalism, we will release a definition of genus gassociators. Next, we will go into the details of the construction of the universal ellipsitomic KZB connection, first over the (M,N)-twisted configuration space of an elliptic curve and then over the moduli space of elliptic curves with a level structure. We will associate this connection to its realized version by means of the use of double affine Hecke algebras and of classical dynamical r-matrices. Finally we will present the applications of this construction, namely : the formality of certain subgroups of the braid group on the torus, the ellipsitomic KZB associator, elliptic multiple zeta values at points of torsion as well as an application in representations of cyclotomic Cherednik algebras.
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Submitted on : Thursday, June 11, 2020 - 5:18:10 PM
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Martin Gonzalez. Contributions to the theory of KZB associators. Quantum Algebra [math.QA]. Sorbonne Université, 2018. English. ⟨NNT : 2018SORUS303⟩. ⟨tel-02865424⟩



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