Polynomial and topological tau functions of the Drinfeld–Sokolov hierarchies

Abstract : This thesis deals with the computation and applications of tau functions of the Drinfeld– Sokolov hierarchies introduced in 1984. The Drinfeld– Sokolov hierarchies are sequences of integrable partial differential equations which one associates to any semisimple Lie algebra. The tau function is a function associated to any solution of a given hierarchy and which contains all the information of the solution. Tau functions are at the heart of the bonds between Drinfeld–Sokolov hierarchies and algebraic geometry. In Chapter 3, we establish an explicit transformation between the polynomial tau functions of the Korteweg–de Vries hierarchy (associated to the algebra sl(2,C)) and the Adler–Moser polynomials (1978). The latter form a sequence of polynomials satisfying a certain differential recursion relation. Chapter 4 is dedicated to the computation of tau functions via Toeplitz determinants; a method introduced by Cafasso and Wu (2015). In collaboration with Cafasso and Yang, we obtained an expansion of the tau function as a sum over all integer partitions. It follows a simple criterion for the polynomiality of the tau function; we give some nontrivial examples. In Chapter 5, in collaboration with Paolo Rossi, we confirm the so-called ‘strong DR/DZ conjecture’ for the algebra o(8,C) (D4). The latter states an equivalence between, in particular, Drinfeld–Sokolov hierarchies and another kind of hierarchies called ‘the double ramification hierarchies’ introduced by Buryak (2015) and constructed from the cohomology of the moduli spaces of stables complex curves Mg,n.
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Ann Du Crest de Villeneuve. Polynomial and topological tau functions of the Drinfeld–Sokolov hierarchies. Operator Algebras [math.OA]. Université d'Angers, 2018. English. ⟨NNT : 2018ANGE0019⟩. ⟨tel-02022215v2⟩

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