HQSMCF

"Homotopy quantum symmetries, monoidal categories and formality"

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 Obiettivo del progetto (Objective)

'This project will be hosted at the MIT (outgoing host) and at the University of Zurich (return host). Its main objectives are : (I) to develop a theory of homotopy quantum groups. This can be understood as the natural the- ory that should sit at the intersection of four important disciplines of mathematics and physics : monoidal categories, homotopy theory, quantum groups and higher categories. This fact gives a clear multidisciplinary aspect to the proposal. (II) to prove the formality of the homotopy Lie algebra governing simultaneous deformations of a Poisson manifold and its coisotropic submanifolds. This is the key step in solving the problem of quantization of symmetries from the point of view of deformation quantization, giving interdisciplinary applications. These two objectives organize themselves into subobjectives : I1 ) define homotopy braided monoidal categories (∞-braided ∞-monoidal category) I2 ) define homotopy quasi-triangular quasi-Hopf algebras (∞-triangular ∞-Hopf algebras) I3 ) define monodromy for higher connections I4 ) give examples of higher Drinfeld associators I5 ) define and give examples of homotopy quantum groups II1 ) construct the homotopy Lie algebra governing simultaneous deformations of a Poisson manifold and its coisotropic submanifolds II2 ) prove the formality of this homotopy Lie algebra. Interdisciplinary aspects come also from tools used which are borrowed from physics (higher holonomies, branes, quantization) or from new rewriting techniques in computer science and operads (Gröbner basis).'