University of Birmingham > Talks@bham > Geometry and Mathematical Physics seminar > Character manifolds and quantum cluster algebras

## Character manifolds and quantum cluster algebrasAdd to your list(s) Download to your calendar using vCal - Leonid Chekhov (Michigan State)
- Wednesday 05 June 2019, 13:30-14:30
- Watson Building (Mathematics, R15 on map) Lecture Room C.
If you have a question about this talk, please contact Andrea Brini. We describe and quantise SL_N character manifold on a surface $\Sigma_{g,s,n}$ of arbitrary genus g, $s>0$ holes and $n>0$ decorated boundary cusps (marked points on hole boundaries). All such manifolds can be constructed by amalgamation procedure from elementary blocks which are ideal triangles $\Sigma_{0,1,3}$ endowed with the Fock-Goncharov cluster algebra structure. Elements of monodromy matrices correspond to sums over weighted paths, and we show that for any planar directed (acyclic) network, elements of this matrices satisfy quantum R-matrix relations. From these elementary relations, under the satisfaction of the groupoid property, we construct general quantum monodromy matrices, which satisfy the Goldman bracket in the semi-classical limit. Moreover, a fresh view on the monodromy algebra in the above triangle allowed us to solve an old problem of finding classical and quantum Darboux coordinates for the groupoid of upper-triangular matrices and presenting the braid-group action in this groupoid via mutations of quantum cluster variables in a special quiver obtained from the triangle $\Sigma_{0,1,3}$. (Forthcoming joint paper with M.Shapiro). This talk is part of the Geometry and Mathematical Physics seminar series. ## This talk is included in these lists:- Bham Talks
- Geometry and Mathematical Physics seminar
- School of Mathematics Events
- Watson Building (Mathematics, R15 on map) Lecture Room C
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