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

Character manifolds and quantum cluster algebras

Add to your list(s) Download to your calendar using vCal

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.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.


Talks@bham, University of Birmingham. Contact Us | Help and Documentation | Privacy and Publicity.
talks@bham is based on from the University of Cambridge.