Topological Quantum Field Theory Seminar   RSS

10/11/2017, 14:00 — 15:00 — Room P3.10, Mathematics Building
Carlos Florentino, Universidade de Lisboa

Geometry, Topology and Arithmetic of character varieties

Character varieties are spaces of representations of finitely presented groups $F$ into Lie groups $G$. When $F$ is the fundamental group of a surface, these spaces play a key role both in Chern-Simons theory and in 2d conformal field theory. In some cases, they are also interpreted as moduli spaces of $G$-Higgs bundles over Kähler manifolds, and were recently studied in connection with the geometric Langlands program, and with mirror symmetry. When $G$ is a complex algebraic group, character varieties are algebraic and have interesting geometry and topology. We can also consider more refined invariants such as Deligne's mixed Hodge structures, which are typically very difficult to compute, but also provide relevant arithmetic information.

In this seminar, we present some explicit computations of the mixed Hodge-Deligne polynomials, and the so-called E-polynomials, of $G$-character varieties of free, and free abelian groups, when $G$ is a group such as $\operatorname{SL}(n,\mathbb{C})$, $(P)\operatorname{GL}(n,\mathbb{C})$ or $\operatorname{Sp}(n,\mathbb{C})$. We also comment on interesting relations between the free case and some explicit formulas by Reineke-Mozgovoy on counting quiver representations over finite fields.

This is joint work with A. Nozad, J. Silva and A. Zamora.

Please note unusual day (Friday).

To join the session on Zoom a password is required. If you are not already on the mailing list, please subscribe the announcements to receive password info and updates.

Current organizers: José MourãoRoger Picken, Marko Stošić


FCT Projects PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry, PTDC/MAT-PUR/31089/2017, Higher Structures and Applications.