Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

Spider evaluation and representations of web groups

The topology of [Formula: see text]-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to [Formula: see text] are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic [Formula: see text] from the root of the tree to the tip of a leaf an irreducible component [Formula: see text] of the representation variety of the web, and a graded subalgebra [Formula: see text] of [Formula: see text]. The spider evaluation of geodesic [Formula: see text] is the symmetrized Poincaré polynomial of [Formula: see text]. The spider evaluation of the web is the sum of the symmetrized Poincaré polynomials of the graded subalgebras associated to all maximal geodesics from the root of the tree to the leaves.

Algebras of Quantum Monodromy Data and Character Varieties

This chapter examines the Poisson structure of the representation variety of the fundamental groupoid of a Riemann surface with punctures and cusps, and the associated decorated character variety.

On deformation spaces of nonuniform hyperbolic lattices

Let Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

THE MODULI PROBLEM OF LOBB AND ZENTNER AND THE COLORED 𝔰𝔩(N) GRAPH INVARIANT

Motivated by a possible connection between the SU (N) instanton knot Floer homology of Kronheimer and Mrowka and 𝔰𝔩(N) Khovanov–Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colorings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yamada in their state-sum interpretation of the quantum 𝔰𝔩(N) knot polynomial. For graphs with two colors, they showed this moduli space can be thought of as a representation variety, and that its Euler characteristic is equal to the 𝔰𝔩(N) polynomial of the graph evaluated at 1. We extend their results to graphs with arbitrary colorings by irreducible anti-symmetric representations of 𝔰𝔩(N).