Hyun Kyu Kim

We prove a conjecture of the first and third named authors relating the Kauffman bracket skein algebra of a genus zero surface with boundary to a quantized $K$-theoretic Coulomb branch. As a consequence, we see that our skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples with monoidal structure defined by the convolution product. We thus give a monoidal categorification of the skein algebra, partially answering a question posed by D. Thurston.

Fock–Goncharov’s moduli spaces \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} of framed \mathrm{PGL}_{3} -local systems on punctured surfaces \mathfrak{S} provide prominent examples of cluster \mathscr{X} -varieties and higher Teichmüller spaces. In a previous paper of the author (2022), building on the works of others, the so-called \mathrm{SL}_{3} quantum trace map is constructed for each triangulable punctured surface \mathfrak{S} and an ideal triangulation \Delta of \mathfrak{S} , as a homomorphism from the stated \mathrm{SL}_{3} -skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster \mathscr{X} -chart for \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} associated to \Delta . We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the \mathrm{SL}_{3} quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum \mathrm{SL}_{3} - \mathrm{PGL}_{3} duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity.