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Naturality of ${\rm SL}_3$ quantum trace maps for surfaces

Hyun Kyu Kim

Abstract

Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called ${\rm SL}_3$ quantum trace map is constructed for each triangulable punctured surface $\frak{S}$ and an ideal triangulation $Δ$ of $\frak{S}$, as a homomorphism from the stated ${\rm SL}_3$-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 $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ associated to $Δ$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\rm SL}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\rm SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

Naturality of ${\rm SL}_3$ quantum trace maps for surfaces

Abstract

Fock-Goncharov's moduli spaces of framed -local systems on punctured surfaces provide prominent examples of cluster -varieties and higher Teichmüller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called quantum trace map is constructed for each triangulable punctured surface and an ideal triangulation of , as a homomorphism from the stated -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 -chart for associated to . We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum - duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

Paper Structure

This paper contains 20 sections, 47 theorems, 237 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Naturality of quantum ${\rm SL}_3$-${\rm PGL}_3$ duality maps for Fock-Goncharov cluster varieties
  3. Naturality of ${\rm SL}_3$ quantum trace maps
  4. ${\rm SL}_3$ quantum trace maps
  5. Surfaces and triangulations
  6. ${\rm SL}_3$-skein algebras and ${\rm SL}_3$-laminations
  7. ${\rm PGL}_3$ Fock-Goncharov algebras for surfaces
  8. ${\rm SL}_3$ quantum trace maps
  9. Quantum coordinate change for flips of ideal triangulations
  10. Classical cluster $\mathscr{X}$-mutations
  11. Quantum mutations for $\mathscr{X}$-seeds and for Fock-Goncharov algebras
  12. The balanced algebras, and the quantum coordinate change maps for them
  13. Naturality of ${\rm SL}_3$ quantum trace maps under changes of triangulations
  14. The base case: crossingless arcs over an ideal quadrilateral
  15. The Weyl-ordering and the classical compatibility statement
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

Let $\frak{S}$ be a triangulable punctured surface. For any two ideal triangulations $\Delta$ and $\Delta'$ of $\frak{S}$ (without self-folded triangles), the ${\rm SL}_3$-${\rm PGL}_3$ quantum duality maps in eq.eq:intro_I_q for $\Delta$ and $\Delta'$, constructed in Kim, are related by the quantum

Figures (6)

  • Figure 1: $n$-triangulation quiver, for one triangle
  • Figure 2: The sequence of four mutations for a flip at an arc, transforming $Q_\Delta=Q_\Delta^{[3]}$ to $Q_{\Delta'}=Q_{\Delta'}^{[3]}$
  • Figure 3: ${\rm SL}_3$-skein relations, drawn locally (${\O}$ means empty) in $\frak{S}$, with the framing pointing toward the eyes of the reader; the regions bounded by a loop, a $2$-gon, or a $4$-gon in (S1), (S2), (S3) are contractible, and $[m]_q = \frac{q^m - q^{-m}}{q-q^{-1}} ~\in~\mathbb{Z}[q^{\pm 1}]$
  • Figure 4: Labels of the nodes of a $3$-triangulation quiver in a triangle
  • Figure 5: Boundary relations for stated ${\rm SL}_3$-skeins (horizontal blue line is boundary); the endpoints in the figure are consecutive in the elevation ordering for that boundary component (i.e. $\nexists$ other endpoint with elevation in between these), and $x\prec y$ means $y$ has a higher elevation than $x$; see eq.\ref{['eq:intro_index_inversion']} for the definition of $r_1,r_2$ appearing in (B1).
  • ...and 1 more figures

Theorems & Definitions (84)

  • Theorem \oldthetheorem: main application: naturality of ${\rm SL}_3$-${\rm PGL}_3$ quantum duality maps
  • Definition \oldthetheorem: Def.\ref{['def:Delta-balanced_elements']}; Kim
  • Definition \oldthetheorem: Def.\ref{['def:balanced_subalgebras']}--\ref{['def:balanced_fraction_algebra']}
  • Proposition \oldthetheorem: the balanced cube-root version of quantum coordinate change maps; §\ref{['subsec:the_balanced_algebras_and_quantum_coordinate_change_maps_for_them']}
  • Theorem \oldthetheorem: main theorem, Thm.\ref{['thm:main']}: naturality of the ${\rm SL}_3$ quantum trace maps
  • Definition \oldthetheorem: Le17 Le18
  • Definition \oldthetheorem: Le17 Le18
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: FST, LF
  • ...and 74 more