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Colliding holes in Riemann surfaces and quantum cluster algebras | | Leonid Chekhov
; Marta Mazzocco
; | | Date: |
23 Sep 2015 | | Abstract: | We introduce the notion of bordered cusped Teichm"uller space, as the
Teichm"uller space of Riemann surfaces with at least one hole and at least one
bordered cusp on the boundary. We propose a combinatorial graph description of
this bordered cusped Teichm"uller space and endow it with a Poisson structure.
This new notion arises in the limit of colliding holes in a Riemann surface, in
which geodesics that originally passed through the "chewing gum" (a domain
between colliding holes) become geodesic arcs between two bordered cusps
decorated by horocycles. The lengths of these arcs are lambda-lengths in
Thurston--Penner terminology, or cluster variables by Fomin and Zelevinsky. We
introduce an extended set of shear coordinates ($Y$-variables in cluster
terminology) on these Riemann surfaces. We demonstrate that on a Riemann
surface $Sigma_{g,s,n}$ of genus $g$ with $sge 1$ holes/orbifold points and
$nge 1$ bordered cusps, we have cusped geodesic laminations that are complete
systems of arcs, or $lambda$-lengths, or $X$-cluster variables together with
closed paths around holes without bordered cusps. (Every such system of arcs
corresponds to a seed of a cluster.) We construct an explicit 1-1 relation
between these variables and the extended shear coordinates and prove that
commutation relations on the set of extended shear coordinates imply
homogeneous $q$-commutation relations between arcs in the same seed. As a
byproduct, we solve the problem of quantum ordering of shear coordinates in
expressions for arcs and give an explicit combinatorial proof of the Laurent
and positivity properties of $X$-cluster variables in any seed. From the
physical point of view, our construction provides an explicit coordinatization
of moduli spaces of open/closed string worldsheets. | | Source: | arXiv, 1509.7044 | | Services: | Forum | Review | PDF | Favorites | |
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