Laminations from the symplectic double

Allegretti, Dylan G. L.

doi:10.1007/s10711-018-0339-0

Cited by 8 publications

(10 citation statements)

References 18 publications

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“…Results of this paper were first reported by the first named author on the Nielsen Retreat of QGM,Århus University, 26-29 October 2014. Simultaneously, the papers [25] and [1] had appeared dealing with similar issues. In particular, Allegretti had also introduced additional shear-type variables associated to external edges of an ideal triangle decomposition of a bordered cusped Riemann surfaces and observed (Lemma 6.3 in [1]) the monoidal relation between exponentiated shear coordinates and λ-lengths.…”

Section: Resultsmentioning

confidence: 94%

Colliding holes in Riemann surfaces and quantum cluster algebras

Chekhov

Mazzocco

2017

Nonlinearity

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In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appear when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesics arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston-Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the Mapping Class Group acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

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Section: Resultsmentioning

confidence: 94%

Colliding holes in Riemann surfaces and quantum cluster algebras

Chekhov

Mazzocco

2017

Nonlinearity

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“…Such a moduli space would have twice the dimension of

X (S, M)

and would be birational to the principal cluster variety . Closely related spaces associated to the symplectic double cluster variety were described in .…”

Section: Introductionmentioning

confidence: 99%

Voros symbols as cluster coordinates

Allegretti

2019

Journal of Topology

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We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock–Goncharov coordinates of framed PGL2false(double-struckCfalse)‐local systems on a marked bordered surface. Using this result, we show that these Borel sums can be extended to meromorphic functions on C∗, and we prove an asymptotic property of the monodromy map introduced in collaboration with Tom Bridgeland.

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“…In [4], Fock and Goncharov showed that this space is closely related to the geometry of the doubled surface S D . In [1], the author studied versions of the Teichmüller space and space of measured laminations on S D which are closely related to the above construction. Definition 3.9 describes the moduli space D P GLm,S when there are no marked points on the boundary of S. To define this space for a general decorated surface, we must modify the definition slightly.…”

Section: The Symplectic Double Moduli Spacementioning

confidence: 99%

“…It was shown in [4] and [1] that this object is related to certain moduli spaces of geometric structures on a doubled surface. Given a compact oriented surface S with boundary, one considers the same surface S • with the opposite orientation.…”

Section: Introductionmentioning

confidence: 99%