The Fenchel-Nielsen Deformation

Wolpert, Scott A.

doi:10.2307/2007011

Cited by 172 publications

(163 citation statements)

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“…As an application of 4.3 we prove the following theorem, due to Wolpert [19] p; a, f3)u( ap, 13p). The purpose of this section is to PEex#P define abstract Lie algebra structures on spaces based on closed curves.…”

Section: Wolpert's Duality Formulamentioning

confidence: 96%

“…This symplectic structure generalizes the Weil-Petersson Kahler form on Teichmiiller space (taking G= PSL(2, lR)), the cup-product linear symplectic structure on H 1 (S, lR) (when G= lR), and the Kahler forms on the Jacobi variety of a Riemann surface M homeomorphic to S (when G= U(l)) and the NarasimhanSeshadri moduli space of semistable vector bundles of rank n and degree 0 on M (when G = U(n)). The purpose of this paper is to investigate the geometry of this symplectic structure with the aid of a natural family of functions on Hom(n, G)/G.The inspiration for this paper is the recent work of Scott Wolpert on the WeilPetersson symplectic geometry of Teichmiiller space [18][19][20]. In particular he showed that the Fenchel-Nielsen "twist flows" on Teichmiiller space are Hamiltonian flows (with respect to the Weil-Petersson Kahler form) whose associated potential functions are the geodesic length functions.…”

mentioning

confidence: 99%

“…The inspiration for this paper is the recent work of Scott Wolpert on the WeilPetersson symplectic geometry of Teichmiiller space [18][19][20]. In particular he showed that the Fenchel-Nielsen "twist flows" on Teichmiiller space are Hamiltonian flows (with respect to the Weil-Petersson Kahler form) whose associated potential functions are the geodesic length functions.…”

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confidence: 99%

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Invariant functions on Lie groups and Hamiltonian flows of surface group representations

1986

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In [7] it was shown that if n is the fundamental group of a closed oriented surface S and G is Lie group satisfying very general conditions, then the space Hom(n, G)/G of conjugacy classes of representation n-+G has a natural symplectic structure. This symplectic structure generalizes the Weil-Petersson Kahler form on Teichmiiller space (taking G= PSL(2, lR)), the cup-product linear symplectic structure on H 1 (S, lR) (when G= lR), and the Kahler forms on the Jacobi variety of a Riemann surface M homeomorphic to S (when G= U(l)) and the NarasimhanSeshadri moduli space of semistable vector bundles of rank n and degree 0 on M (when G = U(n)). The purpose of this paper is to investigate the geometry of this symplectic structure with the aid of a natural family of functions on Hom(n, G)/G.The inspiration for this paper is the recent work of Scott Wolpert on the WeilPetersson symplectic geometry of Teichmiiller space [18][19][20]. In particular he showed that the Fenchel-Nielsen "twist flows" on Teichmiiller space are Hamiltonian flows (with respect to the Weil-Petersson Kahler form) whose associated potential functions are the geodesic length functions. Moreover he found striking formulas which underscore an intimate relationship between the symplectic geometry of Teichmiiller space and the hyperbolic geometry (and hence the topology) of the surface. In particular the symplectic product of two twist vector fields (the Poisson bracket of two geodesic length functions) is interpreted in terms of the geometry of the surface.We will reprove these formulas of Wolpert in our more general context. Accordingly our proofs are simpler and not restricted to Teichmiiller space: while Wolpert's original proofs use much of the machinery of Teichmiiller space theory, we give topological proofs which involve the multiplicative properties of homology with local coefficients and elementary properties of invariant functions.Before stating the main results, it will be necessary to describe the ingredients of the symplectic geometry. Let G be a Lie group with Lie algebra g. The basic property we need concerning G is the existence of an orthogonal structure on G: an orthogonal structure on G is a nondegenerate symmetric bilinear form~: 9 x g-+ lR *

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Section: Wolpert's Duality Formulamentioning

confidence: 96%

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confidence: 99%

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confidence: 99%

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Invariant functions on Lie groups and Hamiltonian flows of surface group representations

1986

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“…In the case of Teichmüller space T . †/, the basic deformation is the Fenchel-Nielsen (F-N) deformation; a thorough study of this has been carried out by Wolpert in [18]. We cut † 0 along a simple closed geodesic˛, rotate one side of the cut relative to the other and attach the sides in their new position.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Another basic formula concerns the mixed variations tˇt l˛; the reader should see for instance Wolpert [19] or [18] for details.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quakebend deformations in complex hyperbolic quasi-Fuchsian space

Platis

2008

Geom. Topol.

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We study quakebend deformations in complex hyperbolic quasi-Fuchsian space Q C . †/ of a closed surface † of genus g > 1, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of † into the group of isometries of complex hyperbolic space. Emanating from an R-Fuchsian point 2 Q C . †/, we construct curves associated to complex hyperbolic quakebending of and we prove that we may always find an open neighborhood U. / of in Q C . †/ containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert-Kerckhoff formulae for the derivatives of geodesic length function in Teichmüller space. 32G05; 32M05

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Off‐Shell Amplitudes in String Theory and Hyperbolic Geometry

Moosavian

Pius

2020

Fortschritte der Physik

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The construction of off‐shell amplitudes in closed string theory using Riemann surfaces having constant curvature −1 is explored in detail. Off‐shell string amplitudes constructed using surfaces with constant curvature −1 fail to satisfy the off‐shell factorization properties, and as a result, scattering amplitudes computed using them are ill‐defined. In this paper, we describe a procedure for systematically modifying this construction and obtain consistent off‐shell amplitudes in closed string theory satisfying the off‐shell factorization properties.

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The Fenchel-Nielsen Deformation

Cited by 172 publications

References 14 publications

Invariant functions on Lie groups and Hamiltonian flows of surface group representations

Invariant functions on Lie groups and Hamiltonian flows of surface group representations

Quakebend deformations in complex hyperbolic quasi-Fuchsian space

Off‐Shell Amplitudes in String Theory and Hyperbolic Geometry

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