Diff (S 1) and the Teichmüller spaces

Nag, Subhashis; Verjovsky, Alberto

doi:10.1007/bf02099878

Cited by 93 publications

(97 citation statements)

References 12 publications

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“…It also allows to use Nag and Verjovsky [108] arguments (summarized in Section 7 of Ref. [3]) for use of QS deformations of S 1 ∞ in order to obtain the Virasoro algebra.…”

Section: Universal Teichmüller Space Kdv and Frobenius Manifoldsmentioning

confidence: 99%

“…Next, we make both ϕ and φ to depend upon the parameter δ , that is we write φ = φ(ξ, δ).Next, we assume that the parameter δ plays role of "time" t and, finally, we write 2) of Segal [113] and, hence, can be called KdV equation. We provided details of derivation in order to emphasize the universality of this equation in problems which involve circular maps or maps of D. Since KdV is effectively dual to the Virasoro algebra surely it also can be obtained via Nag-Verjovsky approach to construction of the Virasoro algebra and the Kirillov-Kostant two-form by using the universal Teichmüller space T(1) [108 ] . Summary of Nag-Verjovsky results can be found in our earlier work, Ref.…”

Section: Universal Teichmüller Space Kdv and Frobenius Manifoldsmentioning

confidence: 99%

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Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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In previous publications ( J. Geom. Phys.38 (2001) 81-139 and references therein ) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23 ) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.

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“…It also allows to use Nag and Verjovsky [108] arguments (summarized in Section 7 of Ref. [3]) for use of QS deformations of S 1 ∞ in order to obtain the Virasoro algebra.…”

Section: Universal Teichmüller Space Kdv and Frobenius Manifoldsmentioning

confidence: 99%

Section: Universal Teichmüller Space Kdv and Frobenius Manifoldsmentioning

confidence: 99%

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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“…As a manifold, T is a complex Banach manifold. It carries a Kähler metric which is, however, not defined on all tangent vectors (see, for details, Nag and Verjovsky [1990]). …”

Section: Using the Kdv Hamiltonianmentioning

confidence: 99%

“…The coadjoint orbit symplectic structure on T 0 is the imaginary part of this Kähler structure; in other words, the structure coming from coadjoint orbit reduction agrees with that coming from the Kähler structure. On T as well as on T 0 , the real part of this Kähler structure is the famous Weil-Petersson metric (see Nag and Verjovsky [1990] for details). Teo [2004, 2006] take a completely different point of view motivated by the fact that in the complex Banach manifold topology, the Weil-Petersson metric on T is not everywhere defined.…”

Section: Classical Teichmüller and Moduli Spacesmentioning

confidence: 99%

Hamiltonian Reduction by Stages

Marsden¹,

Misiołek²,

Ortega³

et al. 2007

Lecture Notes in Mathematics

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PrefaceThis book is about Symplectic Reduction by Stages for Hamiltonian systems with symmetry. Reduction by stages means, roughly speaking, that we have two symmetry groups and we want to carry out symplectic reduction by both of these groups, either sequentially or all at once. More precisely, we shall start with a "large group" M that acts on a phase space P and assume that M has a normal subgroup N . The goal is to carry out reduction of the phase space P by the action of M in two stages; first by N and then by the quotient group M/N . For example, M might be the Euclidean group of R 3 , with N the translation subgroup so that M/N is the rotation group. In the Poisson context such a reduction by stages is easily carried out and we shall show exactly how this goes in the text. However, in the context of symplectic reduction, things are not nearly as simple because one must also introduce momentum maps and keep track of the level set of the momentum map at which one is reducing. But this gives an initial flavor of the type of problem with which the book is concerned.As we shall see in this book, carrying out reduction by stages, first by N and then by M/N , rather than all in "one-shot" by the "large group" M is often not only a much simpler procedure, but it also can give nontrivial additional information about the reduced space. Thus, reduction by stages can provide an essential and useful tool for computing reduced spaces, including coadjoint orbits, which is useful to researchers in symplectic geometry and geometric mechanics.Reduction theory is an old and time-honored subject, going back to the early roots of mechanics through the works of Euler, Lagrange, Poisson, vi Preface Liouville, Jacobi, Hamilton, Riemann, Routh, Noether, Poincaré, and others. These founding masters regarded reduction theory as a useful tool for simplifying and studying concrete mechanical systems, such as the use of Jacobi's elimination of the node in the study of the n-body problem to deal with the overall rotational symmetry of the problem. Likewise, Liouville and Routh used the elimination of cyclic variables (what we would call today an Abelian symmetry group) to simplify problems and it was in this setting that the Routh stability method was developed.The modern form of symplectic reduction theory begins with the works of Arnold [1966a], Smale [1970], Meyer [1973], and Marsden and Weinstein [1974]. A more detailed survey of the history of reduction theory can be found in the first Chapter of the present book. As was the case with Routh, this theory has close connections with the stability theory of relative equilibria, as in Arnold [1969] and Simo, Lewis and Marsden [1991]. The symplectic reduction method is, in fact, by now so well known that it is used as a standard tool, often without much mention. It has also entered many textbooks on geometric mechanics and symplectic geometry, such as Abraham and Marsden [1978], Arnold [1989], Guillemin and Sternberg [1984], Libermann and Marle [1987], and McDuff and Salamon [1995]. De...

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“…In [13] Nag and Verjovsky make the connection between the almost complex structure for Teichmuller theory and the almost complex structure for the Lie algebra of the Lie group of the C°° diffeomorphisms of 51. In the Ahlfors-Bers approach to Teichmuller theory, the holomorphic dependence of a solution to the Beltrami equation on the Beltrami coefficient p leads to the complex structure for Teichmuller space.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimal bending and twisting in one-dimensional dynamics

Gardiner¹

1995

Trans. Amer. Math. Soc.

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Abstract. An infinitesimal theory for bending and earthquaking in onedimensional dynamics is developed. It is shown that any tangent vector to Teichmuller space is the initial data for a bending and for an earthquaking ordinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear operators from a Banach space of measures on the Mobius strip to tangent vectors to Teichmuller space, and the construction of a nonlinear right inverse for these operators. The inverse is constructed by establishing an infinitesimal version of Thurston's earthquake theorem.

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Diff (S 1) and the Teichmüller spaces

Cited by 93 publications

References 12 publications

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Hamiltonian Reduction by Stages

Infinitesimal bending and twisting in one-dimensional dynamics

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