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223
The self-duality equations on a Riemann surface
- Proc. Lond. Math. Soc., III. Ser
, 1987
"... In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action—the so-called 'instanton ..."
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Cited by 502 (6 self)
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In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action—the so-called 'instantons'. The same equations may be
Invariant functions on Lie groups and Hamiltonian flows of surface group representations
, 1986
"... ..."
Lie group valued moment maps
- Preprint, ETH, Yale and M.I.T
, 1997
"... Abstract. We develop a theory of “quasi”-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity ..."
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Cited by 147 (29 self)
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Abstract. We develop a theory of “quasi”-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G 2 × · · · × G 2. 1.
Quantum gravity in 2 + 1 dimensions . . .
- LIVING REVIEWS IN RELATIVITY
, 2005
"... In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body o ..."
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Cited by 137 (0 self)
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In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity in the setting of a spatially closed universe.
Dual Teichmüller spaces
, 1998
"... We describe two spaces related to Riemann surfaces — the Teichmüller space of decorated surfaces and the Teichmüller space of surfaces with holes. We introduce simple explicit coordinates on them. Using these coordinates we demonstrate the relation of these spaces to the spaces of measured laminatio ..."
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Cited by 70 (3 self)
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We describe two spaces related to Riemann surfaces — the Teichmüller space of decorated surfaces and the Teichmüller space of surfaces with holes. We introduce simple explicit coordinates on them. Using these coordinates we demonstrate the relation of these spaces to the spaces of measured laminations, compute Weil-Petersson forms, mapping class group action and study properties of lamination length function. Finally we use the developed technique to construct a noncommutative deformation of the space of functions on the Teichmüller spaces and define a class of unitary projective mapping class group representations (conjecturally a modular functor). One can interpret the latter construction as quantisation of 3D or 2D Liouville gravity. Some theorems concerning Markov numbers as well as Virasoro orbits are given as a by-product.
Ergodic Theory On Moduli Spaces
, 1997
"... Let M be a compact surface with Ø(M ) ! 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group \Gamma M of M acts on the moduli space X(M ) of flat G-bundles over M (possibly twisted by a fixed cent ..."
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Cited by 50 (6 self)
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Let M be a compact surface with Ø(M ) ! 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2)). Then the mapping class group \Gamma M of M acts on the moduli space X(M ) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then \Gamma M preserves a symplectic structure on X(M ) which has finite total volume on M . More generally, the subspace of X(M ) corresponding to flat bundles with fixed behavior over @M carries a \Gamma M -invariant symplectic structure. The main result is that \Gamma M acts ergodically on X(M ) with respect to the measure induced by the symplectic structure.
Finite Dehn surgery on knots
- J. Amer. Math. Soc
, 1996
"... Let K be a knot with a closed tubular neighbourhood N(K) in a connected orientable closed 3-manifold W, such that the exterior of K, M = W −intN(K), is irreducible. We consider the problem of which Dehn surgeries on K, orequivalently, which Dehn fillings on M, can produce 3-manifolds with finite fun ..."
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Cited by 47 (10 self)
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Let K be a knot with a closed tubular neighbourhood N(K) in a connected orientable closed 3-manifold W, such that the exterior of K, M = W −intN(K), is irreducible. We consider the problem of which Dehn surgeries on K, orequivalently, which Dehn fillings on M, can produce 3-manifolds with finite fundamental group.
An analog of a modular functor from quantized Teichmüller theory
"... The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a ..."
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Cited by 45 (7 self)
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The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a
