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232
Conformal blocks and generalized theta functions
 Comm. Math. Phys
, 1994
"... The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as foll ..."
Abstract

Cited by 100 (9 self)
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The aim of this paper is to construct a canonical isomorphism between two vector spaces associated to a Riemann surface X. The first of these spaces is the space of conformal blocks Bc(r) (also called the space of vacua), which plays an important role in conformal field theory. It is defined as follows: choose a point p ∈ X, and let AX be the
Lie group valued moment maps
 Preprint, ETH, Yale and M.I.T
, 1997
"... Abstract. We develop a theory of “quasi”Hamiltonian Gspaces 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 GuilleminSternberg symplectic crosssection theorem and of convexity ..."
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Cited by 96 (23 self)
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Abstract. We develop a theory of “quasi”Hamiltonian Gspaces 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 GuilleminSternberg symplectic crosssection theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2manifold with boundary as quasiHamiltonian quotients of the space G 2 × · · · × G 2. 1.
The topological vertex
, 2003
"... We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the th ..."
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Cited by 92 (17 self)
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We construct a cubic field theory which provides all genus amplitudes of the topological Amodel for all noncompact toric CalabiYau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed CalabiYau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the Bmodel mirror which is the quantum KodairaSpencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the Bbranes on the mirror Riemann
Stable pairs, linear systems and the Verlinde formula
 Invent. Math
, 1994
"... Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a r ..."
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Cited by 82 (8 self)
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Let X be a smooth projective complex curve of genus g ≥ 2, let Λ→X be a line bundle of degree d> 0, and let (E, φ) be a pair consisting of a vector bundle E →X such that Λ 2 E = Λ and a section φ ∈ H 0 (E) − 0. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the wellknown theory of stable
Axiomatic Conformal Field Theory
 Commun. Math. Phys
, 2000
"... A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c = −2 triplet theory and the k = −4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu. Lectures gi ..."
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Cited by 67 (10 self)
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A comprehensive introduction to logarithmic conformal field theory, using an algebraic point of view, is given. A number of examples are explained in detail, including the c = −2 triplet theory and the k = −4/3 affine su(2) theory. We also give some brief introduction to the work of Zhu. Lectures given at the School on Logarithmic Conformal Field Theory and Its Applications, IPM
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
"... ..."
Quantum gravity with a positive cosmological constant
, 2002
"... A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, dis ..."
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Cited by 49 (9 self)
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A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the WheelerdeWitt equation corrections to the energymomentum relations for matter fields of the form E 2 = p 2 +m 2 +αlPlE 3 +... where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary ChernSimons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.
Vector bundles on curves and generalized theta functions: recent results and open problems
 Cambridge University Press
, 1995
"... Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have ..."
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Cited by 43 (2 self)
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Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a nonabelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a new light on these spaces of sections—allowing notably to compute their dimension (Verlinde’s formula). This survey paper is devoted to giving an overview of these ideas and of the most important recent results on the subject.
The Verlinde algebra is twisted equivariant Ktheory
"... Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. ..."
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Cited by 40 (4 self)
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Ktheory in various forms has recently received much attention in 10dimensional superstring theory. Our raised consciousness about twisted Ktheory led to the serendipitous discovery that it enters in a different way into 3dimensional topological field theories, in particular ChernSimons theory. Namely, as the title of the paper reports, the Verlinde algebra is a certain twisted Ktheory group. This assertion, and its proof, is joint work with Michael Hopkins and Constantin Teleman. The general theorem and proof will be presented elsewhere [FHT]; our goal here is to explain some background, demonstrate the theorem in a simple nontrivial case, and motivate it through the connection with topological field theory. From a mathematical point of view the Verlinde algebra is defined in the theory of loop groups. Let G be a compact Lie group. There is a version of the theorem for any compact group G, but here for the most part we focus on connected, simply connected, and simple groups—G = SU2 is the simplest example. In this case a central extension of the free loop group LG is determined by the level, which is a positive integer k. There is a finite set of equivalence classes of positive energy representations of this central extension; let Vk(G) denote the free abelian group they generate. One of the influences of 2dimensional conformal field theory on the theory of loop groups is the construction of an algebra structure on Vk(G), the fusion product. This is the Verlinde algebra [V]. Let G act on itself by conjugation. Then with our assumptions the equivariant cohomology group H3 G (G) is free of rank one. Let h(G) be the dual Coxeter number of G, and define ζ(k) ∈ H3 G (G) to be k + h(G) times a generator. We will see that elements of H3 may be used to twist Ktheory, and so elements of equivariant H 3 twist equivariant Ktheory. Theorem (FreedHopkinsTeleman). There is an isomorphism of algebras