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109
Modular data: the algebraic combinatorics of conformal field theory, preprint math.QA/0103044
"... This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It is essentially selfcontained, apart from some of the background motivation (Section I) and examples (Section III) which are inc ..."
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Cited by 32 (5 self)
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This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It is essentially selfcontained, apart from some of the background motivation (Section I) and examples (Section III) which are included to give the reader a sense of the context.
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 28 (10 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
The MathaiQuillen Formalism and Topological Field Theory
"... These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the MathaiQuillen formalism for finite dimensional ve ..."
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Cited by 27 (2 self)
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These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the MathaiQuillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the AtiyahJeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.
On Finite Type 3Manifold Invariants I
 II, Math. Annalen
, 1996
"... . Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3spheres (ZHS for short). In the present paper we propose another definition of finite type invariants of ZHS and give equivalent reformu ..."
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Cited by 25 (8 self)
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. Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3spheres (ZHS for short). In the present paper we propose another definition of finite type invariants of ZHS and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra and are of finite type in in the sense of Ohtsuki and thus conclude that the associated graded algebra is a priori finite dimensional in each degree. We discover a new set of restrictions that Ohtsuki's invariants satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3manifold invariants with the (Vassiliev) knot invariants. Contents 1. Introduction 2 1.1. History 1.2. A review of Ohtsuki's definition 1.3. Variations for finite type 3manifold invariants 1.4. Statement of the results 1.5. Plan of the proof 1.6. Acknowledgmen...
Topological quantum field theory for CalabiYau threefolds and G_2manifolds
 ADV. THEOR. MATH. PHYS
, 2002
"... In the past two decades we witness many fruitful interactions between mathematics and physics. One example is the DonaldsonFloer theory for oriented four manifolds. Physical considerations leads to the discovery of the SeibergWitten theory which has profound impact to our understandings of four m ..."
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Cited by 16 (2 self)
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In the past two decades we witness many fruitful interactions between mathematics and physics. One example is the DonaldsonFloer theory for oriented four manifolds. Physical considerations leads to the discovery of the SeibergWitten theory which has profound impact to our understandings of four manifolds. Another example is the Mirror Symmetry for CalabiYau manifolds. This duality transformation in the string theory leads to many surprising predictions in the enumerative geometry.
Canonical Wick rotations in 3dimensional gravity
"... We develop a canonical Wick rotationrescaling theory in 3dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structu ..."
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Cited by 16 (8 self)
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We develop a canonical Wick rotationrescaling theory in 3dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structures on arbitrary surfaces, are all different materializations of “more fundamental ” encoding structures. (b) Canonical geometric correlations: this shows how spacetimes of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic 3manifolds, that carry the appropriate asymptotic projective structure. Both Wick rotations and rescalings act along the canonical cosmological time and have universal rescaling functions. These correlations are functorial with respect to isomorphisms of the respective geometric categories. In particular, the theory applies to spacetimes with compact Cauchy surfaces. By Mess classification, for every fixed genus g ≥ 2 of a Cauchy surface S, and
Topological Quantum Field Theories For Surfaces With Spin Structure
, 1995
"... Refined quantum invariants for closed threemanifolds with links and spin structures are extended to a Topological Quantum Field Theory. By a `universal construction ', one associates, to surfaces with structure, modules which are shown to be free of finite rank. These modules satisfy the multiplica ..."
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Cited by 14 (4 self)
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Refined quantum invariants for closed threemanifolds with links and spin structures are extended to a Topological Quantum Field Theory. By a `universal construction ', one associates, to surfaces with structure, modules which are shown to be free of finite rank. These modules satisfy the multiplicativity axiom of TQFT in an extended Z=2graded sense, and their ranks are given by a spin refined version of the `Verlinde formula'. The relationship with the `unspun' theory is given by a natural `transfer map'. Introduction A Topological Quantum Field Theory (TQFT) in dimension 3 is a functor from a 2 + 1dimensional cobordism category to a category of modules, satisfying certain axioms. This terminology was introduced by Atiyah [1] following Witten's [32] interpretation, in terms of quantum field theory, of the Jones polynomial invariant of links in the 3sphere. The TQFTaxioms imply that the functor is determined by its values on closed bordisms. These lie in the ground ring, and are 3...
Lectures on mirror symmetry, derived categories and Dbranes, Uspehi Mat
 arXiv:math.AG/0308173]; Remarks on Abranes, mirror symmetry, and the Fukaya
, 2003
"... Abstract. This is an introduction to Homological Mirror Symmetry, derived categories, and topological Dbranes aimed at a mathematical audience. In the last lecture we explain why it is necessary to enlarge the Fukaya category with coisotropic Abranes and discuss how to extend the definition of Flo ..."
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Cited by 11 (0 self)
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Abstract. This is an introduction to Homological Mirror Symmetry, derived categories, and topological Dbranes aimed at a mathematical audience. In the last lecture we explain why it is necessary to enlarge the Fukaya category with coisotropic Abranes and discuss how to extend the definition of Floer homology to such objects. These lectures were delivered at IPAM, March 2003, as part of a program on Symplectic Geometry and Physics. 1. Mirror Symmetry From a Physical Viewpoint The goal of this lecture is to explain the physicists ’ viewpoint of the Mirror Phenomenon and its interpretation in mathematical terms proposed by Maxim Kontsevich in his 1994 talk at the International Congress of Mathematicians [1]. Another approach to Mirror Symmetry was proposed by A. Strominger, ST. Yau, and E. Zaslow [2], but we will not discuss it here. From the physical point of view, Mirror Symmetry is a relation between 2d conformal field theories with N = 2 supersymmetry. A 2d conformal field theory is a rather complicated algebraic object whose definition will be sketched in a moment. Thus Mirror Symmetry originates in the realm of algebra/analysis. Geometry will appear later, when we specialize to a particular class of N = 2 superconformal field theories related to CalabiYau manifolds.