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164
On the gauge theory/geometry correspondence
 Adv. Theor. Math. Phys
, 1999
"... The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exa ..."
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Cited by 274 (36 self)
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The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on both sides for arbitrary λ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative Dbrane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.
On The MelvinMortonRozansky Conjecture
, 1994
"... . We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the AlexanderConway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial) ..."
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Cited by 100 (22 self)
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. We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the AlexanderConway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of that knot (i.e., coefficients of the "colored" Jones polynomial). We first reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the MelvinMortonRozansky (MMR) conjecture to knot invariants coming from arbitrary semisimple Lie algebras. As side benefits we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial. Contents 1. Introduction 2 1.1. The conjecture 1.2. Preliminaries 1....
Homotopy Galgebras and moduli space operad
 PREPRINT MPI / 9471, MAXPLANCKINSTITUT IN BONN
, 1994
"... This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally ..."
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Cited by 80 (4 self)
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This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex Ω • X of a Kähler manifold X, thereby yielding the most general type of homotopy Galgebra structure on Ω • X. This latter statement is based on a typical construction of supersymmetric sigmamodel, the construction of GromovWitten invariants in Kontsevich’s version.
On Operad Structures of Moduli Spaces and String Theory
, 1994
"... We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a ..."
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Cited by 65 (13 self)
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We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras and BatalinVilkovisky algebras. These algebras appeared recently in the context of string theory, and we give a simple deduction of these algebraic structures from the formal axioms of conformal field theory and string theory.
Integral expressions for the Vassiliev knot invariants
, 1995
"... It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the ChernSimons field theory, converge and yield knot invariants. This was proposed independently by Gaudagnini, Martellini, and Mintche ..."
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Cited by 62 (2 self)
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It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the ChernSimons field theory, converge and yield knot invariants. This was proposed independently by Gaudagnini, Martellini, and Mintchev [11] and BarNatan [4]. The analytic difficulties involved in proving convergence and invariance were reportedly worked out by BarNatan [4, 3], Kontsevich [13, 16], and Axelrod and Singer [1, 2]. But I know of no elementary exposition of this fact. BarNatan [3] only proves invariance for the degree 2 invariant. Kontsevich’s exposition is decidedly nonelementary and leaves many details implicit. Axelrod and Singer’s papers take a physics point of view (and are thus difficult for mathematicians to read) and only discuss the related invariants for 3manifolds. More recently, Bott and Taubes [7] have explained the degree 2 invariant from a purely topological point of view, but again omit the higher degree cases. This thesis is an attempt to remedy this lack. I adopt an almost exclusively topological point of view, rarely mentioning ChernSimons theory. For an explanation
HyperKähler geometry and invariants of threemanifolds
 Selecta Math
, 1997
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An invariant of integral homology 3spheres which is universal for all finite type invariants
, 1996
"... In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type inv ..."
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Cited by 57 (5 self)
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In [LMO] a 3manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite type invariants, i.e. Ωn is an invariant order 3n which dominates all other invariants of the same order. Some corollaries are discussed.
Exact results for perturbative ChernSimons theory with complex gauge group
 Commun. Number Theory Phys
"... Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian irreducible flat connection, perturbative GC invariants turn out to be interestin ..."
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Cited by 54 (17 self)
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Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic TQFT ” and conjecture (with supporting numerical evidence) that SL(2, C) ChernSimons theory is an example of such a theory. CALT682716 Contents
Configuration spaces and Vassiliev classes in any dimension
, 2000
"... The real cohomology of the space of imbeddings of S 1 into R n, n> 3, is studied both by using configuration space integrals and by considering the restriction of classes defined on the corresponding spaces of immersions. Nontrivial classes are explicitly constructed. The cohomology classes obta ..."
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Cited by 53 (5 self)
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The real cohomology of the space of imbeddings of S 1 into R n, n> 3, is studied both by using configuration space integrals and by considering the restriction of classes defined on the corresponding spaces of immersions. Nontrivial classes are explicitly constructed. The cohomology classes obtained by configuration space integrals generalize in a nontrivial way the Vassiliev knot invariants obtained in three dimensions from the Chern–Simons perturbation theory.
Perturbative 3manifolds invariants by cutandpaste topology
, 1999
"... We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that t ..."
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Cited by 50 (0 self)
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We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.