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119
Threedimensional quantum gravity, ChernSimons theory, and the Apolynomial
, 2003
"... We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representati ..."
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Cited by 78 (11 self)
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We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the Apolynomial of a knot. Using this approach, we find some new and rather surprising relations between the Apolynomial, the colored Jones polynomial, and other invariants of hyperbolic 3manifolds. These relations generalize the volume conjecture and the MelvinMortonRozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.
Introduction to Grassmann Manifolds and Quantum
 Computation, J. Applied Math
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Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions
, 2005
"... In this paper, we continue to consider the 2dimensional (open) Toda system (Toda lattice) for SU(N + 1) N∑ ..."
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Cited by 39 (17 self)
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In this paper, we continue to consider the 2dimensional (open) Toda system (Toda lattice) for SU(N + 1) N∑
The Existence Of NonTopological Multivortex Solutions In The Relativistic SelfDual ChernSimons Theory
 Comm. Math. Phys
, 1997
"... . We construct a general type of multivortex solutions of the selfduality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic ChernSimons model with the nontopological boundary condition near infinity. For such construction we use a modified version of the Newton iteration ..."
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Cited by 39 (6 self)
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. We construct a general type of multivortex solutions of the selfduality equations (the Bogomol'nyi equations) of (2+1) dimensional relativistic ChernSimons model with the nontopological boundary condition near infinity. For such construction we use a modified version of the Newton iteration method developed by Kantorovich. Introduction The Lagrangian density of the (2+1)dimensional relativistic ChernSimons gauge field theory is given by L = 4 " ¯ae F ¯ A ae + (D ¯ OE)(D ¯ OE) \Gamma 1 2 jOEj 2 (1 \Gamma jOEj 2 ) 2 ; (1) where A ¯ (¯ = 0; 1; 2) is the gauge field on R 3 ; F ¯ = @ @x ¯ A \Gamma @ @x A ¯ is the corresponding curvature tensor, OE = OE 1 + iOE 2 (i = p \Gamma1) is a complex field on R 3 ; called the Higgs field, D ¯ = @ @x ¯ \Gamma iA ¯ is the gauge covariant derivative associated with A ¯ , " ¯ae is the totally skewsymmetric tensor with " 012 = 1, and finally ? 0 is the ChernSimons coupling constant. Our metric on R 3 is (g ¯ ) = ...
Analytic aspects of the Toda system: I. . . .
, 2000
"... In this paper, we analyze solutions of the open Toda system and establish an optimal MoserTrudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (aij)N×N the Cartan matrix for SU(N + 1), i.e., ..."
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Cited by 22 (3 self)
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In this paper, we analyze solutions of the open Toda system and establish an optimal MoserTrudinger type inequality for this system. Let Σ be a closed surface with area 1 and K = (aij)N×N the Cartan matrix for SU(N + 1), i.e.,
Classification of Solutions of a Toda System in R²
, 2001
"... In this paper, we consider solutions of the following (open) Toda system (Toda lattice) for SU(N + 1) \Gamma 1 2 \Deltau i = N X j=1 a ij e u j in R 2 ; for i = 1; 2; \Delta \Delta \Delta ; N , where K = (a ij ) N \ThetaN is the Cartan matrix for SU(N + 1). We show that any solution u = ( ..."
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Cited by 20 (3 self)
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In this paper, we consider solutions of the following (open) Toda system (Toda lattice) for SU(N + 1) \Gamma 1 2 \Deltau i = N X j=1 a ij e u j in R 2 ; for i = 1; 2; \Delta \Delta \Delta ; N , where K = (a ij ) N \ThetaN is the Cartan matrix for SU(N + 1). We show that any solution u = (u 1 ; u 2 ; \Delta \Delta \Delta ; uN ) with Z R 2 e u i ! 1; i = 1; 2; \Delta \Delta \Delta ; N; can be obtained from a rational curve in C P N .
Supersymmetric Dbrane Bound States with Bfield and Higher
 Dimensional Instantons on Noncommutative Geometry,” Phys. Rev. D
"... We classify supersymmetric D0Dp bound states with a nonzero Bfield by considering Tdualities of intersecting branes at angles. Especially, we find that the D0D8 system with the Bfield preserves 1/16, 1/8 and 3/16 of supercharges if the Bfield satisfies the “(anti)selfdual ” condition in dim ..."
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Cited by 19 (1 self)
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We classify supersymmetric D0Dp bound states with a nonzero Bfield by considering Tdualities of intersecting branes at angles. Especially, we find that the D0D8 system with the Bfield preserves 1/16, 1/8 and 3/16 of supercharges if the Bfield satisfies the “(anti)selfdual ” condition in dimension eight. The D0branes in this system are described by eight dimensional instantons on noncommutative R 8. We also discuss the extended ADHM construction of the eightdimensional instantons and its deformation by the Bfield. The modified ADHM equations admit a sort of the ‘fuzzy sphere ’ (embeddings of SU(2)) solution.
New improved MoserTrudinger inequalities and singular Liouville equations on compact surfaces
 Geom. Funct. Anal
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A variational analysis of the Toda system on compact surfaces
 Comm. Pure Appl. Math
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CLASSIFICATION OF BLOWUP LIMITS FOR SU(3) SINGULAR TODA SYSTEMS
"... ABSTRACT. We prove that for singular SU(3) Toda systems, the weak limits of the energy belong to a finite set. For more general systems we prove a uniform estimate for fully blownup solutions. Our method uses a selection process and a careful study of the interaction of bubbling solutions. 1. ..."
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Cited by 10 (5 self)
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ABSTRACT. We prove that for singular SU(3) Toda systems, the weak limits of the energy belong to a finite set. For more general systems we prove a uniform estimate for fully blownup solutions. Our method uses a selection process and a careful study of the interaction of bubbling solutions. 1.