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343
A path integral approach to the Kontsevich quantization formula
, 1999
"... We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a supercon ..."
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Cited by 306 (21 self)
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We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra.
Higherdimensional algebra and topological quantum field theory
 JOUR. MATH. PHYS
, 1995
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Poisson geometry with a 3 form background
 Prog. Theor. Phys. Suppl. 144 (2001) 145 [math/0107133 [mathsg
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Toeplitz Quantization Of Kähler Manifolds And gl(N), N → ∞ Limits
"... For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann s ..."
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Cited by 134 (10 self)
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For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a welldefined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finitedimensional matrix algebras gl(N), N → ∞.
Nonassociative star product deformations for Dbrane . . .
, 2001
"... We investigate the deformation of D–brane world–volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world ..."
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Cited by 114 (3 self)
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We investigate the deformation of D–brane world–volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world–volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination ω = B + F is constant one obtains the standard Moyal deformation of the brane world–volume. Similarly, when dω = 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H = dω ̸ = 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D–branes. The polarized branes
Kontsevich’s universal formula for deformation quantization
 and the CBH formula, I, math.QA/9811174
"... Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for su ..."
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Cited by 84 (0 self)
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Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such formulae. For the dual of a Lie algebra, the ⋆product given by the universal enveloping algebra via symmetrization is shown to be of this type. In fact this ⋆product is essentially given by the CampbellBakerHausdorff (CBH) formula. We call this the CBHquantization. Next we limn Kontsevich’s construction using a graphical representation for differential calculus. We outline a structure theory for the weighted graphs which encode bidifferential operators in his formula and compute certain weights. We then establish that the Kontsevich and CBH quantizations are identical for the duals of nilpotent Lie algebras. Consequently part of Kontsevich’s ⋆product is determined by the CBH formula. Working the other way, we have a graphical encoding for the
From local to global deformation quantization of Poisson manifolds
, 12
"... To James Stasheff on the occasion of his 65th birthday We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich’s local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a ve ..."
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Cited by 82 (6 self)
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To James Stasheff on the occasion of his 65th birthday We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevich’s local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection. 1.
Pseudodifferential operators on differential groupoids
 Pacific J. Math
, 1999
"... We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of t ..."
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Cited by 71 (7 self)
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We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of noncommutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the PoincaréBirkhoffWitt theorem for Lie algebroids and a concrete quantization of the LiePoisson structure on the dual A ∗ of a Lie algebroid. Introduction. Certain important applications of pseudodifferential operators require variants of the original definition. Among the many examples one can find in the literature are regular or adiabatic families of pseudodifferential operators