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216
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.
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
Reduction of Courant algebroids and generalized complex structures
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Higher derived brackets and homotopy algebras
"... Abstract. We give a construction of homotopy algebras based on “higher derived brackets”. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element ∆. Given this, we introduce an infinite sequence of higher brackets o ..."
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Cited by 77 (5 self)
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Abstract. We give a construction of homotopy algebras based on “higher derived brackets”. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element ∆. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of ∆ 2. This allows to control higher Jacobi identities in terms of the “order ” of ∆ 2. Examples include Stasheff’s strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with ∆ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.
Higherdimensional algebra VI: Lie 2algebras
, 2004
"... The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We ..."
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Cited by 73 (14 self)
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The theory of Lie algebras can be categorified starting from a new notion of ‘2vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, ‘linear functors’ as morphisms and ‘linear natural transformations ’ as 2morphisms. We define a ‘semistrict Lie 2algebra ’ to be a 2vector space L equipped with a skewsymmetric bilinear functor [·, ·]: L × L → L satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the ‘Jacobiator’, which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang–Baxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term L∞algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g � which reduces to g at � = 0. These are closely related to the 2groups G � constructed in a companion paper.
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.
Dbranes on CalabiYau manifolds
, 2004
"... In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to ..."
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Cited by 59 (8 self)
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In this review we study BPS Dbranes on Calabi–Yau threefolds. Such Dbranes naturally divide into two sets called Abranes and Bbranes which are most easily understood from topological field theory. The main aim of this paper is to provide a selfcontained guide to the derived category approach to Bbranes and the idea of Πstability. We argue that this mathematical machinery is hard to avoid for a proper understanding of Bbranes. Abranes and Bbranes are related in a very complicated and interesting way which ties in with the “homological mirror symmetry ” conjecture of Kontsevich. We motivate and exploit this form of mirror symmetry. The examples of the quintic 3fold, flops and orbifolds are discussed at some length. In the latter
Noncommutative differential calculus, homotopy . . .
, 2000
"... We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures. ..."
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Cited by 57 (1 self)
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We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.
Deformation Theory And The BatalinVilkovisky Master Equation
 of the Batalin–Vilkovisky approach,” Secondary Calculus and Cohomological Physics (Moscow, 1997), Contemp. Math. 219, AMS
, 1996
"... The BatalinVilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the BatalinVilkovisky approach is here translated into the language of deformation theory. ..."
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Cited by 48 (0 self)
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The BatalinVilkovisky master equations, both classical and quantum, are precisely the integrability equations for deformations of algebras and differential algebras respectively. This is not a coincidence; the BatalinVilkovisky approach is here translated into the language of deformation theory. The following exposition is based in large part on work by Marc Henneaux (Bruxelles) especially and with Glenn Barnich (Penn State and Bruxelles) and Tom Lada and Ron Fulp of NCSU (The NonCommutative State University). The first statement of the relevance of deformation theory to the construction of interactive Lagrangians, that I am aware of, is due to Barnich and Henneaux [3]: We point out that this problem can be economically reformulated as a deformation problem in the sense of deformation theory [13], namely that of deforming consistently the master equation. The `ghosts' introduced by Fade'ev and Popov [12] were soon incorporated into the BRSTcohomology approach [7] to a variety ...