A path integral approach to the Kontsevich quantization formula

Abstract

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.

22 pages, 2 figures, references added. Conjecture on the center made more precise

Cited by in corpus (241)

• Deformation quantization of Poisson manifolds, I
• Noncommutative Field Theory
• D-branes and Deformation Quantization
• Operads and Motives in Deformation Quantization
• Dilaton Gravity in Two Dimensions
• How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples
• Quantization Methods: A Guide for Physicists and Analysts
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• Non-commutative superspace from string theory
• Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds
• Symmetry, Gravity and Noncommutativity
• Nonabelian noncommutative gauge theory via noncommutative extra dimensions
• AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories
• Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism
• A Noncommutative M-Theory Five-brane
• Noncommutative Yang-Mills from equivalence of star products
• Fuzzy Complex Projective Spaces and their Star-products
• Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds
• Emergent Gravity from Noncommutative Spacetime
• Deformation quantization in the teaching of quantum mechanics
• Perturbative quantum gauge theories on manifolds with boundary
• An Exact Solution to Seiberg-Witten Equation of Noncommutative Gauge Theory
• Deformation quantization and superconformal symmetry in three dimensions
• Holography and String Dynamics in Time-dependent Backgrounds
• Trieste lectures on solitons in noncommutative gauge theories
• Noncommutative homotopy algebras associated with open strings
• Lagrange structure and quantization
• Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models
• *-Trek II: *_n Operations, Open Wilson Lines and the Seiberg-Witten Map
• Generalized complex manifolds and supersymmetry
• Classical BV theories on manifolds with boundary
• Brane Partons and Singleton Strings
• On area and entropy of a black hole
• BRST theory without Hamiltonian and Lagrangian
• The Classical Solutions of the Dimensionally Reduced Gravitational Chern-Simons Theory
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Target space symmetries in topological theories I
• Homotopy algebras inspired by classical open-closed string field theory
• Poisson sigma models and deformation quantization
• From local to global deformation quantization of Poisson manifolds
• Hamiltonian perspective on generalized complex structure
• Double Field Theory and Membrane Sigma-Models
• Dualities, Twists, and Gauge Theories with Non-Constant Non-Commutativity
• Homotopy Algebra Morphism and Geometry of Classical String Field Theory
• Generalized complex geometry, generalized branes and the Hitchin sigma model
• D-branes on Calabi-Yau Manifolds and Superpotentials
• On the Globalization of Kontsevich's Star Product and the Perturbative Poisson Sigma Model
• Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops
• T-duality with H-flux: non-commutativity, T-folds and G x G structure
• A Poincare lemma for sigma models of AKSZ type
• Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model
• A sigma model field theoretic realization of Hitchin's generalized complex geometry
• Deformation quantization with traces
• A minimal BV action for Vasiliev's four-dimensional higher spin gravity
• Lectures on AKSZ Sigma Models for Physicists
• Dirac Sigma Models
• First order parent formulation for generic gauge field theories
• BV Quantization of Topological Open Membranes
• Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
• Seiberg-Witten Transforms of Noncommutative Solitons
• Marginal Deformations of N=4 SYM from Open/Closed Twistor Strings
• Poisson geometry of sigma models with extended supersymmetry
• Effective Field Theories on Non-Commutative Space-Time
• Star products made (somewhat) easier
• Parent formulations, frame-like Lagrangians, and generalized auxiliary fields
• Star Product for Second Class Constraint Systems from a BRST Theory
• An action principle for Vasiliev's four-dimensional higher-spin gravity
• $L_{\infty }$-algebras and the perturbiner expansion
• Batalin-Vilkovisky quantization and the algebraic index
• Hamiltonian BRST and Batalin-Vilkovisky formalisms for second quantization of gauge theories
• Frame-like Lagrangians and presymplectic AKSZ-type sigma models
• Star products and perturbative quantum field theory
• Poisson sigma model on the sphere
• Cyclicity of non-associative products on D-branes
• States and representations in deformation quantization
• Generalized complex geometry and the Poisson Sigma Model
• On the Open-Closed B-Model
• The formulae of Kontsevich and Verlinde from the perspective of the Drinfeld double
• Noncommutative Differential Calculus for D-brane in Non-Constant B Field Background
• Parent formulation at the Lagrangian level
• Generalized Poisson sigma models
• Vertex algebras and quantum master equation
• Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains
• Effective descriptions of branes on non-geometric tori
• Covariant Star Product for Exterior Differential Forms on Symplectic Manifolds
• Multiple zeta values in deformation quantization
• Superspin chains from superstring theory
• Gauge PDE and AKSZ-type Sigma Models
• Chiral Higher Spin Gravity and Convex Geometry
• Two-Dimensional Yang-Mills Theory on Surfaces With Corners in Batalin-Vilkovisky Formalism
• Noncommutative Field Theory on Homogeneous Gravitational Waves
• Topological Poisson Sigma models on Poisson-Lie groups
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Odd Scalar Curvature in Anti-Poisson Geometry
• Hamiltonian BRST deformation of a class of n-dimensional BF-type theories
• Split Chern-Simons theory in the BV-BFV formalism
• Nonassociative Weyl star products
• Towards holography in the BV-BFV setting
• QP-Structures of Degree 3 and 4D Topological Field Theory
• Dirac Sigma Models from Gauging
• Higher U(1)-gerbe connections in geometric prequantization
• Introduction to the BV-BFV formalism
• Topological Field Theories and Geometry of Batalin-Vilkovisky Algebras
• The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
• Gauging the Poisson sigma model
• 2D sigma models and differential Poisson algebras
• Quantization of Emergent Gravity
• Quantization of 2D dilaton supergravity with matter
• Brackets, Sigma Models and Integrability of Generalized Complex Structures
• Finite dimensional AKSZ-BV theories
• Presymplectic AKSZ formulation of Einstein gravity
• Noncommutative gauge theory using covariant star product defined between Lie-valued differential forms
• Long time black hole evaporation with bounded Hawking flux
• Supersymmetric WZ-Poisson sigma model and twisted generalized complex geometry
• Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds
• Double field theory for the A/B-models and topological S-duality in generalized geometry
• Semiclassical quantization of classical field theories
• Path integral quantization of the Poisson-Sigma model
• Constructions of Non Commutative Instantons on $T^4$ and $K_3$
• Gravity theory on Poisson manifold with $R$-flux
• Some Noncommutative Multi-instantons from Vortices in Curved Space
• A Path Integral Approach To Noncommutative Superspace
• Poisson sigma model over group manifolds
• D-branes on Group Manifolds and Deformation Quantization
• The Picard groupoid in deformation quantization
• Consistent interactions of dual linearized gravity in D=5: couplings with a topological BF model
• Lie algebroids, Lie groupoids and TFT
• Action principles for higher and fractional spin gravities
• Non-commutative tachyon action and D-brane geometry
• Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory
• Kontsevich product and gauge invariance
• Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary
• Symplectic connections, Noncommutative Yang Mills theory and Supermembranes
• D-brane on Poisson manifold and Generalized Geometry
• The geometry of variations in Batalin-Vilkovisky formalism
• Holographic duals to Poisson sigma models and noncommutative quantum mechanics
• PT-symmetry breaking hamiltonian interactions in BF models
• Wheeled pro(p)file of Batalin-Vilkovisky formalism
• The Geometry of Quantum Mechanics
• The calculus of multivectors on noncommutative jet spaces
• On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism
• Geometry from Matrices via D-branes
• Formality theorems for Hochschild complexes and their applications
• Deformation of Batalin-Vilkovisky Structures
• An Embedding of the BV Quantization into an N=1 Local Superfield Formalism
• Poisson reduction and branes in Poisson-Sigma models
• On Dequantization of Fedosov's Deformation Quantization
• Classical Solutions for Poisson Sigma Models on a Riemann surface
• Algebraic formulation of higher gauge theory
• Jacobi sigma models
• Relational Symplectic Groupoid Quantization for Constant Poisson Structures
• The BV action of 3D twisted R-Poisson sigma models
• On Globalized Traces for the Poisson Sigma Model
• Four-dimensional couplings among BF and massless Rarita-Schwinger theories: a BRST cohomological approach
• On Coordinate Transformations in Planar Noncommutative Theories
• Kronecker foliation, D1-branes and Morita equivalence of Noncommutative two-tori
• Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures
• Gauge field theories with covariant star-product
• Hamiltonian N=2 Superfield Quantization
• Aspects of noncommutative descriptions of planar systems in high magnetic fields
• Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles
• On higher rank coisotropic A-branes
• Higher prequantum geometry
• The Lie algebroid Poisson sigma model
• Comparing Poisson Sigma Model with A-model
• The Fuzzy Kaehler Coset Space with the Darboux Coordinates
• Covariance in the Batalin-Vilkovisky formalism and the Maurer-Cartan equation for curved Lie algebras
• Towards the deformation quantization of linearized gravity
• Self-interactions in a topological BF-type model in D=5
• First order gauge field theories from a superfield formulation
• On the generalized Freedman-Townsend model
• Superfield Hamiltonian quantization in terms of quantum antibrackets
• Couplings between a collection of BF models and a set of three-form gauge fields
• Covariant star product on symplectic and Poisson spacetime manifolds
• The gauging of BV algebras
• Topological A-Type Models with Flux
• BRST-Invariant Constraint Algebra in Terms of Commutators and Quantum Antibrackets
• Topological Dirac Sigma Models and the Classical Master Equation
• Symmetrization of Berezin Star Product and Path-Integral Quantization
• A Remark on the Deformation of GNS Representations of *-Algebras
• Star products and branes in Poisson-Sigma models
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• Lie algebroids, non-associative structures and non-geometric fluxes
• Closed constraint algebras and path integrals for loop group actions
• Hamiltonian Superfield Formalism with N Supercharges
• Normalized Weyl-type $\star$-product on Kähler manifolds
• Non-commutativity and Open Strings Dynamics in Melvin Universes
• Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization
• Target Space Equivariant Cohomological Structure of the Poisson Sigma Model
• Ground States of Duality-twisted Sigma-Models with K3 Target Space
• Graph complexes in deformation quantization
• The Fuzzy Kaehler Coset Space by the Fedosov Formalism
• Orbites coadjointes et variétés caractéristiques
• On Quantum Obstruction Spaces and Higher Codimension Gauge Theories
• Matter-gravity coupling for fuzzy geometry and the Landau-Hall problem
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• Topological and dynamical aspects of Jacobi sigma models
• Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics
• Boundary coupling of Lie algebroid Poisson sigma models and representations up to homotopy
• Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• 2D Poisson Sigma Models with Gauged Vectorial Supersymmetry
• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization
• On the Kontsevich $\star$-product associativity mechanism
• On Nonlinear Gauge Theories
• Uncertainty Relations in Deformation Quantization
• Deformation quantization of the simplest Poisson Orbifold
• Split Canonical Relations
• Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models
• Dual branes in topological sigma models over Lie groups. BF-theory and non-factorizable Lie bialgebras
• Field Theory with Coordinate Dependent Noncommutativity
• 4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism
• Combinatorial proof of a Non-Renormalization Theorem
• Does the nontrivially deformed field-antifield formalism exist?
• Poisson sigma model and semiclassical quantization of integrable systems
• Jacobi algebroids and Jacobi sigma models
• Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids
• Extended sigma-model in nontrivially deformed field-antifield formalism
• Planar field theories with space-dependent noncommutativity
• The coupling of Poisson sigma models to topological backgrounds
• Conversion of second-class constraints and resolving the zero curvature conditions in the geometric quantization theory
• Quantum gauge fields and flat connections in 2-dimensional BF theory
• Duality and the Equivalence Principle of Quantum Mechanics
• Non-topological non-commutativity in string theory
• The Quantum Darboux Theorem,
• The Background Field Method and the Linearization Problem for Poisson Manifolds
• Poly-symplectic Geometry and the AKSZ Formalism
• BV Quantization
• Convolution algebras for Relational Groupoids and Reduction
• Covariant Star Product on Semi-Conformally Flat Noncommutative Calabi-Yau Manifolds and Noncommutative Topological Index Theorem
• Supersymmetric Poisson and Poisson-supersymmetric sigma models
• Formal Global Perturbative Quantization of the Rozansky-Witten Model in the BV-BFV Formalism
• Instances of Higher Geometry in Field Theory
• Generalized sigma model with dynamical antisymplectic potential and non-Abelian de Rham's differential
• Gauge Theories on Open Lie Algebra Non-Commutative Spaces
• Quantizing Derived Mapping Stacks
• Bulk deformations of open topological string theory
• Target space diffeomorphisms in Poisson sigma models and asymptotic symmetries in 2D dilaton gravities
• On the computation of star products
• The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry II: Noncommutative Quantum Black Holes
• Symmetrization of the Berezin Star Product and Multiple Star Product Method