Operads and Motives in Deformation Quantization

Abstract

This paper is dedicated to the memory of Moshe Flato, and will appear in Lett. Math. Phys. 48 (1) It became clear during last 5-6 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has many deep connections with the geometric world of two-dimensional surfaces. One of manifestations of this is Deligne's conjecture (1993) which says that on the cohomological Hochschild complex of any associative algebra naturally acts the operad of singular chains in the little discs operad. Recently D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.

37 pages, 1 figure

References in corpus (1)

• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group

Cited by in corpus (229)

• Multiple polylogarithms and mixed Tate motives
• Deformation quantization of algebraic varieties
• A Short Survey of Noncommutative Geometry
• Euler's constant: Euler's work and modern developments
• Deformation theory of representations of prop(erad)s
• Deformations of algebras over operads and Deligne's conjecture
• The tangent complex and Hochschild cohomology of E_n-rings
• Shifted Poisson Structures and Deformation Quantization
• M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
• Homological perturbation theory for algebras over operads
• *-TREK: The One-Loop N=4 Noncommutative SYM Action
• Formality of the little N-disks operad
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• From Atiyah Classes to Homotopy Leibniz Algebras
• Hydrodynamic Limit For An Active Exclusion Process
• Perturbative 3-manifold invariants by cut-and-paste topology
• Long knots and maps between operads
• Multi-loop techniques for massless Feynman diagram calculations
• On the Rozansky-Witten weight systems
• Quantum fields and motives
• Holography from quantum cosmology
• Perturbative Corrections to Kahler Moduli Spaces
• The rational homology of spaces of long knots in codimension >2
• Equivalence of formalities of the little discs operad
• On q-deformed gl(l+1)-Whittaker function II
• Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
• Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
• A Prehistory of n-Categorical Physics
• Deformation Quantization of Nonholonomic Almost Kahler Models and Einstein Gravity
• Deformation Quantization of Almost Kahler Models and Lagrange-Finsler Spaces
• Koszul duality of E_n-operads
• Einstein Gravity as a Nonholonomic Almost Kahler Geometry, Lagrange-Finsler Variables, and Deformation Quantization
• Deformation quantization of cosmological models
• Derived coisotropic structures II: stacks and quantization
• The Homotopy Braces Formality Morphism
• States and representations in deformation quantization
• A solution of Deligne's conjecture
• Topological Open Membranes
• The topology of spaces of knots
• On the Multiplicative Structure of Topological Hochschild Homology
• Homotopy Algebras for Operads
• Cardy condition for open-closed field algebras
• Exotic automorphisms of the Schouten algebra of polyvector fields
• Topological conformal field theories and Calabi-Yau categories
• Deformation Quantization of Classical Fields
• Sur la conjecture combinatoire de Kashiwara-Vergne
• Multiple zeta values in deformation quantization
• Dyer-Lashof-Cohen operations in Hochschild cohomology
• Low-Energy Effective Action of N=2 Gauge Multiplet Induced by Hypermultiplet Matter
• Real Homotopy Theory of Semi-Algebraic Sets
• The oriented graph complexes
• Twisted Deformation Quantization of Algebraic Varieties
• Quantization on Curves
• OCHA and Leibniz Pairs, towards a Koszul duality
• On the relation between Nori Motives and Kontsevich Periods
• Homotopical and Higher Categorical Structures in Algebraic Geometry
• A model for configuration spaces of points
• Helix Structures in Quantum Cohomology of Fano Varieties
• Non-formality of the Swiss-Cheese operad
• Some remarks on q-deformed multiple polylogarithms
• Iterated bar complexes of E-infinity algebras and homology theories
• Smooth Version of Johnson's Problem Concerning Derivations of Group Algebras
• Generalized Lagrange Transforms: Finsler Geometry Methods and Deformation Quantization of Gravity
• Proof of Swiss Cheese Version of Deligne's Conjecture
• Renormalization and motivic Galois theory
• Traces for star products on the dual of a Lie algebra
• Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism
• Quantization via Linear homotopy types
• Feynman integrals and motives of configuration spaces
• Real models for the framed little $n$-disks operads
• Quantisation of derived Lagrangians
• Open-closed field algebras
• Formality of Sinha's cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology
• The Algebraic Index Theorem and Fedosov Quantization of Lagrange-Finsler and Einstein Spaces
• Kontsevich's Swiss Cheese Conjecture
• Dalian notes on rational Pontryagin classes
• The Kashiwara-Vergne conjecture and Drinfeld's associators
• The Geometry of Quantum Mechanics
• Branes from Moyal Deformation Quantization of Generalized Yang Mills Theories
• (Non-)formality of the extended Swiss Cheese operads
• Periods and Feynman amplitudes
• The Lambrechts-Stanley Model of Configuration Spaces
• Formality theorems for Hochschild complexes and their applications
• Field theoretic study of electron-electron interaction effects in Dirac liquids
• Hypercommutative operad as a homotopy quotient of BV
• Derived Algebraic Geometry
• The Quantum State Of The Universe From Deformation Quantization and Classical-Quantum Correlation
• Multivariable cochain operations and little n-cubes
• Kontsevich star-product on the dual of a Lie algebra
• Quantization and the tangent groupoid
• Poisson structures on the homology of the space of knots
• Heap and Ternary Self-Distributive Cohomology
• On the formality of the little disks operad in positive characteristic
• A Finite Dimensional Gauge Problem in the WZNW Model
• A model for framed configuration spaces of points
• The Category CNOT
• The $\hat{\Gamma }$-genus and a regularization of an $S^1$-equivariant Euler class
• Commutative hairy graphs and representations of $Out(F_r)$
• Deformation Quantization in Algebraic Geometry
• Flat family of QFTs and quantization of d-algebras
• Quantum heaps, cops and heapy categories
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
• Deformation Quantization of Geometric Quantum Mechanics
• Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators
• Swiss-Cheese operad and Drinfeld center
• Homology and manifolds with corners
• Entropy and Diversity: The Axiomatic Approach
• A Remark on the Deformation of GNS Representations of *-Algebras
• New perspectives on self-linking
• Representations of the Renormalization Group as Matrix Lie Algebra
• Operads and knot spaces
• Quantisation of derived Poisson structures
• Multiple zeta values over global function fields
• The Kontsevich graph orientation morphism revisited
• Algebra of Noncommutative Riemann Surfaces
• Ground-state Wigner functional of linearized gravitational field
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• Permutahedral Structures of $E_2$ Operads
• On the Spectral Sequence of the Swiss-cheese Operad
• The deformation quantization mapping of Poisson- to associative structures in field theory
• On the rational homotopy type of embedding spaces of manifolds in $R^n$
• Tamarkin's construction is equivariant with respect to the action of the Grothendieck-Teichmueller group
• Weyl invariant polynomial and deformation quantization on Kahler manifolds
• Operads, configuration spaces and quantization
• On certain combination of colored multizeta values
• Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads
• Graph complexes in deformation quantization
• On Spineless Cacti, Deligne's Conjecture and Connes--Kreimer's Hopf Algebra
• Continuous and Twisted L_infinity Morphisms
• Moyal-Weyl Star-products as Quasiconformal Mappings
• Configuration Spaces of Surfaces
• The explicit equivalence between the standard and the logarithmic star product for Lie algebras
• Descent properties of the topological chiral homology
• Algebra of the infrared and secondary polytopes
• Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of $E_n$-operads
• On Quantum Obstruction Spaces and Higher Codimension Gauge Theories
• Atiyah classes of strongly homotopy Lie pairs
• On a quantum analog of the Grothendieck-Teichmueller group
• Configuration space integrals and the topology of knot and link spaces
• Non-formality of the odd dimensional framed little balls operads
• The Eckman-Hilton argument and higher operads
• Relative Recognition Principle
• Configuration Spaces of Manifolds with Boundary
• Quadratic Algebras arising from Hopf operads generated by a single element
• A universal A-infinity structure on Batalin-Vilkovisky algebras with multiple zeta value coefficients
• Homotopy-theoretically enriched categories of noncommutative motives
• Weight 11 compactly supported cohomology of moduli spaces of curves
• Frobenius_infinity invariants of homotopy Gerstenhaber algebras I
• Centers and homotopy centers in enriched monoidal categories
• Moduli of framed formal curves
• Periods and Applications
• Cyclic operad formality for compactified moduli spaces of genus zero surfaces
• On mapping spaces of differential graded operads with the commutative operad as target
• On deformation theory of quantum vertex algebras
• Classical Universe Arising from Quantum Cosmology
• Calculus of functors, operad formality, and rational homology of embedding spaces
• Homotopy graph-complex for configuration and knot spaces
• Knots, operads and double loop spaces
• A cubical approach to homotopy orbits of circle actions
• Calculus of the embedding functor and spaces of knots
• Homological Quantum Field Theory
• Coformality and the rational homotopy groups of spaces of long knots
• Curved Koszul duality of algebras over unital versions of binary operads
• Secondary products in supersymmetric field theory
• Manifold theoretic compactifications of configuration spaces
• Formality of a higher-codimensional Swiss-Cheese operad
• When can a formality quasi-isomorphism over rationals be constructed recursively?
• On the universality of string theory
• Quillen cohomology of enriched operads
• Duality and the Equivalence Principle of Quantum Mechanics
• Noncommutative calculus and the Gauss-Manin connection
• Periods of generalized Tate curves
• Grothendieck-Teichmueller group and Poisson cohomologies
• Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem
• The relative lattice path operad
• Spaces of Operad Structures
• Limits of Voronoi Diagrams
• Graph insertion operads
• Deformations and homotopy of Rota-Baxter operators and $\mathcal{O}$-operators on Lie algebras
• Hodge decomposition in the homology of long knots
• Configuration space integrals and Taylor towers for spaces of knots
• Lectures on Noncommutative Geometry
• Forests and the W Construction
• Weyl n-algebras and the Swiss cheese operad
• Homotopy derivations of the framed little discs operads
• Derived deformation theory of algebraic structures
• Multiplicative structure in equivariant cohomology
• The Grothendieck-Teichmüller group action on differential forms and formality morphism of chains
• Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture
• The Quantum Darboux Theorem,
• Torsors, herds and flocks
• From Lie Theory to Deformation Theory and Quantization
• Graph Complexes and higher genus Grothendieck-Teichmüller Lie algebras
• Formal weights in Kontsevich's formality construction and multiple zeta values
• A universal symmetry structure in open string theory
• Rarita-Schwinger Quantum Free Field Via Deformation Quantization
• Formality of little disks and algebraic geometry
• Dynamical Quantization of Contact Structures
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
• Automorphisms of the Weyl manifold
• The Bagger-Lambert model and Type IIA string theory
• Operads for x-physics
• The rational (non-)formality of the non-3-equal manifolds
• Compactified configuration space of points on a line and homotopies of $A_{\infty }$ morphisms
• Non-formality of planar configuration spaces in characteristic two
• Moduli Spaces and Formal Operads
• Operadic Definition of Non-Stricts Cells
• The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
• Quantum Phase Space Description of a Cosmological Minimal Massive Bigravity Model
• A survey of Bott-Taubes integration
• Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}({\hslash }^k)$: $k⩽6$ unlike $k⩾7$
• Symmetrisation of $n$-operads and compactification of real configuration spaces
• Cosimplicial models for spaces of links
• Tannakian twists of quadratic forms and orthogonal Nori motives
• Some notes on string theory
• Multiplicative formality of operads and Sinha's spectral sequence for long knots
• A Kuenneth theorem for configuration spaces
• A Physical Origin for Singular Support Conditions in Geometric Langlands Theory
• Models for the n-Swiss Cheese operads
• Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules
• A Note on the Symmetries and Renormalisability of (Quantum) Gravity
• Associahedron, cyclohedron, and permutohedron as compactifications of configuration spaces
• Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad
• Non-formality of Voronov's Swiss-Cheese operads
• Poisson structures over a complete intersection with isolated singularities
• Hochschild and Simplicial Cohomology
• Brace algebras and the cohomology comparison theorem
• Log Geometric Models for Little Disks Operads in Even Dimensions
• Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory