$L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

Abstract

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_{\infty }$-algebra and how quasi-isomorphisms between $L_{\infty }$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.

v4: 131 pages, minor improvements, typos fixed, references added

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Cited by in corpus (66)

• Double Copy from Homotopy Algebras
• BRST-Lagrangian Double Copy of Yang-Mills Theory
• Kinematic Lie Algebras From Twistor Spaces
• $L_{\infty }$-algebras and the perturbiner expansion
• Loop Amplitudes and Quantum Homotopy Algebras
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Gauge independent kinematic algebra of self-dual Yang-Mills theory
• Mapping between Witten and Lightcone String Field Theories
• Double-Copying Self-Dual Yang-Mills Theory to Self-Dual Gravity on Twistor Space
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• The $L_{\infty }$ structure of gauge theories with matter
• Braided Symmetries in Noncommutative Field Theory
• Towards an M5-Brane Model II: Metric String Structures
• Tree-Level Color-Kinematics Duality Implies Loop-Level Color-Kinematics Duality up to Counterterms
• Double Copy from Tensor Products of Metric BV${}^{■}$-algebras
• Adjusted Parallel Transport for Higher Gauge Theories
• Perspectives on the pure spinor superfield formalism
• Tree-Level Color-Kinematics Duality from Pure Spinor Actions
• Pictures from Super Chern-Simons Theory
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Minimal models of field theories: SDYM and SDGR
• Correlation functions of scalar field theories from homotopy algebras
• Homotopy double copy and the Kawai-Lewellen-Tye relations for the non-abelian and tensor Navier-Stokes equations
• A toy model for background independent string field theory
• Symmetry Factors of Feynman Diagrams and the Homological Perturbation Lemma
• Courant sigma model and $L_{\infty }$-algebras
• Global Double Field Theory is Higher Kaluza-Klein Theory
• On-Shell Action for Type IIB Supergravity and Superstrings on $Ad{S}_5\times S^5$
• Super Chern-Simons Theory: BV-formalism and $A_{\infty }$-algebras
• Towards an M5-Brane Model III: Self-Duality from Additional Trivial Fields
• Higher Gauge Theory
• Higher Gauge Theory and Integrability
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Perturbative path-integral of string field and the $A_{\infty }$ structure of the BV master equation
• Batalin-Vilkovisky quantization of fuzzy field theories
• Field Theory Equivalences as Spans of $L_{\infty }$-algebras
• Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex
• Symplectic structure in open string field theory II: Sliding lump
• Basic curvature and the Atiyah cocycle in gauge theory
• Covariant phase space and $L_{\infty }$ algebras
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• Correlation Functions Involving Dirac Fields from Homotopy Algebras I: The Free Theory
• Correlation functions involving Dirac fields from homotopy algebras II: the interacting theory
• Algebraic Structures In Closed Superstring Field Theory, Homotopy Transfer And Effective Actions
• On the $L_{\infty }$ formulation of Chern-Simons theories
• Duality Anomalies in Linearized Gravity
• Generalised wavefunction coefficients and acyclonesto-cosmohedra
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Non-commutative gauge symmetry from strong homotopy algebras
• Gerbes in Geometry, Field Theory, and Quantisation
• Generalized symmetries as homotopy Lie algebras
• Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes
• Gravity from AKSZ-Manin theories in two, three, and four dimensions
• Full S-matrices and Witten diagrams with (relative) L-infinity algebras
• Homotopy Manin Theories: Generalising Third-Way, Yang-Mills and Integrable Sigma Models
• Explicit Non-Abelian Gerbes with Connections
• Extended Bargmann FDA and non-relativistic gravity
• Ambitwistor Yang-Mills Theory Revisited
• Twisting BFSS & IKKT
• Singular field redefinition between Witten's string field theory and Witten's theory deformed by Ellwood invariant
• The standard cohomology of regular Courant algebroids
• Lagrangian Relations and Quantum $L_{\infty }$ Algebras
• Adjusting Higher Chern-Simons Theory
• Homotopy representations of extended holomorphic symmetry in holomorphic twists
• Extremal Black Holes from Homotopy Algebras