Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory

Abstract

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes", i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the $A_{\infty }$-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of $A_{\infty }$-algebras. The associated generalized Maurer-Cartan equation gives the analogue of the Yang-Mills equation for a wide class of VOAs. Applying this construction to an example of VOA generated by $\beta$-$\gamma$ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.

33 pages, minor revisions due to referees' comments, Communications in Mathematical Physics, in press

References in corpus (16)

• Double Field Theory
• The gauge algebra of double field theory and Courant brackets
• Two-Dimensional Models With (0,2) Supersymmetry: Perturbative Aspects
• Non-geometric Backgrounds and the First Order String Sigma Model
• Conformal Field Theory and Algebraic Structure of Gauge Theory
• Homotopy Lie Superalgebra in Yang-Mills Theory
• Formal Maurer-Cartan Structures: from CFT to Classical Field Equations
• BV Yang-Mills as a Homotopy Chern-Simons via SFT
• Perturbed Beta-Gamma Systems and Complex Geometry
• BRST, Generalized Maurer-Cartan Equations and CFT
• SFT-inspired Algebraic Structures in Gauge Theories
• Homotopy Batalin-Vilkovisky algebras
• Vertex Algebroids II
• Lagrangian approach to sheaves of vertex algebras
• Beta-gamma systems and the deformations of the BRST operator
• The BV double of a Courant algebroid

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• Homotopy Algebras of Differential (Super)forms in Three and Four Dimensions
• A-infinity structure in string theory and the Yang-Mills equation
• Homotopy Relations for Topological VOA
• On higher order Leibniz identities in TCFT
• Sigma-models and Homotopy Algebras