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

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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 -subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the category of VOA with a formal parameter to the category of A -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 β - γ systems, we find a remarkable relation between the Courant algebroid and the homotopy algebra of the Yang-Mills theory.

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Correspondence to Anton M. Zeitlin.

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Communicated by Y. Kawahigashi

To Gregg J. Zuckerman for his 60th birthday.

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Zeitlin, A.M. Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory. Commun. Math. Phys. 303, 331–359 (2011). https://doi.org/10.1007/s00220-011-1206-0

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