Mathematics – Quantum Algebra
Scientific paper
2009-10-19
Comm.Math.Phys.303:331-359, 2011; Commun.Math.Phys.303:331-359,2011
Mathematics
Quantum Algebra
33 pages, minor revisions due to referees' comments, Communications in Mathematical Physics, in press
Scientific paper
10.1007/s00220-011-1206-0
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.
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