Background independent algebraic structures in closed string field theory

Abstract

We construct a Batalin-Vilkovisky (BV) algebra on moduli spaces of Riemann surfaces. This algebra is background independent in that it makes no reference to a state space of a conformal field theory. Conformal theories define a homomorphism of this algebra to the BV algebra of string functionals. The construction begins with a graded-commutative free associative algebra C built from the vector space whose elements are orientable subspaces of moduli spaces of punctured Riemann surfaces. The typical element here is a surface with several connected components. The operation Δ of sewing two punctures with a full twist is shown to be an odd, second order derivation that squares to zero. It follows that (C, Δ) is a Batalin- Vilkovisky algebra. We introduce the odd operator δ=∂+ħΔ, where ∂ is the boundary operator. It is seen that δ²=0, and that consistent closed string vertices define a cohomology class of δ. This cohomology class is used to construct a Lie algebra on a quotient space of C. This Lie algebra gives a manifestly background independent description of a subalgebra of the closed string gauge algebra.