BV Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of (W) Strings

Abstract

Given a simple, simply laced, complex Lie algebra g corresponding to the Lie group G, let n+ be the subalgebra generated by the positive roots. In this Letter we construct a BV algebra BV[g] whose underlying graded commutative algebra is given by the cohomology, with respect to n+, of the algebra of regular functions on G with values in Λ(n+\g). We conjecture that BV[g] describes the algebra of all physical (i.e., BRST invariant) operators of the noncritical W[g] string. The conjecture is verified in the two explicitly known cases, g = sl2 (the Virasoro string) and g = sl3 (the W3 string). Show more