  We study deformations of topological closed strings. A well-known example is
the perturbation of a topological closed string by itself, where the
associative OPE product is deformed, and which is governed by the WDVV
equations. Our main interest will be closed strings that arise as the boundary
theory for topological open membranes, where the boundary string is deformed by
the bulk membrane operators. The main example is the topological open membrane
theory with a nonzero 3-form field in the bulk. In this case the Lie bracket of
the current algebra is deformed, leading in general to a correction of the
Jacobi identity. We identify these deformations in terms of deformation theory.
To this end we describe the deformation of the algebraic structure of the
closed string, given by the BRST operator, the associative product and the Lie
bracket. Quite remarkably, we find that there are three classes of deformations
for the closed string, two of which are exemplified by the WDVV theory and the
topological open membrane. The third class remains largely mysterious, as we
have no explicit example.
