Deformations of closed strings and topological open membranes

We study deformations of topological closed strings. A well-known example is the perturba- tion of a topological closed string by itself, where the associative OPE product is deformed, and which isgoverned by the WD VV equations. Our main in terest will be closed strings that arise as the boundary theory for topological open mem branes, where the boundary string is deformed b y the bulk mem brane operators. The main example is the topological open mem brane theory with a nonzero 3-form field in the bulk. In this case the Lie brac k et of the current algebra is deformed, leading in general to a correction of the Jacobi identity. W e identify these deformations in terms of deformation theory. To thisendwe describe the deformation of the algebraic structure of the closed string, giv en by the BRST operator, the associative product and the Lie bracket. Quite remark ably, we find that there are three classes of deformations for the closed string, t wo ofwhich are exemplified b y the WD VV theory and the topological open mem brane. The third class remains largely m ysterious, as we have no explicit example