Deformations of Closed Strings and Topological Open Membranes

TL;DR

This work develops a deformation-theory framework for topological strings, linking deformations of closed-string algebraic structures to the physics of boundaries of topological open membranes. It identifies three distinct deformation classes for closed strings and provides explicit analysis of the second class via a topological open membrane with a bulk $3$-form, showing how the Lie bracket deforms into higher operations consistent with an $L_\infty$-type structure. The authors connect BRST, product, and bracket deformations to Hochschild cohomology, $A_\infty$/$L_\infty$ formalisms, and the geometry of multi-vector fields, culminating in an effective target-space action with Courant algebroid-like features. The results bridge 2D topological string deformations, 3D topological membrane dynamics, BV quantisation, and geometric structures, suggesting deep links to M-theory objects and generalized gauge theories. Overall, the paper provides a comprehensive algebraic and geometric account of how bulk membranes deform boundary string theories and lays groundwork for future explorations of higher algebroid structures in string/M-theory contexts.

Abstract

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.

Figures (8)

• Figure 1: The correlation functions on the sphere defining the $A_\infty$ structure constants. The first descendants are integrated along the indicated path in path order.
• Figure 2: Factorisation expressing the action of the BRST operator.
• Figure 3: Factorisation for the bracket.
• Figure 4: A typical diagram corresponding to an element of the Hochschild cohomology for a deformation of the associative structure, the three-point function giving the deformation of the BRST operator, and the four-point function giving the deformation of the product.
• Figure 5: A typical diagram corresponding to an element of the Hochschild cohomology for a deformation of the differential Lie structure, the three-point function giving the deformation of the BRST operator, and the four-point function giving the deformation of the bracket.