Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings

Abstract

We give a systematic derivation of the consistency conditions which constrain open-closed disk amplitudes of topological strings. They include the A-infinity relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulk-boundary crossing symmetry and Cardy constraint. We discuss integrability of amplitudes with respect to bulk and boundary deformations, and write down the analogs of WDVV equations for the space-time superpotential. We also study the structure of these equations from a string field theory point of view. As an application, we determine the effective superpotential for certain families of D-branes in B-twisted topological minimal models, as a function of both closed and open string moduli. This provides an exact description of tachyon condensation in such models, which allows one to determine the truncation of the open string spectrum in a simple manner.

Figures (5)

• Figure 1: Boundary and bulk insertions for disk amplitudes. (a) Three boundary fields $\psi_{a_0}$, $\psi_{a_1}$ and $\psi_{a_m}$ are at fixed positions, the others are integrated in a path ordered way between $\psi_{a_1}$ and $\psi_{a_m}$. (b) One bulk and one boundary field are fixed. In both cases additional bulk operators may be present, which are integrated over the whole disk.
• Figure 2: The integration domain $\mathbb{S}_5(\tau_2,\tau_5)$ and its boundary components (through a magnifying glass) for the correlation function $\bigl\langle \psi_a(\tau_0) \psi_b(\tau_1)~ P\!\int\!\psi_c^{(1)}(\tau_2)\!\int\!\psi_d^{(1)}(\tau_3)~ \psi_e(\tau_4)\bigr\rangle$.
• Figure 3: The integration domain $\mathbb{S}_3(\tau_1)$ and its boundary components (through a magnifying glass) for the correlation function $\bigl\langle \phi_i(w,\bar{w})~ \psi_a(\tau_1)~ P\!\int\!\psi_b(\tau_2)\!\int\!\psi_c(\tau_3) \bigr\rangle$. The real line, as boundary of the disk, was compactified to a circle by identifying $\tau_1^+$ and $\tau_1^-$.
• Figure 4: The two contributions to the factorizations leading to equation (\ref{['eq:linpert']}). Figure (a) shows a summand of the first term in this equation, while Figure (b) shows a summand of the second term.
• Figure 5: The factorization associated with the stringy version of the second bulk-boundary crossing constraint. Configuration $(A)$ corresponds to the topological bulk product and $(B)$ to the factorization at the boundary. Configuration $(C)$ connects these channels. The quantity $b$ is the equal distance of the bulk fields from the boundary, while $t$ is the horizontal separation of the bulk fields.