D-brane effective action and tachyon condensation in topological minimal models

TL;DR

This paper develops a comprehensive framework for D-brane moduli and tachyon condensation in B-type topological minimal models and their massive deformations by leveraging the Landau-Ginzburg realization and Kontsevich–Orlov category theory. It proposes a closed-form expression for the open-string effective potential ${\mathcal W}_{\rm eff}$ as a generating function of tree-level amplitudes, linking it to matrix factorizations, and shows that the D-brane moduli space locally coincides with the critical locus of ${\mathcal W}_{\rm eff}$; this provides a concrete handle on obstruction theory in open strings. A central result is that any D-brane is isomorphic to a direct sum of minimal branes, yielding a stratified moduli space whose strata encode different minimal-brane content and tachyon-condensation pathways, including explicit examples that realize algebro-geometric moduli and flows between composites. The effective potential admits a holomorphic (complex-matrix-model) interpretation, with brane positions encoded by matrix eigenvalues and ${\mathcal W}_{\rm eff}$ as a classical holomorphic action; this connects open-string dynamics to holomorphic matrix models and clarifies how D0-brane constituents organize inside branes. Overall, the work provides a tractable, exact handle on topological D-brane dynamics in minimal models, offering a foundation for extending to more general Landau-Ginzburg setups and exploring stability and gravity couplings in topological contexts.

Abstract

We study D-brane moduli spaces and tachyon condensation in B-type topological minimal models and their massive deformations. We show that any B-type brane is isomorphic with a direct sum of `minimal' branes, and that its moduli space is stratified according to the type of such decompositions. Using the Landau-Ginzburg formulation, we propose a closed formula for the effective deformation potential, defined as the generating function of tree-level open string amplitudes in the presence of D-branes. This provides a direct link to the categorical description, and can be formulated in terms of holomorphic matrix models. We also check that the critical locus of this potential reproduces the D-branes' moduli space as expected from general considerations. Using these tools, we perform a detailed analysis of a few examples, for which we obtain a complete algebro-geometric description of moduli spaces and strata.

Figures (3)

• Figure 1: Schematic depiction of the 'total' joint deformation space ${\cal Z}^{(\ell+1)}$. Each branch is associated with a minimal brane, and carries a ${\Bbb C}^*$ bundle corresponding to the constant $C$ in (\ref{['minimal_gen']}).
• Figure 2: Schematic description of the moduli space for the composite of ${\mathbb M}_0$ and ${\mathbb M}_{\ell-1}$.
• Figure 3: Realization of the processes ${\mathbb M}_1\oplus {\mathbb M}_1\longrightarrow {\mathbb M}_0\oplus {\mathbb M}_2$ and ${\mathbb M}_1\oplus {\mathbb M}_1\longrightarrow {\mathbb M}_{-1}\oplus {\mathbb M}_3$ in the moduli space.