Holomorphic potentials for graded D-branes

TL;DR

The paper develops a holomorphic potential framework for the moduli of graded D-branes in Calabi–Yau type II compactifications, extending the ungraded brane superpotentials via an A∞/L∞-structured string field theory on a graded Chern–Simons-type action. It demonstrates that the low-energy moduli problem is equivalent to a Maurer–Cartan deformation problem governed by a tree-level holomorphic potential W defined on the harmonic sector H^1_d(H), and it proves acyclicity for unit relative-grade brane pairs under certain flat-isomorphism conditions. The authors instantiate the construction with topological D-brane pairs on T^3, obtaining explicit descriptions of the moduli space including an acyclic branch and a branch parameterized by a two-form, and they show how W encodes obstructions and effective symmetries via an A∞-structure. These results lay a rigorous groundwork for open-string deformation theory in graded D-brane systems and point toward an enhanced derived-category framework that preserves string-field-theoretic data.

Abstract

We discuss gauge-fixing, propagators and effective potentials for topological A-brane composites in Calabi-Yau compactifications. This allows for the construction of a holomorphic potential describing the low-energy dynamics of such systems, which generalizes the superpotentials known from the ungraded case. Upon using results of homotopy algebra, we show that the string field and low energy descriptions of the moduli space agree, and that the deformations of such backgrounds are described by a certain extended version of `off-shell Massey products' associated with flat graded superbundles. As examples, we consider a class of graded D-brane pairs of unit relative grade. Upon computing the holomorphic potential, we study their moduli space of composites. In particular, we give a general proof that such pairs can form acyclic condensates, and, for a particular case, show that another branch of their moduli space describes condensation of a two-form.

Figures (7)

• Figure 1: The bundle structure of harmonic states.
• Figure 2: Virtual dimensions of the various strata.
• Figure 3: Massive, harmonic and Goldstone modes. After gauge fixing and integrating out the massive modes, one obtains a potential $W$ for the harmonic modes, whose critical set characterizes the true moduli.
• Figure 4: Physical vertices and propagator for the perturbative expansion of $W$. Dashed lines represent massless modes. Beyond tree level, the perturbation expansion also involves vertices and propagators for ghost and antighosts, which are not shown in the figure.
• Figure 5: Action of $G$ along the generic (i.e. ${\bf C}^2$) fibers of ${\cal Z}_1$. The vector ${\bf g}=(G_1,G_2,G_3)$ is defined by the components $G_i$ of the one-form $g_1=G_idx^i$.