D-Brane Superpotentials in Calabi-Yau Orientifolds

TL;DR

Problem: compute tree-level D-brane superpotentials in Type IIB Calabi-Yau orientifolds. Approach: extend the Aspinwall–Katz geometric framework to orientifolds by constructing a graded orientifold functor on the D-brane derived category and developing a cochain model that yields a minimal $L_\infty$ description for invariant deformations; show the orientifold superpotential $W^+$ equals the restriction $W|_{H^+}$ of the unprojected potential. Contributions: (i) a precise categorical construction of orientifold projections in $D^b(X)$ with parity data and $P^2 \simeq \mathrm{Id}$ via an isomorphism $J$; (ii) a practical scheme using $ abla\mathcal{C}(P(\mathfrak{E}),\mathfrak{E})$ to compute $W^+$; (iii) concrete computations for obstructed curves in O5 and for the local conifold in O3/O7, including $SO/Sp$ gauge data and anomaly consistency. Significance: provides a systematic, algebraic method to obtain orientifolded D-brane superpotentials and clarifies the role of $SO/Sp$ projections and obstruction structure in string vacua with D-branes.

Abstract

We develop computational tools for the tree-level superpotential of B-branes in Calabi-Yau orientifolds. Our method is based on a systematic implementation of the orientifold projection in the geometric approach of Aspinwall and Katz. In the process we lay down some ground rules for orientifold projections in the derived category.

Theorems & Definitions (11)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3