[Paper Review] Orientifolds, Mirror Symmetry and Superpotentials

B. S. Acharya, Mina Aganagic+2 more|ArXiv.org|Feb 28, 2002

Black Holes and Theoretical PhysicsPhysics and Astronomy27 references79 citations

TL;DR

This paper establishes a mirror symmetry framework to compute superpotentials in Type IIA and IIB orientifolds of Calabi-Yau threefolds, showing that worldsheet instantons with disc and real projective plane (RP²) topology contribute to the superpotential. The key result is a derivation of the superpotential for the O(-1)+O(-1)→ℙ¹ orientifold, confirming predictions from large N Chern-Simons duality via mirror symmetry and identifying contributions from non-orientable surfaces.

ABSTRACT

We consider orientifolds of Calabi-Yau 3-folds in the context of Type IIA and Type IIB superstrings. We show how mirror symmetry can be used to sum up worldsheet instanton contributions to the superpotential for Type IIA superstrings. The relevant worldsheets have the topology of the disc and ${\bf RP^2}$.

Motivation & Objective

• To extend mirror symmetry to orientifold compactifications of Type IIA and IIB strings on Calabi-Yau threefolds.
• To compute the spacetime superpotential in N=1 supersymmetric orientifold compactifications, including contributions from worldsheet instantons with disc and RP² topology.
• To confirm a prediction from large N Chern-Simons duality for the superpotential in the O(-1)+O(-1)→ℙ¹ orientifold using mirror symmetry.

Proposed method

• Uses mirror symmetry to map Type IIA orientifolds to their Type IIB mirror counterparts, enabling computation of superpotentials via holomorphic 3-form integrals.
• Applies the formula W = ∫H ∧ ω, where H is the RR 3-form field strength, to compute superpotentials from D5-brane and O5-plane sources.
• Identifies contributions to the superpotential from unoriented worldsheet maps to ℙ¹ with RP² topology, corresponding to non-orientable Riemann surfaces with a single crosscap.
• Relies on the fact that for Type IIA orientifolds, the orientifold action IΩ projects D6-branes and their images, leading to disc instanton contributions.
• Uses the topological B-model on the mirror to compute superpotential terms, interpreting them as holomorphic Chern-Simons functionals.
• Compares results with large N Chern-Simons theory predictions, showing agreement in genus-zero amplitudes and identifying contributions from odd and even powers of the string coupling.

Experimental results

Research questions

• RQ1 How can mirror symmetry be used to compute superpotentials in orientifold compactifications of Type IIA and IIB strings on Calabi-Yau threefolds?
• RQ2 What is the role of worldsheet instantons with RP² and disc topology in generating the spacetime superpotential in orientifold backgrounds?
• RQ3 How do the superpotential contributions from non-orientable surfaces (e.g., RP²) relate to large N Chern-Simons duality predictions?
• RQ4 Why do certain orientifold planes (e.g., O5) generate a superpotential while others (e.g., O7) do not, and how does this relate to the holomorphic 3-form transformation under the involution?
• RQ5 Can mirror symmetry be generalized to compute higher-genus amplitudes in orientifold compactifications, particularly those involving both orientable and non-orientable instantons?

Key findings

• The superpotential for the O(-1)+O(-1)→ℙ¹ orientifold is computed via mirror symmetry, and the result matches the prediction from large N Chern-Simons duality.
• The genus-zero contribution from the RP² instanton (single crosscap) is identified as the coefficient of g_s^{-1} in the free energy, corresponding to the first term in the large N expansion.
• The second term in the large N free energy, proportional to ∑ e^{-nt}/n[2sin(ng_s/2)]², is interpreted as arising from oriented maps and corresponds to N=2 amplitudes with RR flux.
• The superpotential vanishes for orientifolds with O7 or O9 planes due to the holomorphic 3-form transforming as ω → ω, while it is non-zero for O3 and O5 planes where ω → -ω.
• The method confirms that only 'half'-orientifolds—those with fixed-point sets topologically R² in the B-model—generate non-vanishing superpotentials in Type IIB compactifications.
• The correspondence between disc and RP² instantons in Type IIA and Euclidean M2-brane instantons in M-theory lifts the construction to G₂-holonomy manifolds with singularities.