[Paper Review] Extending Mirror Conjecture to Calabi-Yau with Bundles

Cumrun Vafa|ArXiv.org|Apr 20, 1998

Geometry and complex manifoldsMathematics51 citations

TL;DR

This paper extends mirror symmetry to Calabi-Yau manifolds equipped with stable vector bundles by identifying the mirror as a supersymmetric Lagrangian cycle in the mirror manifold, relating the variation of Hodge structure for bundle cohomologies to holomorphic maps with boundaries on the mirror. The key result is a generalized mirror map linking open string amplitudes in the B-model to disk instantons in the A-model, with explicit correspondence between Chern classes and cycle homology classes.

ABSTRACT

We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi-Yau manifolds in terms of supersymmetric cycles on the mirror.

Motivation & Objective

• To generalize mirror symmetry to include vector bundles on Calabi-Yau manifolds.
• To establish a correspondence between stable bundles on a Calabi-Yau threefold and supersymmetric Lagrangian cycles in its mirror.
• To relate the moduli space of stable bundles to the moduli space of supersymmetric cycles via Hodge theory.
• To derive a generalized mirror map connecting open string B-model amplitudes with A-model disk instantons.
• To provide a physical and mathematical framework for studying bundles via mirror-supersymmetric cycles.

Proposed method

• Define the mirror of a Calabi-Yau manifold with a stable U(N) bundle as a supersymmetric n-cycle C in the mirror manifold M, where n is the complex dimension of the Calabi-Yau.
• Identify the homology class of C with the Chern classes of the bundle via a map from H^{k,k}(M) to H_n(W), including c_0 through c_n.
• Use T-duality on T^n fibers to motivate the duality between D-branes wrapping the bundle and Lagrangian cycles in the mirror.
• Construct a correspondence between the Chern-Simons action on the mirror cycle C and the B-model action on the bundle, with instanton corrections from holomorphic maps with boundary on C.
• Derive a generalized mirror formula equating the triple intersection of (0,1)-forms on the bundle side to the generating function of disk maps with boundary in C and insertions at marked points.
• Include multi-wrapped instanton corrections and Wilson line insertions to account for moduli dependence and enumerative invariants.

Experimental results

Research questions

• RQ1 How can mirror symmetry be extended to include vector bundles on Calabi-Yau manifolds?
• RQ2 What is the mirror dual of a stable vector bundle on a Calabi-Yau threefold in terms of D-branes and supersymmetric cycles?
• RQ3 How do the Chern classes of a bundle relate to the homology class of its mirror cycle in the mirror manifold?
• RQ4 What is the physical and mathematical correspondence between open string amplitudes in the B-model and A-model disk instantons with boundary on the mirror cycle?
• RQ5 How does the variation of Hodge structure for bundle cohomologies map to enumerative invariants of holomorphic maps with boundary?

Key findings

• The mirror of a stable U(N) bundle on a Calabi-Yau threefold is a supersymmetric Lagrangian n-cycle C in the mirror manifold M, with its homology class determined by the Chern classes of the bundle.
• The moduli space of stable bundles with fixed Chern classes is isomorphic to the complex moduli space of the mirror cycle C, both of dimension H^1(C).
• The generalized mirror map equates the triple intersection of (0,1)-forms on the bundle side to the generating function of holomorphic disk maps from Riemann surfaces with boundary mapped to C, weighted by exponentials of area and Wilson lines.
• The left-hand side of the generalized mirror formula involves derivatives of the holomorphic connection A and the holomorphic 3-form Ω, representing the variation of Hodge structure in the open string B-model.
• The right-hand side includes instanton corrections from holomorphic maps with boundary in C, labeled by homology classes in H_1(C) and H_2(M), with marked points mapped to Poincaré dual 2-cycles in C.
• The correspondence holds even for non-trivial cycles, with the classical limit (no instantons) recovering the triple intersection number on C, and higher-order terms accounting for multi-wrapped disks and quantum corrections.