[Paper Review] D-branes on Stringy Calabi-Yau Manifolds

Duiliu-Emanuel Diaconescu, Michael R. Douglas|ArXiv.org|Jun 28, 2000

Black Holes and Theoretical Physics36 references61 citations

TL;DR

This paper establishes a direct correspondence between rational B-type boundary states in Gepner models and coherent sheaves on Calabi–Yau manifolds via the generalized McKay correspondence, using fractional branes in the Landau–Ginzburg orbifold phase of the linear sigma model. It computes K-theory classes of these branes in the large volume limit without mirror symmetry, confirming results for the weighted projective space $\mathbb{P}^{1,1,2,2,2}$ hypersurface.

ABSTRACT

We argue that D-branes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau-Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their K-theory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi-Yau hypersurface in $\WP^{1,1,2,2,2}$.

Motivation & Objective

• To identify rational B boundary states in Gepner models as coherent sheaves in the large volume limit of Calabi–Yau manifolds.
• To derive K-theory classes of these branes using the generalized McKay correspondence in the Landau–Ginzburg orbifold phase.
• To provide a systematic method for computing tautological line bundles and K-theory generators on the ambient toric variety.
• To verify the correspondence against mirror symmetry results for the $\mathbb{P}^{1,1,2,2,2}$ Calabi–Yau hypersurface.
• To establish a framework for relating quiver gauge theories of fractional branes to mathematical constructions of stable holomorphic bundles.

Proposed method

• Realize Gepner model boundary states as fractional branes in the Landau–Ginzburg orbifold phase of the linear sigma model.
• Apply the generalized McKay correspondence to map fractional branes to coherent sheaves on the resolved orbifold.
• Use the K-theory intersection pairing on the ambient toric variety to determine the basis of K-theory generators $S_k$.
• Restrict the $S_k$ classes to the Calabi–Yau hypersurface to obtain the K-theory classes of the Gepner model boundary states.
• Construct a toric algorithm to compute tautological line bundles $R_k$ from the quiver diagram.
• Verify the results by comparing the computed K-theory classes with those obtained via mirror symmetry for the $\mathbb{P}^{1,1,2,2,2}$ example.

Experimental results

Research questions

• RQ1 How can rational B boundary states in Gepner models be systematically mapped to coherent sheaves on Calabi–Yau manifolds in the large volume limit?
• RQ2 What is the role of the generalized McKay correspondence in identifying fractional branes with holomorphic bundles in the linear sigma model framework?
• RQ3 How do the K-theory classes of D-branes in the Landau–Ginzburg phase relate to those in the geometric phase without invoking mirror symmetry?
• RQ4 Why does working with unphysical $S_k$ classes on the ambient toric variety yield correct physical results on the Calabi–Yau hypersurface?
• RQ5 Can the quiver gauge theory of fractional branes be directly linked to mathematical constructions of stable coherent sheaves, as in Beilinson’s theorem?

Key findings

• The paper successfully identifies rational B boundary states in Gepner models as coherent sheaves in the large volume limit via the generalized McKay correspondence.
• The K-theory classes of these branes are computed directly in the Landau–Ginzburg phase using the intersection pairing on the ambient toric variety, without requiring mirror symmetry.
• For the $\mathbb{P}^{1,1,2,2,2}$ Calabi–Yau hypersurface, the derived K-theory classes match the results obtained through mirror symmetry.
• The construction reveals that the intersection form on the ambient toric variety differs from that on the Calabi–Yau hypersurface, yet the restriction process yields correct physical results.
• A systematic toric algorithm is developed to compute tautological line bundles $R_k$ from the quiver diagram, enabling explicit computation of K-theory generators.
• The framework suggests a deep correspondence between moduli spaces of ${\cal N}=1$ quiver gauge theories and moduli spaces of stable coherent sheaves on Calabi–Yau manifolds.