[Paper Review] D-Branes And Mirror Symmetry

Kentaro Hori, Amer Iqbal+1 more|ArXiv.org|May 25, 2000

Black Holes and Theoretical Physics63 references385 citations

TL;DR

This paper establishes a geometric and algebraic framework linking D-branes in N=2 supersymmetric theories to mirror symmetry, demonstrating that holomorphic D-branes on K"ahler manifolds correspond to Lagrangian D-branes in Landau-Ginzburg (LG) mirror theories. It provides a geometric realization of the Verlinde algebra for SU(2) WZW models via D-brane intersection numbers and links soliton numbers in massive LG theories to R-charges via Picard-Lefschetz monodromy.

ABSTRACT

We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of Kähler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.

Motivation & Objective

• To develop a systematic description of D-branes preserving half-supersymmetry in (2,2) supersymmetric field theories with boundaries.
• To establish a mirror correspondence between holomorphic D-branes on K"ahler manifolds and Lagrangian D-branes in Landau-Ginzburg theories.
• To provide a geometric interpretation of the Verlinde algebra and modular S-matrix using D-brane intersection numbers in LG models.
• To clarify the connection between soliton numbers in massive LG theories and R-charges of chiral fields at the UV fixed point.
• To explain the helix structure of exceptional bundles on Fano varieties through mirror symmetry and Picard-Lefschetz theory applied to LG mirrors.

Proposed method

• Analyzes BPS boundary conditions in N=2 supersymmetric sigma models, Landau-Ginzburg (LG) models, and gauged linear sigma models using supersymmetry constraints.
• Applies Picard-Lefschetz theory to compute monodromy and intersection numbers of vanishing cycles in LG theories with non-compact cycles.
• Uses period integrals and boundary entropy to characterize boundary states in LG models, linking them to Ramond charges.
• Derives brane creation mechanisms in massive LG theories via monodromy and R-charge flow under parameter deformation.
• Constructs mirror maps between holomorphic cycles in K"ahler manifolds and Lagrangian submanifolds in LG mirrors, particularly for Fano varieties.
• Applies the LG description of N=2 minimal models to realize the Verlinde algebra geometrically as intersection numbers of D-branes.

Experimental results

Research questions

• RQ1 How do D-branes preserving half of the bulk supercharges arise in (2,2) supersymmetric field theories with boundaries?
• RQ2 What is the precise mirror correspondence between holomorphic D-branes on K"ahler manifolds and Lagrangian D-branes in LG theories?
• RQ3 How is the Verlinde algebra for SU(2) level k realized geometrically via D-brane intersections in LG models?
• RQ4 What is the origin of the link between soliton numbers in massive LG theories and R-charges of chiral fields?
• RQ5 How does Picard-Lefschetz theory on the mirror LG side explain the helix structure of exceptional bundles on Fano varieties?

Key findings

• The Verlinde algebra for the SU(2) WZW model at level k is geometrically realized as the intersection matrix of D-branes in the LG mirror of the minimal model.
• The modular S-matrix of the SU(2) WZW model is directly computed as the transition matrix between D-brane basis states, providing a geometric interpretation of its role in diagonalizing the fusion algebra.
• For Fano varieties, the helix structure of exceptional bundles is explained via Picard-Lefschetz monodromy acting on vanishing cycles in the mirror LG theory.
• Soliton numbers in massive LG theories are computed via intersection matrices of vanishing cycles, with explicit matrices derived for $\mathbb{P}^2$, $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$, and $F_0 = \mathbb{P}^1 \times \mathbb{P}^1$.
• The intersection matrix for $\mathbb{P}^2$ is $\begin{pmatrix} 0 & -3 & 3 \\ 0 & 0 & -3 \\ 0 & 0 & 0 \end{pmatrix}$, matching the Ramond charge structure of the chiral ring.
• Brane creation in massive LG theories is shown to arise from monodromy-induced changes in the cycle structure under parameter flow, linking to R-charge evolution.