D-Branes And Mirror Symmetry

TL;DR

This work develops a comprehensive D-brane framework for N=(2,2) theories on 2D worldsheets with boundaries, linking holomorphic (B-type) branes in sigma-models to Lagrangian (A-type) branes in Landau-Ginzburg mirrors. It establishes a consistent picture of boundary conditions, ground-state indices, and boundary entropies, and shows how brane creation under monodromy encodes R-charge data of UV fixed points. By applying these ideas to N=2 minimal models and their LG descriptions, the authors derive a geometric realization of Verlinde algebra, Cardy data, and period-integral relations, and connect them to exceptional bundles and helices on Fano varieties via mirror symmetry. The paper further develops boundary linear sigma models and demonstrates explicit brane/mirror correspondences in both toric and non-toric examples, including IP^N and toric del Pezzo surfaces, providing a unified geometric language for D-branes in massive and conformal settings. The results illuminate the deep interplay between soliton numbers, monodromy, and algebraic-geometry structures (exceptional collections, mutations, and Verlinde coefficients) within the D-brane/mirror symmetry framework, with implications for local F-theory and non-compact Calabi–Yau geometries.

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.

Figures (49)

• Figure 1: BPS soliton map to straight line in the $W$-plane. Soliton solutions exist for each intersection point of vanishing cycles. Lines in the $W$-plane which are homotopic to the straight line (doted lines) can also be used to calculate soliton numbers.
• Figure 2: As the positions of critical values change in the $W$-plane, the choice of the vanishing cycles relevant for computing the soliton numbers change.
• Figure 3: The cycles emanating from the critical points. The images in the $W$-plane are the straight lines emanating from the critical values and extending to the infinity in the real positive direction.
• Figure 4: The images in the $W$-plane of $\hbox{\large $\gamma$}_a$ and $\hbox{\large $\gamma$}_b$ (left); and $\hbox{\large $\gamma$}_a$ and $\hbox{\large $\gamma$}_b^{\prime}$ (right). The second will give rise to"intersection number". As we will see in the next section, this contains a certain information on D-branes in the LG model.
• Figure 5: The two solitons of the $\hbox{I}\!\hbox{P}^{1}$ model.