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D-branes as Defects in the Calabi-Yau Crystal

Natalia Saulina, Cumrun Vafa

TL;DR

This work extends the A-model/topological string–melting crystal duality by identifying non-compact Lagrangian D-branes with specific defects in the Calabi–Yau crystal. Using a fermionic operator framework, brane insertions are encoded as defect operators that modify the crystal partition function in a controlled way, yielding exact open-string amplitudes that agree with the topological vertex formalism up to universal renormalization factors, and revealing non-perturbative contributions. The analysis covers branes at zero and general framings, as well as anti-branes, and shows how the melting rules adapt to more complex brane configurations. In the semiclassical limit, the brane-induced defects reproduce genus-zero open-string data via changes in the limiting crystal shape, connecting to the Calabi–Yau mirror geometry and supporting a non-perturbative completion of the open-string sector within the crystal picture.

Abstract

We define the notion of A-model Lagrangian D-branes as introducing defects in the Calabi-Yau crystal. The crystal melting in the presence of these defects reproduces all genus string amplitudes as well as leads to additional non-perturbative terms.

D-branes as Defects in the Calabi-Yau Crystal

TL;DR

This work extends the A-model/topological string–melting crystal duality by identifying non-compact Lagrangian D-branes with specific defects in the Calabi–Yau crystal. Using a fermionic operator framework, brane insertions are encoded as defect operators that modify the crystal partition function in a controlled way, yielding exact open-string amplitudes that agree with the topological vertex formalism up to universal renormalization factors, and revealing non-perturbative contributions. The analysis covers branes at zero and general framings, as well as anti-branes, and shows how the melting rules adapt to more complex brane configurations. In the semiclassical limit, the brane-induced defects reproduce genus-zero open-string data via changes in the limiting crystal shape, connecting to the Calabi–Yau mirror geometry and supporting a non-perturbative completion of the open-string sector within the crystal picture.

Abstract

We define the notion of A-model Lagrangian D-branes as introducing defects in the Calabi-Yau crystal. The crystal melting in the presence of these defects reproduces all genus string amplitudes as well as leads to additional non-perturbative terms.

Paper Structure

This paper contains 9 sections, 84 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Review of the closed topological strings/melting crystal duality
  3. String/Crystal duality with D-branes
  4. D-branes in topological A-model string theory on ${{\bf C}^3}$
  5. D-branes as defects in the melting crystal
  6. Crystal amplitudes for more general D-brane configurations
  7. Anti-D-brane amplitudes
  8. The D-brane framing in crystal picture
  9. D-branes in the crystal picture in the limit $g_s\to 0$

Figures (15)

  • Figure 1: We draw 2d partition $\mu$ in a diagonal slice at $t=-2.$ The numbers of rows are $\mu_1=4,\quad \mu_2=2,\quad \mu_3=1$. The fermion state is $\vert \mu>=\psi^*_{-7/2}\psi_{-5/2}\psi^*_{-1/2}\psi_{-1/2}\vert 0>.$
  • Figure 2: Projection to the base of a D-brane ending on y-axis at y=a.
  • Figure 3: The D-brane with zero framing. Plane $P_D$ contains the projection $D$ of the D-brane to ${\cal O}^+.$
  • Figure 4: The half-lines $L_1$ and $L_2$ enter the definition of the 2-cycle $\Sigma.$ A cross stands for vacancy and dot for atom.
  • Figure 5: The shape of the melting crystal in the presence of D-branes. The arrows carry negative (positive) sign and show the evolution with $\Gamma_{-}$ ($\Gamma_{+}$). The crosses are vacant sites inside the cylinder ${\cal M}_{\nu}.$
  • ...and 10 more figures