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Derivation of Calabi-Yau Crystals from Chern-Simons Gauge Theory

Takuya Okuda

TL;DR

The paper derives Calabi-Yau crystal models from Chern-Simons theory on $S^3$ and shows, via large $N$ duality, that the crystal reproduces the A-model amplitudes on the resolved conifold with a wall-bound structure encoding the Kähler modulus $t=g_s N$. It establishes a unitary matrix-model formulation of CS and links the open-string picture to a crystal via an open-string slicing, up to a renormalization factor tied to non-compact D-branes. D-branes are incorporated as crystal defects, producing a Kähler modulus shift $ ilde{t}=t+g_s M$, and the resulting amplitudes align with known D-brane physics in the resolved conifold. The work further sketches how these crystal methods could generalize to prove broader large $N$ dualities, including orbifold cases and connections to Nekrasov counting and 5D gauge theories, to all orders in $g_s$.

Abstract

We derive new crystal melting models from Chern-Simons theory on the three-sphere. Via large N duality, these models compute amplitudes for A-model on the resolved conifold. The crystal is bounded by two walls whose distance corresponds to the Kahler modulus of the geometry. An interesting phenomenon is found where the Kahler modulus is shifted by the presence of non-compact D-branes. We also discuss the idea of using the crystal models as means of proving more general large N dualities to all order in $g_s$.

Derivation of Calabi-Yau Crystals from Chern-Simons Gauge Theory

TL;DR

The paper derives Calabi-Yau crystal models from Chern-Simons theory on and shows, via large duality, that the crystal reproduces the A-model amplitudes on the resolved conifold with a wall-bound structure encoding the Kähler modulus . It establishes a unitary matrix-model formulation of CS and links the open-string picture to a crystal via an open-string slicing, up to a renormalization factor tied to non-compact D-branes. D-branes are incorporated as crystal defects, producing a Kähler modulus shift , and the resulting amplitudes align with known D-brane physics in the resolved conifold. The work further sketches how these crystal methods could generalize to prove broader large dualities, including orbifold cases and connections to Nekrasov counting and 5D gauge theories, to all orders in .

Abstract

We derive new crystal melting models from Chern-Simons theory on the three-sphere. Via large N duality, these models compute amplitudes for A-model on the resolved conifold. The crystal is bounded by two walls whose distance corresponds to the Kahler modulus of the geometry. An interesting phenomenon is found where the Kahler modulus is shifted by the presence of non-compact D-branes. We also discuss the idea of using the crystal models as means of proving more general large N dualities to all order in .

Paper Structure

This paper contains 7 sections, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Crystal melting model for the resolved conifold
  3. Large $N$ dual open string theory
  4. Unitary matrix model for Chern-Simons theory
  5. Crystal from Chern-Simons theory
  6. Adding D-branes
  7. More general large $N$ dualities, instanton counting, and geometric engineering

Figures (3)

  • Figure 1: (a) The crystal melting model for the resolved conifold ${\mathcal{O}}(-1)\oplus{\mathcal{O}}(-1)\rightarrow {\bf P}^1$. The edges, drawn as solid lines, of the positive octant bounded by the wall at $y=N$ form the toric diagram of the resolved conifold. (b) Many atoms have been removed from the crystal. Atoms cannot be removed from the region beyond the wall at $y=N$.
  • Figure 2: (a)Closed string slicing: This slicing by planes $x=y+j$ allows one to compute the closed string amplitude as in eq. (\ref{['closedstringslicing']}). (b)Open string slicing: Another slicing of the crystal by planes $x=z+j$ corresponds to the free field representation eq. (\ref{['openstringslicing']}) obtained from Chern-Simons theory.
  • Figure 3: The initial configuration of the crystal with defects representing multiple non-compact D-branes intersecting ${\bf P}^1$ in the resolved conifold. The defects introduce faces at $y=\mu^t_1=N_1-M+1,\mu^t_2=N_2-M+2,...,\mu_{M-1}^t=N_{M-1}-1,\mu_M^t=N_M$.