Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex

Abstract

We conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov-Witten theory of Calabi-Yau fivefolds with a torus action. We prove the conjecture for skeletal, locally anti-diagonal torus actions by establishing a vertex formalism evaluating the Gromov-Witten invariants via the topological vertex of Aganagic, Klemm, Marino and Vafa. We apply the formalism in several examples.

Figures (4)

• Figure 1: Illustration of (A) the embedding of the $\mathsf{T}_{\mathrm{A}}$-diagram in $\mathrm{Lie} \mathsf{T}_{\mathrm{A},\bR} \cong \bR^3$ and (B) the $\mathsf{T}_{\mathrm{B}}$-diagram in $\mathrm{Lie} \mathsf{T}_{\mathrm{B},\bR} \cong \bR^2$. The blue, green and orange lines indicate the three torus preserved coordinate lines of $\mathbb{C}$. The circle highlights the choice of a distinct direction at each vertex as will be introduced in \ref{['sec: sign and order']}.
• Figure 2: The $\mathsf{T}$-diagram of $\mathop{\mathrm{Tot}}\nolimits \cO(-1,0) \oplus \cO(-1,0) \oplus \cO(0,-2)$ with a choice of distinct direction at each vertex and half-edges decorated by partitions. The half-edges associated with a line bundle are coloured blue, green and orange respectively. All vertical lines on the left should be parallel. The tilt of the coloured half-edges is solely for display purposes.
• Figure 3: The $\mathsf{T}$-diagram of $\mathop{\mathrm{Tot}}\nolimits \cO_{\bP^2}(-1)^{\oplus 3}$ with a choice of a distinct direction at each vertex and half-edges decorated by partitions.
• Figure 4: The $\mathsf{T}$-diagram of $\mathop{\mathrm{Tot}}\nolimits \cO_{\bP^3}(-2)^{\oplus 2}$ with a choice of a distinct direction at each vertex and half-edges decorated by partitions.

Theorems & Definitions (44)

• Conjecture A
• Theorem B
• Theorem C
• Definition 1.1
• Example 1.2
• Example 1.3
• Example 1.4
• Definition 1.5
• Example 1.6
• Theorem 1.7