Membranes and Sheaves

TL;DR

This work proposes a conjectural correspondence between curve-counting in Calabi–Yau 5-folds and 1-dimensional sheaves on fixed-point 3-folds, formulated in equivariant $K$-theory. The central idea is that the 3D DT parameter $q$ becomes a $\,\mathbb{C}^ imes_q$-weight controlling the 5D fixed locus geometry, enabling a framework in which the DT q-expansion is naturally summed by a 5D theory. The authors construct the DT integrand via a modified virtual structure sheaf and a detailed interaction term $\boldsymbol{\Phi}$, analyze membranes through proposed membrane moduli, and connect to PT/DT theory through localization on Chow varieties. They develop the machinery of square roots of virtual canonical bundles, establish refined invariants and κ-symmetries, and introduce index and refined vertices in toric Calabi–Yau settings, culminating in a cohesive program that unifies DT theory, M-theory indices, and refined vertex combinatorics. The work provides both conceptual foundations and concrete computational tools (index vertex, refined vertex) for exploring higher-dimensional DT-type correspondences and their physical interpretations.

Abstract

Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds $Z$ and 1-dimensional sheaves on 3-folds $X$ that are embedded in $Z$ as fixed points of certain $\mathbb{C}^\times$-actions. In both cases, the enumerative information is taken in equivariant $K$-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson-Thomas theories, one sums up over all Euler characteristics with a weight $(-q)^χ$, where $q$ is a parameter, informally referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter $q$ becomes in $5$ dimensions the equivariant parameter for the $\mathbb{C}^\times$-action that defines $X$ inside $Z$. The 5-dimensional theory effectively sums up the $q$-expansion in the Donaldson-Thomas theory. In particular, it gives a natural explanation of the rationality (in $q$) of the DT partition functions. Other expected as well as unexpected symmetries of the DT counts follow naturally from the 5-dimensional perspective. These involve choosing different $\mathbb{C}^\times$-actions on the same $Z$, and thus relating the same 5-dimensional theory to different DT problems. The important special case $Z=X \times \mathbb{C}^2$ is considered in detail in Sections 7 and 8. If $X$ is a toric Calabi-Yau threefold, we compute the theory in terms of a certain index vertex. We show the refined vertex found combinatorially by Iqbal, Kozcaz, and Vafa is a special case of the index vertex.

Figures (5)

• Figure 1: The $\mathbb{C}^\times_q$-fixed locus $X$ may be disconnected and the map $\pi_{\textsf{M2}}$ keeps those components of $\mathbb{C}^\times_q$-invariant curves that lie in $X$ and discards the $\mathbb{C}^\times_q$-orbits that are drawn vertically in the picture.
• Figure 2: The moment map for the fiberwise $(\mathbb{C}^\times)^2$-action offers a schematic representation of $Z_4$. The 3-folds $X_i$ and the divisors $D_s$ are mapped to vertices and edges, respectively. The arrows indicate $q\to 0$ limits.
• Figure 3: A 3-legged 3-dimensional partition
• Figure 4: An $\mathsf{A}$-fixed ideal sheaf on $\mathscr{O}(-1)\oplus\mathscr{O}(-1)\to \mathbb{P}^1$
• Figure 5: The function $\xi_\lambda(\textup{❒})$

Theorems & Definitions (32)

• Conjecture 1
• Proposition 2.1: Zth
• Conjecture 2: Zth
• Definition 1
• Proposition 5.1
• Lemma 6.1
• proof
• Proposition 6.2
• Lemma 6.3
• proof