Membranes and Sheaves

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 $(-q)^\chi$, 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.