The Categorical Count

Enumerative geometry counts curves — how many rational curves of degree d pass through the right number of points, how many holomorphic maps from a surface to a Calabi-Yau threefold satisfy given constraints. String theory provides one framework for organizing these counts: closed strings sweep out worldsheets, and the partition function encodes the enumerative invariants.

Ulmer extends this from closed to open-closed string field theory using Calabi-Yau A∞-categories as the input data.

The mathematical structure is layered. Graph complexes — combinatorial objects encoding how surfaces can degenerate — connect to the algebraic structure of Calabi-Yau categories through Kontsevich’s cocycle construction. A formality L∞-morphism relates the algebraic structures built from a Calabi-Yau category to those built from a single object within it. This morphism is the bridge: it extends categorical enumerative invariants (which count closed curves) to the open-closed setting (which counts curves with boundary on a specified object).

The physical interpretation: closed strings propagate freely; open strings end on branes. The brane is the object in the category. The open-closed partition function counts both kinds of curves simultaneously, and the categorical framework generates it from the same input that produces the closed invariants alone.

The goal beyond the current paper — a categorical formulation of Twisted Holography at the partition function level — would connect the enumerative geometry of Calabi-Yau categories to large-N gauge theory. The mathematics generates physics from algebra.