The Operator Rings of Topological Symmetric Orbifolds and their Large N Limit

TL;DR

This paper analyzes the structure constants of topological symmetric orbifold theories in the large N limit, revealing how interactions evolve from simple contractions to complex joinings and genus one contributions.

Contribution

It provides a detailed computation of structure constants up to third order in large N, encompassing a broad class of Frobenius algebras including K3 cohomology.

Findings

Leading order dominated by topological metric contractions

First order involves single cycle joining

Third order includes genus one contributions

Abstract

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise.