# Structure Constants of a Single Trace Operator and Determinant Operators from Hexagon

## TL;DR

This paper conjectures a hexagon form factor approach to compute structure constants involving single trace and determinant operators in ${
m extbf{N}=4}$ super Yang-Mills theory, validated by tree-level calculations.

## Contribution

It introduces a novel hexagon-based method for calculating structure constants involving open and closed strings in holography, extending previous techniques.

## Key findings

- Conjecture that structure constants are expressed via hexagon form factors.
- Validation of the conjecture through explicit tree-level computations.
- Matching results support the proposed hexagon approach.

## Abstract

We study the structure constant of a single trace operator and two determinant operators in ${\cal N}=4$ super Yang-Mills theory. Holographically such a quantity corresponds to the interaction vertex between a closed string and two open strings attached to the spherical $D$-branes. Relying on diagrammatic intuition, we conjecture that the structure constant at the finite coupling is nicely written by the hexagon form factors. Precisely we need to prepare two hexagon twist operators and appropriately glue edges together by integrating mirror particles contributions and by contracting boundary states. The gluing generates the worldsheet for a closed string and two open strings attached to the $D$-branes. At the weak coupling, the asymptotic expression simply reduces to sum over all possible partitions not only for the edge related to the closed string but also for the edges representing the half of the open string together with reflection effects for the opposite open string edges. We test the conjecture by directly computing various tree level structure constants. The result is nicely matched with our conjecture.