# Free quantum fields in 4D and Calabi-Yau spaces

## TL;DR

This paper develops counting formulas for primary fields in 4D free scalar conformal field theory, revealing a connection to Calabi-Yau spaces through permutation orbifolds and holomorphic polynomial functions.

## Contribution

It introduces a duality map linking primary operators to polynomial functions, identifies Calabi-Yau structures in orbifolds, and extends previous primary field constructions.

## Key findings

- Holomorphic primary fields correspond to polynomial functions on permutation orbifolds.
- Permutation orbifolds have palindromic Hilbert series, indicating Calabi-Yau geometry.
- Constructed the top-dimensional holomorphic form consistent with Calabi-Yau properties.

## Abstract

We develop general counting formulae for primary fields in free four dimensional (4D) scalar conformal field theory (CFT). Using a duality map between primary operators in scalar field theory and multi-variable polynomial functions subject to differential constraints, we identify a sector of holomorphic primary fields corresponding to polynomial functions on a class of permutation orbifolds. These orbifolds have palindromic Hilbert series, which indicates they are Calabi-Yau. We construct the top-dimensional holomorphic form expected from the Calabi-Yau property. This sector includes and extends previous constructions of infinite families of primary fields. We sketch the generalization of these results to free 4D vector and matrix CFTs.