Free Quantum Fields in 4D and Calabi-Yau Spaces

Area of Science:

• Theoretical Physics
• High Energy Physics
• Mathematical Physics

Background:

• Conformal Field Theories (CFTs) are crucial in understanding critical phenomena and quantum gravity.
• Counting primary fields in CFTs is essential for characterizing these theories.
• Previous work has identified some infinite families of primary fields.

Purpose of the Study:

• To develop general counting formulas for primary fields in free four-dimensional (4D) scalar CFTs.
• To identify a specific sector of holomorphic primary fields.
• To explore connections between CFTs and Calabi-Yau manifolds.

Main Methods:

• Utilizing a duality map between primary operators and polynomial functions.
• Analyzing multivariable polynomial functions with differential constraints.
• Investigating permutation orbifolds with palindromic Hilbert series.

Main Results:

• Derived general counting formulas for primary fields in 4D scalar CFTs.
• Identified a sector of holomorphic primary fields linked to Calabi-Yau orbifolds.
• Constructed the unique top-dimensional holomorphic form for these Calabi-Yau orbifolds.
• Extended existing infinite families of primary fields.

Conclusions:

• The study establishes a clear connection between primary fields in 4D scalar CFTs and Calabi-Yau orbifolds.
• The developed formulas provide a systematic way to count and construct these fields.
• The findings suggest potential generalizations to other types of CFTs, such as vector and matrix CFTs.