Counting and construction of holomorphic primary fields in free CFT4 from rings of functions on Calabi-Yau orbifolds

• Published in: The journal of high energy physics Vol. 2017; no. 8; pp. 1 - 48
• Main Authors: de Mello Koch, Robert, Rabambi, Phumudzo, Rabe, Randle, Ramgoolam, Sanjaye
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2017; Springer Nature B.V; SpringerOpen
• Subjects: AdS-CFT Correspondence; Classical and Quantum Gravitation; Conformal Field Theory; Counting; Elementary Particles; Field theory; Functions (mathematics); High energy physics; Mathematical analysis; Matrix algebra; Matrix methods; Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Scalars; Space-Time Symmetries; String Theory; AdS-CFT Correspondence; Conformal Field Theory; Space-Time Symmetries
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP08(2017)077

A bstract Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.