TITLE:
Hecke-Langlands Duality and Witten’s Gravitational Moonshine
AUTHORS:
Igor Yu. Potemine
KEYWORDS:
Conformal Field Theory, Modular Invariance, Quantum Gravity, S-Duality, Vertex Algebra
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.7 No.2,
March
31,
2021
ABSTRACT: The purpose of this research is to give a dual description of conformal blocks of d=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related d=2 spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten’s idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of c=24 Monster vertex algebra of Frenkel-Lepowsky- Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references). The main new result in this paper is the construction of number-theoretical lattice vertex superalgebras in Section 5 and applications to conformal field theories and quantum gravity.