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.
Highlights
Number-Theoretical Dualities, S-Dualities and Gravitational CosmologyIn this paper, we discuss an amazing interplay between number-theoretical dualites and black holes in Anti-de Sitter spaces
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
We describe the current state of art and define vertex superalgebras for number fields giving remarkable applications to conformal field theories (CFT) and quantum gravity
Summary
We discuss an amazing interplay between number-theoretical dualites and black holes in Anti-de Sitter spaces. It passes through modular invariance of conformal field theories and AdS/CFT correspondences. It extends to S-dualities in CFT and string theories. In recent years [1] [2], it was realized that the Langlands correspondence is the number-theoretical counterpart of the electromagnetic duality in gauge field theories. Witten [3] has associated, via AdS/CFT correspondence, the pure d = 3 gravity to the Monster Moonshine Module V# It transports the Hecke-Langlands duality for the extremal c = 24 CFT to modular dualities for BTZ black holes on AdS3 We describe the current state of art and define vertex superalgebras for number fields (cf. sect. 5) giving remarkable applications to CFT and quantum gravity
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