Sketch of a Program for Universal Automorphic Functions to Capture Monstrous Moonshine
| Autor: | Frenkel, Igor, Penner, Robert |
|---|---|
| Přispěvatelé: | Institut des Hautes Etudes Scientifiques (IHES), IHES |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
High Energy Physics - Theory dimension: 3 [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Weil-Petersson form FOS: Physical sciences algebra: Lie three dimensional quantum gravity Monstrous Moonshine Maurer-Cartan form Mathematics - Geometric Topology Thompson group T FOS: Mathematics capture Mathematical Physics field theory: conformal universal Teichmüller space [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] 58D05 22E65 11F55 20H10 Geometric Topology (math.GT) Mathematical Physics (math-ph) group: modular boson: yield Differential Geometry (math.DG) High Energy Physics - Theory (hep-th) quantum gravity monster spin: 1 holography loop algebra for sl 2 |
| Popis: | We review and reformulate old and prove new results about the triad $ {\rm PPSL}_2({\mathbb Z})\subseteq{\rm PPSL}_2({\mathbb R})\circlearrowright ppsl_2({\mathbb R}) $, which provides a universal generalization of the classical automorphic triad ${\rm PSL}_2({\mathbb Z})\subseteq{\rm PSL}_2({\mathbb R})\circlearrowright psl_2({\mathbb R})$. The leading P or $p$ in the universal setting stands for $piecewise$, and the group ${\rm PPSL}_2({\mathbb Z})$ plays at once the role of universal modular group, universal mapping class group, Thompson group $T$ and Ptolemy group. We produce a new basis of the Lie algebra $ppsl_2({\mathbb R})$, compute its structure constants, define a central extension which is compared with the Weil-Petersson 2-form, and discuss its representation theory. We construct and study new framed holographic coordinates on the universal Teichm\"uller space and its symmetry group ${\rm PPSL}_2({\mathbb R})$, and construct an invariant 1-form as its Maurer-Cartan form analogous to the invariant Eisenstein 1-form $E_2(z)dz$, which gives rise to the spin 1 representation of $psl_2({\mathbb R})$ extended by the trivial representation. This suggests the full program for developing the theory of universal automorphic functions conjectured to yield the bosonic CFT$_2$. Relaxing the automorphic condition to the commutant leads to our ultimate conjecture on realizing the Monster CFT$_2$ via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichm\"uller and the Monster CFT$_2$ theories to the three-dimensional quantum gravity. Comment: 54 pages, 7 figures; elaboration of proof of Theorem 7.1 in v.2 |
| Databáze: | OpenAIRE |
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