Sketch of a Program for Automorphic Functions from Universal Teichmüller Theory to Capture Monstrous Moonshine.

We review and reformulate old and prove new results about the triad , which provides a universal generalization of the classical automorphic triad. The leading P or p in the universal setting stands for piecewise, and the group PPSL 2 (Z) plays at once the role of universal modular group, universal mapping class group, Thompson group T and Ptolemy group. In particular, we construct and study new framed holographic coordinates on the universal Teichmüller space and its symmetry group PPSL 2 (R) , the group of piecewise PSL 2 (R) homeomorphisms of the circle with finitely many pieces, which is dense in the group of orientation-preserving homeomorphisms of the circle. We produce a new basis of its Lie algebra p p s l 2 (R) and compute the structure constants of the Lie bracket in this basis. We define a central extension of p p s l 2 (R) and compare it with the Weil-Petersson form. Finally, we construct a PPSL 2 (Z) -invariant 1-form on the universal Teichmüller space formally as the Maurer-Cartan form of p p s l 2 (R) , which suggests the full program for developing the theory of automorphic functions for the universal triad which is analogous, as much as possible, to the classical triad. In the last section we discuss the representation theory of the Lie algebra p p s l 2 (R) and then pursue the universal analogy for the invariant 1-form E 2 (z) d z , which gives rise to the spin 1 representation of p s l 2 (R) extended by the trivial representation. We conjecture that the corresponding automorphic representation of p p s l 2 (R) yields the bosonic CFT 2 . Relaxing the automorphic condition from PSL 2 (Z) to its commutant allows the increase of the space of 1-forms six-fold additively in the classical case and twelve-fold multiplicatively in our universal case. This leads to our ultimate conjecture that we can realize the Monster CFT 2 via the automorphic representation for the universal triad. This conjecture is also bolstered by the links of both the universal Teichmüller and the Monster CFT 2 theories to the three-dimensional quantum gravity. [ABSTRACT FROM AUTHOR]