Generalized Mcshane’s identity via Fock–Goncharov A modul space and triple ratios
Abstract: (Joint work with Yi Huang) Goncharov and Shen introduced a Landau-Ginzberg potential on the Fock-Goncharov A_{G,S} moduli space, where G is a semisimple Lie group and S is a ciliated surface. They used the potential to formulate a mirror symmetry via Geometric Satake Correspondence. This potential is the markoff equation for A_{ PSL(2,R), S_{1,1} }. When S=S_{g,m}, such potential can be written as a sum of rank G*m partial potentials. We obtain a family of generalized Mcshane’s identities by splitting these partial potentials for A_{PSL(n,R),S_{g,m}} by certain pattern of cluster transformations with geometric meaning. We also find some interesting new phenomena in higher rank case, like triple ratio is bounded in mapping class group orbit. As applications, we find a generalized collar lemma which involves lambda1/lambda2 length spectral, discreteness of that spectral etc. In further research, we would like to ask how can we integrate to obtain the generalized Mirzakhani’s topological recursion with Wn constraint?