Communications in Number Theory and Physics
Volume 3 (2009)
New percolation crossing formulas and second-order modular forms
Pages: 677 – 696
We consider the three crossing probability densities for percolation recently found via conformal field theory . We prove that all three of them (i) may be simply expressed in terms of Cardy’s  and Watts’  crossing probabilities, (ii) are (weakly holomorphic) second-order modular forms of weight 0 (and a single particular type) on the congruence group Γ(2), and (iii) under some technical assumptions (similar to those used in ) are completely determined by their transformation laws.
The only physical input in (iii) is Cardy’s crossing formula, which suggests an unknown connection between all crossing-type formulas.