New percolation crossing formulas and second-order modular forms

We consider the three crossing probability densities for percolation recently found via conformal field theory [23]. We prove that all three of them (i) may be simply expressed in terms of Cardy’s [4] and Watts’ [24] 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 [19]) 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