New percolation crossing formulas and second-order modular forms

TL;DR

This paper proves that three new percolation crossing probabilities can be expressed through known formulas, are second-order modular forms, and are uniquely determined by their transformation properties, revealing a deep mathematical structure.

Contribution

It establishes that the new crossing probabilities are second-order modular forms and links them to existing formulas, providing a unified mathematical framework.

Findings

All three crossing probabilities are expressible via Cardy's and Watts' formulas.

They are weakly holomorphic second-order modular forms on a0Gamma(2).

The probabilities are uniquely determined by their modular transformation laws.

Abstract

We consider the three new crossing probabilities for percolation recently found via conformal field theory by Simmons, Kleban and Ziff. 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 $\Gamma(2)$, and (iii) under some technical assumptions (similar to those used by Kleban and Zagier, 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.