New percolation crossing formulas and second-order modular forms

Abstract

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.

Rectangular amplitudes, conformal blocks, and applications to loop models

“…This equation is simply stating the associativity of the fusion product of boundary fields, the so-called sewing constraint coming from crossing symmetry of correlators [28]. It has been shown in [34] that its solution is given by (13). Then, since we defined directly our boundary states in terms of physical boundary OPEs, consistency under modular inversion does not give any additional constraint.…”

“…The action of the modular group on a rectangle has been discussed in[13,27] in the case of percolation.…”

Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms

“…The range of covered topics includes combinatorics, geometry, representation theory, and string theory [3,49]. Higher order modular forms, first introduced with this name by Chinta, Diamantis, and O'Sullivan [18] and Kleban and Zagier [37], have served as a handle on the distribution of modular symbols [52,53] and their connection to the ABC-conjecture [32], and made appearance in conformal field theory [22,37]. Recently, iterated Eichler-Shimura integrals have received a modular interpretation analogous to the one of mock modular forms [10], and at the same time were equipped with a motivic-geometric interpretation [2,11].…”

Iterated Integrals and Higher Order Invariants

“…Modular forms of higher order have been studied extensively in recent years [4,5,[8][9][10][11][12][13][14]18]. To construct them, one often uses iterated integrals.…”