Differential algebra on lattice Green functions and Calabi–Yau operators

Abstract

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups. We show that these irreducible operators are not only globally nilpotent, but are such that they are homomorphic to their (formal) adjoints. Considering these operators, or, sometimes, equivalent operators, we show that they are also such that, either their symmetric square or their exterior square, have a rational solution. This is a general result: an irreducible linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group. We thus define the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many Derived From Geometry n-fold integrals ("Periods") occurring in physics, are seen to be diagonals of rational functions, we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, systematically, associated with irreducible factors homomorphic to their adjoint. -adjoint operators, homomorphism or equivalence of differential operators, Special Geometry, globally bounded series, diagonal of rational functions. † Sorbonne Universités (previously the UPMC was in Paris Universitas). ¶ Their corresponding linear differential operators are necessarily globally nilpotent [8]. ‡ One shows that there are no rational solutions of symmetric powers in degree 2, 3,4,6,8,9,12, using an algorithm in M. van Hoeij et al. [9] ♯ This operator is actually homomorphic to its adjoint (see below) with non-trivial order-two intertwiners. † This operator has an irregular singularity at infinity. At x = ∞ the solutions behave like: t · (1 + 77/72 t 2 + · · · ), and exp(−2/t)/t · (1 + 13/36 t + · · · ) where t = ±1/ √ x. † In maple the exterior power is normalised to be a monic operator (the head polynomial is normalised to 1). § The intertwiner T is given by the command Homomorphisms(L,L) of the DEtools package in Maple [33]. ♯ Note that the constraint on the order rules out the "tautological" intertwining relation, satisfied by any operator, like L · adjoint(T ) = T · adjoint(L) with T = L.¶ It is easy to show, in the case of an homomorphism of an operator L with its adjoint, that the intertwiner on the right-hand-side of (8) is necessarily equal to the adjoint of the intertwiner on the left-hand-side. Actually, from the equivalence L · T = S · adjoint(L), taking adjoint on both sides gives adjoint(T ) · adjoint(L) = L · adjoint(S). For irreducible L, the intertwiner is unique, so S = adjoint(T ). ¶ Corresponding to a change of variable: F (x) → F (x/(1 − 18 x))/(1 − 18 x).

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…One easily verifies that this is the case for the previous modular form example where W (x) reads (93), as well as for all the other modular forms emerging in physics or enumerative combinatorics we mentioned in previous papers [29,30,31,35,37].…”

“…The fact that being the symmetric cube of an underlying order-two operator verifies automatically the new condition (34) emerging from a compatibility condition of an order-four linear differential operator by pullback is far less obvious. The "parametrization" (35) necessarily yields the Calabi-Yau condition (32) and the new condition (34), and, conversely, (32) and (34) can be parametrized by (35). Our large calculations thus show that the pullback-compatibility of an order-four linear differential operator which verifies the Calabi-Yau condition (32), amounts to saying that this order-four linear differential operator reduces to (the symmetric cube of) an underlying order-two linear differential operator.…”

“…along with the existence of n F n−1 analogues of the previous relation. Such remarkable identities correspond to modular forms that emerged in the analysis of multiple integrals related to the square Ising model [29,30,31,35] or in other enumerative combinatorics context [37]. They can be seen as a simple occurence of infinite order † † covariance symmetries in physics [2] or enumerative combinatorics.…”

“…One finds straightforwardly that the coefficients given by (35) verify the Calabi-Yau condition (32), as well as the new condition (34). In this case the differential Galois group is no longer the symplectic differential Galois group Sp(4, C), but actually reduces ‡ to the differential Galois group of the underlying order-two linear differential operator, namely SL(2, C).…”

“…Let us consider the condition corresponding to imposing the symmetric square of L 3 to be of order five instead of the generic order six. This ("symmetric" Calabi-Yau [35]) † The D 3 x coefficient is normalized to 1.…”