The classification of orthogonally rigid $G_2$-local systems and related differential operators

Dettweiler, Michael; Reiter, Stefan

doi:10.1090/s0002-9947-2014-06042-x

Cited by 7 publications

(22 citation statements)

References 19 publications

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“…Remark: For n even the order-n linear differential operatorL n , equivalent to the symmetric (n − 1)-th power of operator L 2 (see (21)), gives decompositions of the formL n = (M · N + 1) · r(x), corresponding to symplectic Galois groups, where M is of order two and N are of even order n − 2. This corresponds to the fact that the differential Galois group of the order-two operator L 2 , namely SL(2, C), is also † a symplectic group SL(2, C) ≃ Sp(2, C).…”

Section: Similar Decompositions For Simple Order-n Operatorsmentioning

confidence: 99%

Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

Boukraa¹,

Hassani²,

Maillard³

et al. 2015

J. Phys. A: Math. Theor.

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We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

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Section: Similar Decompositions For Simple Order-n Operatorsmentioning

confidence: 99%

Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

Boukraa¹,

Hassani²,

Maillard³

et al. 2015

J. Phys. A: Math. Theor.

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“…Here the e i ∈ Z ≥0 are fixed and the constants a i yet unspecified. We looked at many (e 1 , e 3 , e 4 , e 5 , e 6 , e 7 , e 9 ) near-proportional to (1,3,4,5,6,7,9) so as to ensure that D(a 1 t e1 , . .…”

Section: 3mentioning

confidence: 99%

“…Centralizer dimensions are calculated in [6, §3] and the numerics associated with (6.1) are as follows. Dettweiler and Reiter classify tuples of classes in G 2 (C) which are SO 7 (C) rigid in [6]. Thus ([a], [b], [c], [d]) is in their classification.…”

Section: Orthogonal Rigidity Of a Lift Ofmentioning

confidence: 99%

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Division Polynomials with Galois Group $$SU_{3}(3).2\mathop{\cong}G_{2}(2)$$

Roberts

2015

Fields Institute Communications

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We use a rigidity argument to prove the existence of two related degree twenty-eight covers of the projective plane with Galois group SU 3 (3).2 ∼ = G 2 (2). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasablity. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G 2 motives in the classification of Dettweiler and Reiter. We conclude by specializing our two covers to get interesting three-point covers and number fields which would be hard to construct directly.

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“…the globally bounded character) property of the corresponding series [13].Inside this "Geometry" framework [14], the multifold integrals of Ising model seem to be even more "selected". This justifies to explore § these "Special Geometries".Actually, in a previous paper [16,17], and with a learn-by-example approach, we displayed a set of enumerative combinatorics examples corresponding to miscellaneous lattice Green functions [18,19,20,21,22,23,24], as well as Calabi-Yau examples, together with order-seven operators [25,26] associated with differential Galois groups which are exceptional groups. On the irreducible operators of these examples, two differential algebra properties occur simultaneously [16].…”

mentioning

confidence: 99%

The Ising model and special geometries

Boukraa¹,

Hassani²,

Maillard³

2014

J. Phys. A: Math. Theor.

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Abstract. We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals χ (n) of the magnetic susceptibility of the Ising model (n ≤ 6) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries".Beyond the small order factor operators (occurring in the linear differential operators associated with χ (5) and χ (6) ), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group Sp(12, C). The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group SO(21, C).

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The classification of orthogonally rigid $G_2$-local systems and related differential operators

Cited by 7 publications

References 19 publications

Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

Division Polynomials with Galois Group $$SU_{3}(3).2\mathop{\cong}G_{2}(2)$$

The Ising model and special geometries

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