On symplectically rigid local systems of rank four and Calabi–Yau operators

Abstract

We classify all Sp 4 (C)-rigid, quasi-unipotent local systems and show that all of them have geometric origin. Furthermore, we investigate which of those having a maximal unipotent element are induced by fourth order Calabi-Yau operators. Via this approach, we reconstruct all known Calabi-Yau operators inducing a Sp 4 (C)-rigid monodromy tuple and obtain closed formulae for special solutions of them. (4, 8, 4) := Λ 2 L a+ 1 4 ⋆ H L 1 4 −a z 1 2 ⋆ H L b+ 3 4 ⋆ H L 3 4 −b z − 3 2 ⋆ H L 3 2 = 64 ϑ 4 + z −128 ϑ 4 − 256 ϑ 3 + ϑ 2 (128(a 2 + b 2 ) − 304) + z ϑ (128(a 2 + b 2 ) − 176) + 48(a 2 + b 2 ) + 256 a 2 b 2 − 39 + 64z 2 (a + 1 + ϑ − b) (a + 1 + ϑ + b) (a − 1 − ϑ − b) (−1 + a − ϑ + b) .

Calabi–Yau manifolds realizing symplectically rigid monodromy tuples

“…, 1)) into the convolution of n + 1 local systems of rank one, each with a holomorphic solution of the type in Equation (3.5) with α = α i . This is a special case of a general classification result by Katz [51] that applies to every linearly rigid local system; see also [12]. In fact, Katz proved that every linearly rigid local system is obtained as tensor products and convolutions of rank-one local systems associated with the holomorphic solution (3.5).…”

“…We know that the elements of monodromy tuples induced by a rank-four Calabi-Yau operator must lie in Sp(4, C). Bogner and Reiter showed that all of the symplectically rigid monodromy tuples of quasi-unipotent elements admit a decomposition into a sequence of middle convolutions and tensor products of Kummer sheaves of rank one [12]. In particular, they are constructible using only tuples of rank-one.…”

“…There is a corresponding notion of the Hadamard product (cf. [12,Def. 4.11]) for the differential operators involved: the Hadamard product of the differential operator L (1) t (α 1 ; ) with the operator L (n) t ((α 2 , .…”

A differential equation with monodromy group 2.J2

“…[5, Section 2] and [2]. The following result is a consequence of the numerology of the middle convolution MC λ (cf.…”

“…Then we end up with a symplectically rigid triple in dimension 4 containing a bi-reflection. This arises from a monodromy triple of a hypergeometric differential equation of order 4 by taking the wedge product and applying a suitable middle convolution as shown in [2,Theorem 3.3]. Thus we are in the linearly rigid (hypergeometric) case.…”