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Algebraic characterization of differential operators of Calabi-Yau type

Michael Bogner

TL;DR

The article provides a rigorous algebraic framework to characterize Picard-Fuchs operators of Calabi-Yau families with a maximally unipotent monodromy point, introducing CY-type differential operators defined by self-duality, integer exponents, and N-integral arithmetics. It shows how the Poincaré pairing and MUM structure constrain the differential Galois group to a small list (including SL2, Sp, SO, and G2 in a rare 7th-order case) and develops a local normal form and q-coordinate that yield strong invariants (Y-invariants) for classification. The SL2- and G2-cases illustrate how CY-type operators arise from symmetric powers and exceptional monodromy, respectively, and highlight a bridge between geometric periods and arithmetic properties via G-function behavior and Lambert-series expansions. Overall, the work provides tools to identify, construct, and distinguish CY-type Picard-Fuchs operators and connects their algebraic structure to monodromy and arithmetic features relevant to mirror symmetry and enumerative geometry.

Abstract

We give an algebraic characterization of Picard-Fuchs operators attached to families of Calabi-Yau manifolds with a point of maximally unipotent monodromy and discuss possibilities for their differential Galois groups.

Algebraic characterization of differential operators of Calabi-Yau type

TL;DR

The article provides a rigorous algebraic framework to characterize Picard-Fuchs operators of Calabi-Yau families with a maximally unipotent monodromy point, introducing CY-type differential operators defined by self-duality, integer exponents, and N-integral arithmetics. It shows how the Poincaré pairing and MUM structure constrain the differential Galois group to a small list (including SL2, Sp, SO, and G2 in a rare 7th-order case) and develops a local normal form and q-coordinate that yield strong invariants (Y-invariants) for classification. The SL2- and G2-cases illustrate how CY-type operators arise from symmetric powers and exceptional monodromy, respectively, and highlight a bridge between geometric periods and arithmetic properties via G-function behavior and Lambert-series expansions. Overall, the work provides tools to identify, construct, and distinguish CY-type Picard-Fuchs operators and connects their algebraic structure to monodromy and arithmetic features relevant to mirror symmetry and enumerative geometry.

Abstract

We give an algebraic characterization of Picard-Fuchs operators attached to families of Calabi-Yau manifolds with a point of maximally unipotent monodromy and discuss possibilities for their differential Galois groups.

Paper Structure

This paper contains 14 sections, 23 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Some basics
  3. Geometric setting
  4. Differential setting
  5. Algebraic characterization
  6. The Poincaré pairing
  7. Exponents at the MUM point
  8. Solutions at the MUM point
  9. The local normal form at the MUM point
  10. Differential operators of CY-type
  11. Differential Galois groups of CY-type operators
  12. General possibilities
  13. The $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{C})$-case
  14. The $G_2(\mathbb{C})$-case

Key Result

Lemma 3.3

Let $B=\{e,\dots,\partial^{n}e\}$ and $B^{\vee}=\left\{e^{\vee},\dots,\left(\partial^{n}e\right)^{\vee}\right\}$ the basis of the dual differential module $M_L^{\vee}$ which is dual to $B$. Then $\left(M_L^{\vee}, \left(\partial^{n}e\right)^{\vee}\right)$ is a marked differential module with minimal

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • proof
  • Corollary 3.6
  • ...and 43 more