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Motives meet SymPy: studying $λ$-ring expressions in Python

Daniel Sanchez, David Alfaya, Jaime Pizarroso

TL;DR

The paper tackles the challenge of manipulating motivic expressions in the Grothendieck ring by introducing motives, a general-purpose Python package integrated with SymPy that implements λ-ring calculus via Adams operations. It presents a robust simplification workflow that rewrites expressions as polynomials in Adams operations and provides a modular architecture with Curves, Groups, and moduli motives. The authors demonstrate significant performance gains and succeed in verifying Mozgovoy's conjectural formula for the motive of twisted Higgs bundle moduli spaces up to genus $g\le 18$ and rank $r\le 3$, illustrating the practical impact of scalable symbolic tools in algebraic geometry. The work lays a foundation for broader motivic computations and future extensions to additional moduli spaces and λ-ring structures.

Abstract

We present a new Python package called "motives", a symbolic manipulation package based on SymPy capable of handling and simplifying motivic expressions in the Grothendieck ring of Chow motives and other types of $λ$-rings. The package is able to manipulate and compare arbitrary expressions in $λ$-rings and, in particular, it contains explicit tools for manipulating motives of several types of commonly used moduli schemes and moduli stacks of decorated bundles on curves. We have applied this new tool to advance in the verification of Mozgovoy's conjectural formula for the motive of the moduli space of twisted Higgs bundles, proving that it holds in rank 2 and 3 for any curve of genus up to 18 and any twisting bundle of small degree.

Motives meet SymPy: studying $λ$-ring expressions in Python

TL;DR

The paper tackles the challenge of manipulating motivic expressions in the Grothendieck ring by introducing motives, a general-purpose Python package integrated with SymPy that implements λ-ring calculus via Adams operations. It presents a robust simplification workflow that rewrites expressions as polynomials in Adams operations and provides a modular architecture with Curves, Groups, and moduli motives. The authors demonstrate significant performance gains and succeed in verifying Mozgovoy's conjectural formula for the motive of twisted Higgs bundle moduli spaces up to genus and rank , illustrating the practical impact of scalable symbolic tools in algebraic geometry. The work lays a foundation for broader motivic computations and future extensions to additional moduli spaces and λ-ring structures.

Abstract

We present a new Python package called "motives", a symbolic manipulation package based on SymPy capable of handling and simplifying motivic expressions in the Grothendieck ring of Chow motives and other types of -rings. The package is able to manipulate and compare arbitrary expressions in -rings and, in particular, it contains explicit tools for manipulating motives of several types of commonly used moduli schemes and moduli stacks of decorated bundles on curves. We have applied this new tool to advance in the verification of Mozgovoy's conjectural formula for the motive of the moduli space of twisted Higgs bundles, proving that it holds in rank 2 and 3 for any curve of genus up to 18 and any twisting bundle of small degree.
Paper Structure (17 sections, 8 theorems, 65 equations, 2 figures)

This paper contains 17 sections, 8 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. lambda-rings
  3. lambda-ring structures in the Grothendieck ring of Chow motives
  4. The motives package
  5. The simplification algorithm
  6. Core classes in the package
  7. Integration with
  8. Base objects
  9. Structure of the main lambda-rings implemented in the motives package
  10. Base lambda-ring expressions
  11. Grothendieck motives
  12. Curves
  13. Groups
  14. Moduli schemes and moduli stacks
  15. Usage example
  16. ...and 2 more sections

Key Result

Theorem 1.1

Mozgovoy's conjectural formula for the motive of the moduli spaces of $L$-twisted Higgs bundles holds in the Grothendieck ring of Chow motives in rank at most 3 for any smooth complex projective curve of genus $g$ such that $2\le g \le 18$ and any line bundle $L$ on the curve such that $2g-1\le \deg

Figures (2)

  • Figure 1: Size comparison between the polynomials generated for $M_{g,3,20}^{\operatorname{BB}}=M_{g,3,20}^{\operatorname{ADHM}}$ after simplification for different $g$, all taken with $r=3$ and $p=20$.
  • Figure 2: Time comparison between algorithm Alf22 and package for the computational verification of Mozgovoy's conjectural formula. Alf22 data is limited to $g\le 11$ because the program reached the memory limit for the machine for that $g$.

Theorems & Definitions (15)

  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 5 more