Rigidity via Modular Properties of Theta Functions

2025/05/05 by Indraneel Tambe, Tambe, Indraneel
Computer Science · Mathematics · #Advanced Mathematical Identities #Advanced Mathematical Theories #Computability, Logic, AI Algorithms #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2505.02792

openalex publication_date 2025/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we use methods of Liu to show that the twisted Dirac operators D on certain bundles Φ considered by Guan and Wang are rigid. To do so, we use a Lefschetz formula and Atiyah-Bott localization to obtain formulas for the Lefschetz numbers L of these operators D in terms of Jacobi theta functions; then, using the translational and modular transformation properties of theta functions and the properties of their zeros, we prove L is constant provided certain conditions on characteristic classes hold, thus showing the rigidity of D on Φ under these conditions.

Citations

• SL(2,Z) Modular Forms and Anomaly Cancellation Formulas II
• On the Witten Rigidity Theorem for Odd Dimensional Manifolds
• Rigidity theorems for SpinC-manifolds
• On Family Rigidity Theorems I
• On elliptic genera and theta-functions
• The index of the dirac operator in loop space

Related

• E₈ Bundles and Rigidity
• On Family Rigidity Theorems II
• Rigidity of differential operators and Chern numbers of singular varieties
• Dirac operators twisted by ramified Euclidean line bundles
• Local Fourier transforms and rigidity for D-modules