[Paper Review] On degree zero semistable bundles over an elliptic curve

Calin Iuliu Lazaroiu|ArXiv.org|Dec 30, 1997

Geometry and complex manifolds13 references3 citations

TL;DR

This paper presents a computational algorithm to test semistability and determine the decomposition type of degree zero holomorphic vector bundles over a nonsingular elliptic curve, relying on explicit knowledge of a basis of sections of an associated twisted bundle. The key contribution is a practical criterion for semistability and decomposition that is applicable in heterotic string compactifications on elliptically fibered Calabi-Yau manifolds.

ABSTRACT

Motivated by the study of heterotic string compactifications on elliptically fibered Calabi-Yau manifolds, we present a procedure for testing semistability and identifying the decomposition type of degree zero holomorphic vector bundles over a nonsingular elliptic curve. The algorithm requires explicit knowledge of a basis of sections of an associated `twisted bundle'.

Motivation & Objective

• To develop a systematic method for testing semistability of degree zero holomorphic vector bundles over elliptic curves.
• To determine the decomposition type of such bundles into direct sums of stable bundles.
• To provide a computational tool applicable in the context of heterotic string compactifications on elliptically fibered Calabi-Yau threefolds.
• To establish a procedure based on explicit section data of a twisted bundle associated to the original bundle.
• To bridge algebraic geometry techniques with physical applications in string theory, particularly in moduli space analysis.

Proposed method

• The method relies on constructing a twisted bundle associated to a given degree zero holomorphic vector bundle on an elliptic curve.
• It requires explicit knowledge of a basis of holomorphic sections of this twisted bundle as input.
• Semistability is tested via a criterion involving the vanishing of certain cohomological obstructions derived from the section data.
• The decomposition type is determined by analyzing the Jordan-Hölder filtration and the structure of the associated graded bundle.
• The approach uses techniques from algebraic geometry, including the theory of moduli spaces of semistable bundles and the Harder-Narasimhan filtration.
• The algorithm is formulated in terms of linear algebra over the base field, leveraging the elliptic curve's group structure and line bundle properties.

Experimental results

Research questions

• RQ1 How can one algorithmically test whether a degree zero holomorphic vector bundle over an elliptic curve is semistable?
• RQ2 What is the decomposition type of a degree zero semistable bundle into stable subbundles?
• RQ3 How can the structure of the section space of a twisted bundle be used to infer properties of the original bundle?
• RQ4 What conditions on the section basis ensure semistability of the original bundle?
• RQ5 In what way does this method facilitate the study of moduli spaces in heterotic string compactifications?

Key findings

• The paper establishes a necessary and sufficient criterion for semistability of degree zero holomorphic vector bundles on elliptic curves based on section data of an associated twisted bundle.
• It provides a constructive method to determine the decomposition of such bundles into direct sums of stable bundles of degree zero.
• The algorithm is effective and computable, relying only on linear algebra over the base field once a section basis is known.
• The method is particularly suited for applications in heterotic compactifications where such bundles arise as gauge bundles on elliptically fibered Calabi-Yau threefolds.
• The results are consistent with known classification results for semistable bundles on elliptic curves, but provide an explicit computational pathway.
• The procedure is invariant under isogenies of the elliptic curve, reflecting the geometric nature of the invariants involved.