[Paper Review] A GENERALIZATION OF WITTEN'S CONJECTURE RELATING DONALDSON AND SEIBERG-WITTEN INVARIANTS

Adrian Vajiac|arXiv (Cornell University)|Mar 23, 2000

Homotopy and Cohomology in Algebraic Topology36 references3 citations

TL;DR

This paper generalizes Witten's conjecture linking Donaldson and Seiberg–Witten invariants to manifolds of non-simple type using equivariant localization on the Donaldson–Witten and Seiberg–Witten moduli spaces. It provides a formal derivation via an infinite-dimensional version of the abelian localization theorem, bypassing non-abelian monopole theory and offering a direct geometric approach to the conjecture's extension.

ABSTRACT

Abstract. We generalize Witten’s conjectured formula relating Donaldson and Seiberg– Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson–Witten and Seiberg–Witten moduli spaces. We give a formal derivation of Witten’s conjecture and its generalization, making use of an infinite dimensional version of the abelian localization theorem. 1.

Motivation & Objective

• To extend Witten’s conjectured relation between Donaldson and Seiberg–Witten invariants beyond simple-type 4-manifolds.
• To provide a formal derivation of the generalized conjecture without relying on the theory of non-abelian monopoles.
• To establish a direct geometric framework using moduli spaces of Donaldson–Witten and Seiberg–Witten theories.
• To apply infinite-dimensional abelian localization techniques to non-simple-type manifolds.
• To unify the treatment of invariants across different types of 4-manifolds through equivariant methods.

Proposed method

• Utilizes equivariant localization techniques on the infinite-dimensional moduli spaces of Donaldson–Witten and Seiberg–Witten theories.
• Applies an infinite-dimensional version of the abelian localization theorem to relate invariants across different theories.
• Works directly on the moduli spaces without invoking auxiliary structures like non-abelian monopoles.
• Derives the generalized formula through formal manipulations in the context of equivariant cohomology.
• Relies on the structure of the gauge-theoretic moduli spaces and their equivariant localization properties.
• Treats the conjecture as a formal identity in the context of infinite-dimensional geometry and localization.

Experimental results

Research questions

• RQ1 How can Witten’s conjecture relating Donaldson and Seiberg–Witten invariants be extended to 4-manifolds of non-simple type?
• RQ2 What role does equivariant localization play in deriving the generalized conjecture without non-abelian monopole theory?
• RQ3 Can the infinite-dimensional abelian localization theorem be applied meaningfully to gauge-theoretic moduli spaces in this context?
• RQ4 What is the geometric and cohomological structure underlying the generalized relation between the two invariants?
• RQ5 How does the formal derivation via localization compare to existing physical and mathematical approaches?

Key findings

• The paper provides a formal derivation of Witten’s conjecture and its generalization using equivariant localization on the moduli spaces of Donaldson–Witten and Seiberg–Witten theories.
• The approach avoids the use of non-abelian monopole theory, offering a direct geometric route to the generalized conjecture.
• An infinite-dimensional version of the abelian localization theorem is applied to establish the relation between the invariants.
• The method applies uniformly to manifolds of non-simple type, extending the scope of the original conjecture.
• The derivation is formal but consistent with the known structure of the invariants and their transformation properties.
• The results suggest a deeper geometric unification of Donaldson and Seiberg–Witten invariants through localization techniques.