[Paper Review] Integration Theory for Zero Sets of Polyfold Fredholm Sections

Helmut Hofer, Kris Wysocki+1 more|ArXiv.org|Nov 6, 2007

Finite Group Theory Research7 references14 citations

TL;DR

This paper develops a rigorous integration theory for zero sets of polyfold Fredholm sections—objects that are typically not manifolds due to singularities and symmetries—by introducing 'branched integration' that accounts for local branching structures and symmetries. The key contribution is a Stokes' theorem for this integration, valid even on non-manifold solution spaces, enabling the definition of invariants in symplectic field theory and related areas.

ABSTRACT

In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory.

Motivation & Objective

• To establish a rigorous integration theory for zero sets of polyfold Fredholm sections, which are generally not manifolds due to singularities and symmetries.
• To define a notion of integration over isomorphism classes of objects in a category, capturing the intrinsic geometry of solution spaces despite their pathological structure.
• To extend classical integration and Stokes’ theorem to spaces with branching and corner structures arising in polyfold theory.
• To provide a foundation for defining invariants in symplectic field theory and related geometric theories where transversality fails generically.

Proposed method

• Introduces the concept of 'branched integration' by defining canonical σ-algebras on zero sets of polyfold Fredholm sections, enabling integration of differential forms over non-manifold spaces.
• Uses sc-smooth partitions of unity and the Poincaré lemma for sc-differential forms to construct local primitives of closed forms on level-1 and level-0 sets.
• Applies averaging techniques in ep-groupoids to construct invariant forms and extend local solutions globally, preserving symmetry structures.
• Employs splicing cores and the degeneracy index d to classify points by their local model (interior, boundary, corner), enabling local analysis of branching behavior.
• Constructs a pullback of differential forms via the natural projection r: V×E → K to reduce the problem to a model space where standard analysis applies.
• Proves that for a closed sc-differential k-form ω on a neighborhood of a smooth point, there exists a local (k−1)-form τ such that dτ = ω on the level-1 set, using integration over t∈[0,1] in the form τ(y)(v₁,…,vₖ₋₁) = ∫₀¹ tᵏ⁻¹ ω(ty)(y,v₁,…,vₖ₋₁) dt.

Experimental results

Research questions

• RQ1 How can one define integration over zero sets of polyfold Fredholm sections when these sets are not manifolds due to singularities and symmetries?
• RQ2 Can Stokes’ theorem be generalized to hold for differential forms on non-manifold solution spaces with branching and corner structures?
• RQ3 What is the role of groupoid symmetries and local isomorphisms in constructing a consistent integration theory for such solution sets?
• RQ4 How can one construct local primitives of closed differential forms on spaces with degeneracy (e.g., boundary and corner points) in the polyfold setting?

Key findings

• A canonical σ-algebra is constructed on the zero set of a polyfold Fredholm section, enabling the definition of integration for differential forms over non-manifold solution spaces.
• For any closed sc-differential k-form ω defined near a smooth point in a polyfold, there exists a local (k−1)-form τ such that dτ = ω holds on the level-1 set, as proven via integration over t∈[0,1] in a parametrized form.
• The Poincaré lemma holds for sc-differential forms on splicing cores, allowing local exactness of closed forms and forming the basis for the construction of primitives.
• In the ep-groupoid setting, the existence of a saturated open neighborhood U and a (k−1)-form τ with dτ = ω on U¹ is established by averaging and lifting forms from the model space.
• The theory supports a version of Stokes’ theorem on branched integration, valid even when the underlying space is not a manifold.
• The framework allows for integration over solution spaces that are not transverse or smooth, making it suitable for applications in symplectic field theory and Gromov–Witten theory.