[Paper Review] Diffeological Dirac operators and diffeological gluing

Ekaterina Pervova|arXiv (Cornell University)|Jan 24, 2017

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper introduces a diffeological framework for defining Dirac operators on non-smooth spaces by generalizing key geometric structures—such as pseudo-bundles, pseudo-metrics, Clifford modules, and connections—using diffeology. It establishes a gluing procedure for combining Dirac operators on diffeological spaces, showing that the result remains a Dirac operator under specific compatibility conditions, thus extending Atiyah-Singer theory to singular and non-manifold settings.

ABSTRACT

This manuscript attempts to present a way in which the classical construction of the Dirac operator can be carried over to the setting of diffeology. A more specific aim is to describe a procedure for gluing together two usual Dirac operators and to explain in what sense the result is again a Dirac operator. Since versions of cut-and-paste (surgery) operations have already appeared in the context of Atiyah-Singer theory, we specify that our gluing procedure is designed to lead to spaces that are not smooth manifolds in any ordinary sense, and since much attention has been paid in recent years to Dirac operators on spaces with singularities, we also specify that our approach is more of a piecewise-linear nature (although, hopefully, singular spaces in a more analytic sense will enter the picture sooner or later; but this work is not yet about them). Most of it is devoted to the diffeological versions of the components that go into the standard definition of a Dirac operator as the composition of a Clifford connection with Clifford action by sections of the cotangent bundle; a diffeological Dirac operator is then standardly defined.

Motivation & Objective

• To extend the classical theory of Dirac operators to diffeological spaces that are not smooth manifolds, particularly those arising from singular or piecewise-linear constructions.
• To define a diffeological analogue of the Dirac operator using pseudo-bundles, pseudo-metrics, and Clifford connections, replacing standard Riemannian and tangent bundle structures.
• To formalize a gluing procedure for combining two Dirac operators on diffeological spaces, ensuring the resulting object remains a valid Dirac operator under compatibility conditions.
• To explore the behavior of geometric structures—like connections, metrics, and differential forms—under this gluing operation, especially in non-smooth settings.
• To lay foundational groundwork for future applications in analytic geometry and mathematical physics by identifying open problems and limitations in the current framework.

Proposed method

• Define a diffeological pseudo-bundle as a generalization of a vector bundle, equipped with a pseudo-metric to replace the Riemannian metric.
• Introduce the pseudo-bundle of differential 1-forms, $\Lambda^1(X)$, and its dual $ (\Lambda^1(X))^* $, to replace the cotangent and tangent bundles in the classical setting.
• Construct a diffeological Clifford algebra bundle using the pseudo-metric on $ \Lambda^1(X) $, enabling the definition of Clifford actions.
• Define a diffeological connection as a generalization of the standard connection, leading to Levi-Civita and Clifford connections in the diffeological context.
• Formally define the diffeological Dirac operator as the composition $ D = c \circ \nabla^E $, where $ c $ is the Clifford action and $ \nabla^E $ is a compatible connection on a Clifford module pseudo-bundle.
• Implement a gluing procedure via quotient diffeology on the underlying set, assigning a standard quotient diffeology to the resulting space, and analyze how geometric structures transform under this operation.

Experimental results

Research questions

• RQ1 Under what conditions does the gluing of two diffeological spaces via a diffeomorphism preserve the structure of a Dirac operator?
• RQ2 Which diffeological vector pseudo-bundles admit pseudo-metrics, and can such metrics be constructed via gluing operations?
• RQ3 Do the spaces of sections of pseudo-bundles and the space of differential 1-forms $ \Omega^1(X) $ satisfy the necessary extendability conditions under gluing?
• RQ4 Are Levi-Civita and Clifford connections on diffeological spaces compatible with the gluing procedure, and when are they uniquely determined?
• RQ5 What is the relationship between compatibility of pseudo-metrics and compatibility of induced connections on glued spaces?

Key findings

• The diffeological Dirac operator is defined as the composition of a Clifford action and a compatible connection on a pseudo-bundle of Clifford modules, generalizing the classical construction.
• The gluing procedure preserves the Dirac operator structure when the gluing map satisfies certain compatibility conditions, such as preserving the pseudo-metric and connection data.
• The resulting diffeology on the glued space is a quotient diffeology, which is weak but sufficient for the formal framework, though potentially too weak for future analytic applications.
• The paper identifies that the existence of partition of unity, pseudo-metrics, and connections on diffeological spaces remains an open problem, with no general existence theorems established.
• The behavior of sections and differential forms under gluing depends on the condition $ \mathcal{D}_1^\Omega = \mathcal{D}_2^\Omega $, which is not generally satisfied and requires additional assumptions.
• Several open questions remain, particularly regarding the existence of pseudo-metrics and connections on general diffeological spaces, and the compatibility of induced geometric structures after gluing.