Differential Forms on Loop Spaces

• Differential forms on loop spaces are a framework that unites classical Cartan calculus with rational homotopy theory and smooth geometric structures.
• The homotopy Cartan calculus extends Lie derivatives and contractions to free loop spaces, interconnecting André–Quillen cohomology with Hochschild homology.
• The Γ₁ map provides a geometric realization of algebraic operations, linking Sullivan’s isomorphism to explicit homology classes in loop spaces.

Differential forms on loop spaces are a fundamental concept at the intersection of algebraic topology, smooth (and diffeological) geometry, and rational homotopy theory. The classical Cartan calculus—comprising the Lie derivative, contraction with vector fields, and the de Rham differential—admits a sophisticated extension to the setting of free loop spaces, encapsulating geometric, algebraic, and homotopy-theoretic structures. Recent developments have revealed deep ties between the Cartan calculus, Andrè–Quillen cohomology, Hochschild homology, and the rational homotopy of self-homotopy equivalences, culminating in geometric refinements of Sullivan’s isomorphism via the “Γ₁ map” of Félix and Thomas (Kuribayashi et al., 2022). This article details these connections, organizing the subject around foundational constructions, homotopy Cartan calculus, the geometric–algebraic interface, and the pivotal role of evaluation maps on loop spaces.

1. Cartan Calculus: Classical and Homotopy Extensions

The classical Cartan calculus involves operations such as the Lie derivative and contraction, acting on the de Rham complex of a manifold. For a manifold $M$, these operators satisfy fundamental identities: for vector fields $X,Y$,

$$[L_X, \iota_Y] = \iota_{[X,Y]}, \qquad [d,\iota_X] = L_X.$$

The “homotopy Cartan calculus” generalizes this structure to complexes and algebras appearing in homotopical algebra. Fiorenza and Kowalzig, in particular, formulated a version in the context of commutative differential graded algebras (CDGAs), where operators like the Lie derivative and contraction are realized via algebraic or Hochschild cohomological operations (Kuribayashi et al., 2022).

In the setting of the free loop space $LM$ of a simply connected manifold $M$, the “second stage” Cartan calculus refers to endomorphisms acting on the derivation ring of the de Rham complex, or more generally on the Hochschild complex of a CDGA model of $M$. These operators inherit homotopical and combinatorial structure, reflecting invariants of both the underlying space and its function space counterparts.

2. Algebraic Models: Hochschild and Andrè–Quillen Cohomology

For a commutative differential graded algebra $A$ modeling a topological space $M$ (for example, a Sullivan minimal model $\mathcal{M}_M$ for $M$), its Hochschild homology $HH_*(A)$ captures information akin to differential forms on the free loop space $LM$. Andrè–Quillen cohomology $H^*_{AQ}(A)$, meanwhile, provides a receptacle for infinitesimal deformations (measured by the derivation Lie algebra Der$(A)$).

The connection between the two is mediated by the homotopy Cartan calculus. Specifically, operators defined in terms of the Andrè–Quillen complex act naturally on the Hochschild homology, and important algebraic identifications—such as Sullivan’s isomorphism—relate the rational homotopy groups of the monoid of self-homotopy equivalences of $M$ to $H^*_{AQ}(A)$, realized as the Lie algebra homology of Der$(A)$.

3. Geometric Realization: The Γ₁ Map of Félix and Thomas

An essential innovation is the geometric realization of the algebraic Cartan calculus via the “Γ₁ map,” constructed by Félix and Thomas. This map bridges the algebraic and geometric worlds by producing explicit elements in the homology of $LM$ corresponding to algebraic derivations.

Given a simply connected, closed manifold $M$ of dimension $m$, consider the based loop space component $\Omega M_0$ and the monoid $\mathrm{Aut}_1(M)$ of self-homotopy equivalences. The map

$g(\gamma, x)(t) = \gamma(t)(x)$

for $\gamma \in \Omega M_0$, $x \in M$, and $t \in S^1$, gives a “loop construction” from $\Omega M_0 \times M$ to $LM$.

For $n \geq 1$, the Γ₁ map is given by

$$\Gamma_1 \colon \pi_n(\Omega M_0) \otimes \mathbb{R} \to H_{n+m}(LM; \mathbb{R}),$$

sending a homotopy class, via the Hurewicz map, cross product with the fundamental class $[M]$, and pushforward along $g$, to a concrete homology class in $LM$. This image coincides with the first Hodge component $H_{(1)}(LM)$ in the Hodge decomposition.

The significance of this construction is that it “lifts” Sullivan’s isomorphism—establishing an isomorphism between rational homotopy groups of self-homotopy equivalences and Andrè–Quillen cohomology—to a statement about tangible, geometric cycles in the loop space.

4. Commutative Diagrams: Relating Algebraic and Geometric Calculi

The paper establishes commutative diagrams elucidating how geometric and algebraic Cartan calculi interact. At one level, the algebraic Cartan calculus is defined via Lie derivatives and contractions on the Sullivan model; at another, a corresponding calculus is defined “fiberwise” on the free loop space using the “loop construction,” integration along fibers, and evaluation maps.

Of particular importance is Theorem 5.12, presenting the commutative diagram: $$\begin{array}{ccc} \pi_n(M) \otimes \mathbb{R} & \xrightarrow{\Phi} & H_n(\operatorname{Der}(\mathcal{M}_M)) \ {\scriptstyle \delta} \downarrow & & \downarrow {\scriptstyle \mathrm{PD}^{-1}} \ H_{n+m}^{(1)}(LM) & \xrightarrow{\mathrm{PD}^{-1}} & HH_{(1)}^{-n+1}(\mathcal{M}_M) \end{array}$$ where $\Phi$ is Sullivan’s isomorphism, $\delta$ arises from the adjunction of self-homotopy equivalences, and $\mathrm{PD}^{-1}$ is (model-level) Poincaré duality. The commutativity demonstrates that derivations in the cohomological model correspond to explicit geometric classes in $LM$ via the $\Gamma_1$ map.

5. Implications: From Rational Homotopy to Loop Cohomology

The geometric interpretation of Sullivan’s isomorphism via the $\Gamma_1$ map has far-reaching implications:

• It confirms that operations at the algebraic level (e.g., derivations in the minimal model) have direct geometric counterparts in the topology of loop spaces.
• The image of the $\Gamma_1$ map identifies the first Hodge summand in the homology of $LM$, providing a concrete realization of abstract invariants.
• The commutative diagrammatic framework ensures that natural operations—such as Lie derivatives and contractions—respect both the algebraic structure (expressed in Andrè–Quillen/homotopy calculi) and the geometric structure (cycles and evaluations in loop spaces).

This perspective underpins more advanced structures in string topology and noncommutative geometry, where the interaction of operations on loop (and mapping) spaces with algebraic deformation theory is central.

6. Extensions and Broader Context

The approaches discussed here extend naturally to broader settings, including:

• Free loop spaces of non-simply connected manifolds and related mapping spaces.
• Stronger functoriality with respect to group actions, as in the construction of equivariant cohomology on loop spaces (Ungheretti, 2016, Kuribayashi, 2023).
• Models for iterated loop spaces, exploiting higher operations and their representations through “higher” Cartan calculi and iterated integral approaches.
• Cohomological operations on diffeological loop spaces, leveraging generalizations of differential forms and Mayer-Vietoris techniques (Iwase et al., 2015, Iwase, 2022).

The fusion of algebraic and geometric Cartan calculi, realized concretely via the $\Gamma_1$ map and loop space evaluation, substantiates foundational links between rational homotopy theory, derived algebraic geometry, and advanced string-topological operations.