[Paper Review] Derived Differentiable Manifolds

Kai Behrend, Hsuan-Yi Liao+1 more|arXiv (Cornell University)|Jun 2, 2020

Homotopy and Cohomology in Algebraic Topology58 references5 citations

TL;DR

This paper develops derived differential geometry using bundles of curved $L_{ u}[1]$-algebras, establishing derived manifolds as a category of fibrant objects to enable homotopy fibered products and derived intersections. It constructs a finite-dimensional factorization of the diagonal via path spaces and homotopy transfer, proving the inverse function theorem and clarifying the relationship between weak equivalences and quasi-isomorphisms in derived geometry.

ABSTRACT

We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore, we can make sense of "homotopy fibered product" and "derived intersection" of submaifolds in a smooth manifold in the homotopy category of derived manifolds. We construct a factorization of the diagonal using path spaces. First we construct an infinite-dimensional factorization using actual path spaces motivated by the AKSZ construction, then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. We also prove the inverse function theorem for derived manifolds, and investigate the relationship between weak equivalence and quasi-isomorphism for derived manifolds.

Motivation & Objective

• To develop a framework for derived differential geometry using bundles of curved $L_{\infty}[1]$-algebras.
• To establish derived manifolds as a category of fibrant objects, enabling homotopy-theoretic constructions.
• To construct a finite-dimensional factorization of the diagonal using path spaces and homotopy transfer.
• To prove the inverse function theorem in the derived setting and clarify the link between weak equivalences and quasi-isomorphisms.
• To provide a geometric realization of derived intersections of submanifolds in smooth manifolds via homotopy fibered products.

Proposed method

• Models derived manifolds as triples $(M, L, \lambda)$, where $M$ is a $C^\infty$-manifold, $L$ is a graded vector bundle, and $\lambda$ defines a curved $L_{\infty}[1]$-algebra structure on each fiber.
• Uses the homotopy transfer theorem for curved $L_{\infty}[1]$-algebras to reduce infinite-dimensional path space constructions to finite dimensions.
• Constructs the derived path space of a manifold using actual path spaces and then applies the Fiorenza-Manetti method to truncate to finite dimensions.
• Applies the AKSZ construction as motivation for the path space factorization, leading to a homological vector field on the path space of the shifted tangent bundle.
• Defines the derived intersection of submanifolds as a homotopy fibered product in the homotopy category of derived manifolds.
• Uses the tautological vector field $\tau$ and its interaction with the homological vector field $PQ$ to define a dg structure on the tangent bundle of the path space.

Experimental results

Research questions

• RQ1 How can derived intersections of submanifolds be meaningfully defined in classical differential geometry, where fiber products often fail to exist?
• RQ2 What is the role of path spaces in constructing a factorization of the diagonal in derived geometry?
• RQ3 How can infinite-dimensional path space constructions be reduced to finite-dimensional models while preserving homotopical structure?
• RQ4 What is the precise relationship between weak equivalences and quasi-isomorphisms in the category of derived manifolds?
• RQ5 To what extent does the inverse function theorem hold in the derived setting, and how does it relate to the homotopy theory of derived manifolds?

Key findings

• The category of derived manifolds is a category of fibrant objects, enabling the construction of homotopy fibered products and derived intersections in the homotopy category.
• A finite-dimensional factorization of the diagonal is constructed via the derived path space using the Fiorenza-Manetti method and homotopy transfer.
• The derived path space of a derived manifold carries a natural dg structure, realized as $T(P{\mathscr{M}})[-1]$ with homological vector field $\iota_\tau + \widehat{PQ}[-1]$, where $\tau$ is the tautological vector field.
• The inverse function theorem holds for derived manifolds, generalizing the classical result to the homotopical setting.
• Weak equivalences in derived manifolds coincide with quasi-isomorphisms of the underlying $L_{\infty}[1]$-algebras, establishing a key link between homotopy theory and cohomological algebra.
• For any path $a: I \to M$, the pullback complex $\Gamma(I, a^*T{\mathscr{M}}[-1])$ carries a curved $L_{\infty}[1]$-structure induced by $a'\,dt + a^*\mu$, matching the structure from Proposition 2.5.