L-infinity algebra connections and applications to String- and Chern-Simons n-transport

TL;DR

The paper develops a comprehensive differential-geometric framework that generalizes Cartan-Ehresmann connections from Lie algebras to L∞-algebras and uses it to study obstruction problems for lifts through String-like extensions. By formulating descent data, connections, and characteristic forms entirely in terms of L∞-algebras and their Chevalley-Eilenberg/Weil algebras, the authors derive a general obstruction theory: lifts are obstructed by Chern-Simons-type higher bundles whose curvature encodes Pontrjagin classes, linking String and Fivebrane structures to higher n-bundles. They also construct generalized parallel transport functionals (Chern-Simons, BF) arising as action functionals from transported higher connections, and discuss transgression to mapping spaces and loop spaces, relating topological data to loop-group cocycles. The work provides a unified algebraic-topological language for brane structures, higher gauge theories, and their quantization, with concrete examples in string theory, supergravity, and higher BF-type theories. Overall, the framework connects higher algebraic structures to physically relevant obstructions, transports, and topological actions across brane theories.

Abstract

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) -> U(H) -> PU(H) to higher categorical central extensions, like the String-extension BU(1) -> String(G) -> G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures" whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

Figures (12)

• Figure 1: The universal $G$-bundle in its various incarnations. That the ordinary universal $G$ bundle is the realization of the nerve of the groupoid which we denote here by $\mathrm{INN}(G)$ is an old result by Segal (see RS for a review and a discussion of the situation for 2-bundles). This groupoid $\mathrm{INN}(G)$ is in fact a 2-group. The corresponding Lie 2-algebra (2-term $L_\infty$-algebra) we denote by $\mathrm{inn}(\mathfrak{g})$. Regarding this as a codifferential coalgebra and then dualizing that to a differential algebra yields the Weil algebra of the Lie algebra $\mathfrak{g}$. This plays the role of differential forms on the universal $G$-bundle, as already known to Cartan. The entire table is expected to admit an $\infty$-ization. Here we concentrate on discussing $\infty$-bundles with connection in terms just of $L_\infty$-algebras and their dual dg-algebras. An integration of this back to the world of $\infty$-groupoids should proceed along the lines of GetzlerHenriques, but is not considered here.
• Figure 2: A $\mathfrak{g}$-connection descent object and its interpretation. For $\mathfrak{g}$-any $L_\infty$-algebra and $X$ a smooth space, a $\mathfrak{g}$-connection on $X$ is an equivalence class of pairs $(Y,(A,F_A))$ consisting of a surjective submersion $\pi: Y \to X$ and dg-algebra morphisms forming the above commuting diagram. The equivalence relation is concordance of such diagrams. The situation for ordinary Cartan-Ehresmann (1-)connections is described in \ref{['examples for connection descent objects']}.
• Figure 3: A remarkable coincidence of concepts relates the notion of tangency to the notion of universal bundles. The leftmost equality is discussed in RS. The second one from the right is the identification \ref{['Weil and inner']}. The rightmost equality is equation \ref{['Weil algebra as functions on tangent space']}.
• Figure 4: Interpretation of vertical derivations on $\mathrm{W}(\mathfrak{g})$. The algebra $\mathrm{CE}(\mathfrak{g})$ plays the role of the algebra of differential forms on the Lie $\infty$-group that integrates the Lie $\infty$-algebra $\mathfrak{g}$. The coadjoint action of $\mathfrak{g}$ on these forms corresponds to Lie derivatives along the fibers of the universal bundle. These vertical derivatives leave the forms on the base of this universal bundle invariant. The diagram displayed is in the 2-category $\mathbf{Ch}^\bullet$ of cochain complexes, as described in the beginning of \ref{['homotopies and concordances']}.
• Figure 5: Lie algebra cocycles, invariant polynomials and transgression forms in terms of cohomology of the universal $G$-bundle. Let $G$ be a simply connected compact Lie group with Lie algebra $\mathfrak{g}$. Then invariant polynomials $P$ on $\mathfrak{g}$ correspond to elements in the cohomology $H^\bullet(BG)$ of the classifying space of $G$. When pulled back to the total space of the universal $G$-bundle $EG \to BG$, these classes become trivial, due to the contractability of $EG$: $p^* P = d(\mathrm{cs})$. Lie algebra cocycles, on the other hand, correspond to elements in the cohomology $H^\bullet(G)$ of $G$ itself. A cocycle $\mu \in H^\bullet(G)$ is in transgression with an invariant polynomial $P \in H^\bullet(BG)$ if $\mu = i^* \mathrm{cs}$.

Theorems & Definitions (96)

• Theorem 1: characteristic classes
• Theorem 2: string-like extensions and their properties
• Theorem 3: obstructions to lifts through String-like extensions
• Definition 1
• Definition 2
• Definition 3
• Definition 4: forms on spaces of maps
• Definition 5: currents
• Proposition 1
• Definition 6: superpoint