Symplectic $A_infty$-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar'e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for $C_\infty$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

💡 Research Summary

The paper establishes the existence of the three fundamental string‑topology operations – the loop product, the loop bracket, and the string bracket – on both ordinary and equivariant homology of the free loop space of a manifold, and more generally on any formal Poincaré duality space. The authors’ strategy combines obstruction theory for C∞ (or A∞) algebras with rational homotopy theory, producing a symplectic A∞‑algebra structure that encodes the Poincaré duality pairing and is compatible with a cyclic symmetry.

First, given a chain complex A that satisfies Poincaré duality, a non‑degenerate, closed, graded‑symmetric 2‑form ω is introduced. This form provides a symplectic pairing between homology and cohomology and serves as the datum required to define a “symplectic” A∞‑algebra. Using the minimal model of A, the authors construct an underlying C∞‑structure (operations μ₂, μ₃, …) and then systematically adjust each higher operation so that it becomes cyclic with respect to ω. At each stage the obstruction to cyclicity lives in a certain cohomology group of the Koszul complex; for a Poincaré duality space these obstruction groups vanish, allowing the induction to proceed indefinitely. Consequently a full symplectic A∞‑structure exists on A.

With this algebraic structure in place, the paper exploits the well‑known quasi‑isomorphism between the chains on the free loop space LM and the Hochschild chain complex CH₍₎(A,A). Under this identification the loop product corresponds to the cup product in Hochschild cohomology HH⁎(A,A), while the loop bracket matches the Gerstenhaber bracket on HH⁎(A,A). The S¹‑action on LM gives rise to Connes’ B‑operator on the Hochschild complex; passing to S¹‑equivariant homology H₍₎^{S¹}(LM) the induced string bracket is shown to be isomorphic to the Lie bracket on the cyclic (co)homology HC₍*₎(A). Thus all three string‑topology operations acquire a purely algebraic description in terms of Hochschild and cyclic (co)homology.

A crucial consequence of the construction is homotopy invariance. Because the symplectic A∞‑structure is built from the rational minimal model, any rational homotopy equivalence between two Poincaré duality spaces induces an A∞‑quasi‑isomorphism that respects ω and therefore transports the entire suite of string‑topology operations. Hence the operations depend only on the rational homotopy type of the underlying space, not on a particular manifold presentation.

The authors also discuss computational advantages: for formal spaces the symplectic A∞‑structure can be read off from the cohomology algebra, making the string‑topology operations accessible via algebraic models rather than geometric loop‑space constructions. Moreover, the paper highlights the deep link between symplectic A∞‑algebras (or cyclic C∞‑algebras) and string topology, providing a conceptual bridge that unifies geometric loop‑space phenomena with classical algebraic topology tools such as Hochschild and cyclic homology.

In summary, the work delivers a rigorous, homotopy‑invariant algebraic framework for the loop product, loop bracket, and string bracket, extending the Chas‑Sullivan theory to a broad class of formal Poincaré duality spaces and illuminating the underlying symplectic A∞‑structure that governs these operations.