[Paper Review] Twisted IBL-infinity-algebra and string topology: First look and examples

Pavel Hajek|arXiv (Cornell University)|Nov 13, 2018

Black Holes and Theoretical Physics24 references4 citations

TL;DR

This paper investigates a twisted IBL∞-algebra construction on the cyclic cochains of the de Rham complex of a closed oriented manifold, using a Maurer-Cartan element derived from Chern-Simons-like integrals over trivalent ribbon graphs. The authors compute the twist explicitly for the n-sphere and show it is often trivial, supporting the conjecture that this construction models Chas-Sullivan string topology via a chain-level realization.

ABSTRACT

We study the application of IBL-infinity-algebras to string topology and explicitly compute the case of spheres. This involves finding a Green kernel and computing integrals associated to trivalent ribbon graphs, which are similar to integrals from perturbative Chern-Simons theory. We generalize the computation and show that the twisted IBL-infinity-structure in question is in many cases trivial.

Motivation & Objective

• To test the conjecture that a twisted IBL∞-algebra construction on the de Rham complex of a closed oriented manifold models Chas-Sullivan string topology.
• To compute the formal Maurer-Cartan element n built from trivalent ribbon graphs with Chern-Simons-like integrals.
• To investigate whether the twist with n is non-trivial or often trivial in specific cases.
• To provide explicit calculations for the n-sphere and complex projective space as test cases.
• To clarify the algebraic and geometric structure of the dIBL- and IBL∞-algebras in the infinite-dimensional de Rham setting.

Proposed method

• The paper constructs a canonical dIBL-structure on the degree-shifted dual cyclic bar complex of the de Rham cochain complex of a closed oriented manifold M.
• It introduces a Green kernel G as a formal inverse of the de Rham differential, used to define a homotopy equivalence between the full complex and its cohomology.
• The Maurer-Cartan element n is built from trivalent ribbon graphs with m+2 (the cyclic product) at internal vertices and components of cohomology classes at boundary components.
• The twist of the IBL∞-algebra is computed via contributions from labeled ribbon graphs, using formal integration over M×k with Schwartz kernels.
• The computation is carried out explicitly for Sn using the formal pushforward of the canonical Maurer-Cartan element m from the full de Rham complex.
• The paper uses a formal completion and filtration to handle infinite sums arising from the infinite-dimensional setting, even though the Green kernel G is not in Ω(M)⊗2.

Experimental results

Research questions

• RQ1 Does the twisted IBL∞-algebra constructed from Chern-Simons-like integrals on trivalent ribbon graphs on Sn yield a non-trivial deformation of the canonical dIBL-structure?
• RQ2 Is the Maurer-Cartan element n, built from trivalent ribbon graphs, non-vanishing for the n-sphere?
• RQ3 Can the twist with n be shown to be trivial in general, as suggested by the explicit computation on Sn?
• RQ4 How does the formal pushforward of the Maurer-Cartan element m from Ω(M) to H*(M) behave in the infinite-dimensional de Rham setting?
• RQ5 To what extent does the twisted IBL∞-algebra on H*(M) model the Chas-Sullivan string topology algebra?

Key findings

• For the n-sphere, the formal pushforward Maurer-Cartan element n is trivial, meaning the twist does not alter the underlying dIBL-structure.
• The computation of the Green kernel G for Sn is explicitly carried out, showing it is well-defined in the formal sense despite not lying in Ω(Sn)⊗2.
• The contribution of each trivalent ribbon graph to the Maurer-Cartan element n is computed via integration over M×k with appropriate signs and symmetry factors.
• The sign structure in the evaluation of ribbon graphs is derived consistently using Koszul signs and cyclic symmetry, confirming the formal consistency of the construction.
• The paper shows that the twist with n is often trivial, suggesting that the canonical dIBL-structure on cohomology may already capture the string topology operations.
• The formal computation for Sn supports the conjecture that the twisted IBL∞-algebra construction models Chas-Sullivan string topology, even though the Green kernel is not in the de Rham complex.