Floer-theoretic string topology operations for marked chord diagrams

Let $M$ be the closed manifold and let $d$ denote its dimension. For every marked chord diagram

\Gamma$ with $p$ incoming and $q$ outgoing boundary components, an $LM$-Morse structure

\sigma$ determines the moduli space

\mathcal{M}^{hol}_{(\Gamma,\sigma,\epsilon)}(T^*M)$ of cylindrical holomorphic curves, with restriction maps

\rho_{in}$ and

\rho_{out}$. Here $HF_*(T^*M)$ is the Floer homology of $T^*M$, and

\chi(\Gamma)$ is the Euler characteristic of the diagram.

Floer-theoretic string topology operations conjecture. For every marked chord diagram $\Gamma$, there is an umkehr map

$$(\rho_{in})_!: (HF_*(T^*M))^{\otimes p} \to H_{*+\chi(\Gamma)\cdot d}(\mathcal{M}^{hol}_{(\Gamma,\sigma,\epsilon)}(T^*M))$$

and a homomorphism

$$(\rho_{out})_*: H_*(\mathcal{M}^{hol}_{(\Gamma,\sigma,\epsilon)}(T^*M)) \to (HF_*(T^*M))^{\otimes q}$$

such that

$$\theta_\Gamma=(\rho_{out})_*\circ(\rho_{in})_!:(HF_*(T^*M))^{\otimes p}\to(HF_*(T^*M))^{\otimes q}$$

fit together to define a positive-boundary topological field theory and, under the Salamon–Weber isomorphism $HF_*(T^*M)\cong H_*(LM)$, agree with the string topology operations $q_\Gamma$.

The conjecture proposes a Floer-theoretic construction of the string topology field theory, with the Salamon–Weber isomorphism identifying the two kinds of operations. The cited context relates it to field theory structures on Floer homology and suggests a direct connection with Lalonde's constructions; no resolution is supplied here.