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Symplectic cohomology is the closed-string sector of the open-closed TCFT associated to the A-side topologically twisted theory.

  • For general reasons, it should carry the structure of an algebra over the chain operad of framed little discs $H_{-\bullet}(f \mathcal{D}_2)$. In particular, if we believe the open-closed string map from the wrapped Fukaya category to be an isomorphism, we must have a BV-structure.
  • In characteristic zero, these algebras are known to be BV-algebras. The analytic underpinning (more precisely, the domains of the moduli spaces of pseudo-holomorphic maps used) seems very close to the framed Kimura-Stasheff-Voronov operad (which is a framed little discs operad).

Is there a known analytic construction of such an operator $\Delta$ and a proof that it satisfies the required properties (including the 7-term relation)?

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Yes it does. Geometrically the BV operator is gotten by considering the parametrized moduli space of solutions to the Floer equation on the cylinder obtained by rotating one end of the cylinder. This is explained (rather briefly) in Section (8a) of Seidel's "Biased View of Symplectic Cohomology," see also Remark 5.7 of Bourgeois and Oancea's "The Gysin exact sequence for $S^1$-equivariant symplectic homology"

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