Compatibility between the cyclotomic structure and framed E₂ algebra in the cotangent bundle case

Characterize the compatibility relations between the cyclotomic structure on spectral symplectic cohomology SH^•(M; S) and the framed E₂-algebra structure on Σ^{-n}Σ^∞_+ LQ in the case M = T^*Q. Provide a precise description of how the cyclotomic operations induced by the p-fold cover map interact with the framed E₂ structure, and establish the resulting algebraic relations.

Background

In the cotangent bundle setting M = T*Q, spectral symplectic cohomology is expected to model string-topological structures on the free loop space LQ, including a framed E₂-algebra coming from the combination of an E₂ structure and an S¹ action.

The cyclotomic structure constructed in the paper is induced by the p-fold covering of loops and should interact in a controlled way with the framed E₂ operations. A precise description of this interaction in the spectral setting is currently missing, even for M = T*Q, and would clarify the structural algebraic relations between these two rich operations.

References

A description, even in this relatively simple setting, of the compatibility relations between the cyclotomic structure (which is induced by the very geometric $p$-fold cover map eq:p-fold-cover-map) and the framed $E_2$-algebra structure on $\Sigma{-n}\Sigma\infty_+ LQ$, is not known to the author, and we leave this to future work.

— Cyclotomic Structures in Symplectic Topology  (2405.18370 - Rezchikov, 2024) in String topology (subsection in Introduction)

This now leads to the following question: what is the statement of the MUP-Viterbo isomorphism when taking into account the framed E_2-structures? I.e., what is the MUP-local system on L Q which does not break S1-symmetry?

— Spectral Viterbo isomorphism: complex-oriented versus framed  (2608.20289 - Blakey, 20 Aug 2026) in Question 1.4, Section 1.3 (Speculations and symmetry-breaking), subsection Symmetry-breaking