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.