Given an $E_n$-algebra $A$, one can define its $E_n$-Hochschild complex $CH_{E_n}(A,A)$ by the formula $$Ch_{E_n}(A,A)=RHom_{Mod_A^{E_n}}(A,A)$$ where $Mod_A^{E_n}$ is the category of $A$-modules over $E_n$ (following the operadic notion of module over a $P$-algebra: for example, the case $P=Ass$ gives us the usual notion of bimodule over an associative algebra), and $RHom_{Mod_A^{E_n}}(-,-)$ is the derived hom bifunctor in $Mod_A^{E_n}$.

My question concerns what happens when one replaces the $E_n$-operad by its framed version $frE_n$. Does a notion of framed higher Hochschild complex in the spirit of the definition above have been studied somewhere ? Something that could be defined maybe by taking homotopy fixed points $$RHom_{Mod_A^{E_n}}(A,A)^{hSO(n)}$$ under the action of $SO(n)$, or using a definition based on factorization homology ?


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