Fock Space and Strict Orientations

• The free $E_1$ algebra on a (connective for now) spectrum $X$ could be thought of as Fock space of states in $X$. The underlying spectrum is a direct sum of 1-particle states, 2-particles states, 3-particle states, etc. The $E_1$ structure is “concatenation of states”. That is, a 1-particle state followed by a 2-particle state concactenates to a 3 particle state. Observe that order matters, a priori.
• A strict orientation on a $X$ is a map $\theta :H\mathbb{Z}\to X$ gives a lift of the free $E_1$ algebra to an $E{-}_{\infty }$-algebra. This data exhibits the concatenation as equivariant with respect to all permutations of the labels.
• This has to do with trivializing symmetric group actions, therefore perhaps the statement is like “when spacetime is an $\infty$/derived, being “bosonic” or “fermionic” is extra data rather than property”.