Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C \otimes C^{op}\quad$ ($2n$ copies of $C$, half of them "op").]

Let $F_{even}$ be the $E$ module with underlying vector space $C^{\otimes n}$, where the $(2i)$-th factor of $E$ acts on the left side of the $i$-th factor of $F_{even}$, and the $(2i+1)$-th factor of $E$ acts on the right side (i.e. left action of $C^{op}$) of the $i$-th factor of $F_{even}$. ($0 \le i \le n-1$ .)

Define $F_{odd}$ similarly (same underlying vector space $C^{\otimes n}$), but with the $(2i)$-th factor of $E$ acting on the left side of the $i$-th factor of $F_{odd}$, and the $(2i+1)$-th factor of $E$ acting on the right side of the $(i+1)$-th (modulo $n$) factor of $F_{odd}$. (Note that "modulo $n$" means the $n$-th factor is the 0-th factor.)

I'm interested in the derived $Hom_E(F_{even}, F_{odd})$. We can think of this as living in a disk with its boundary divided into $2n$ segments, alternating between incoming and outgoing segments. For $n=1$ this is just the Hochschild cohomology of $C$. For $n=2$ it's what one might associate to a saddle bordism in a 1+1-dimensional TQFT, or rather, the space in which the TQFT invariant of the saddle bordism lives.

Does $Hom_E(F_{even}, F_{odd})$ have a name? Is it mentioned in the literature anywhere?