Let $M$ be a closed oriented manifold. Chas and Sullivan (https://arxiv.org/abs/math/0212358) introduced a Lie bialgebra structure on $H_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop space $LM=\mathrm{Map}(S^1, M)$ relative to constant loops $M\rightarrow LM$.

It seems an "explanation" of the string topology operations comes from topological field theories. I will switch to chains for simplicity and ignore some shifts. Then $C_\bullet(\Omega M)$ is a (smooth, not proper) Calabi-Yau dg algebra, so it defines a positive boundary oriented 2d TFT (this construction is reviewed at the end of https://arxiv.org/abs/0905.0465). This gives rise to some operations on $CH_\bullet(C_\bullet(\Omega M)) = C_\bullet(LM)$ which are supposed to be the string topology operations. Here $CH_\bullet(-)$ are Hochschild chains which is the value of the TFT on $S^1$.

For instance, you get a structure of an $E_2^{fr}$-algebra on $C_\bullet(LM)$, where $E_2^{fr}$ is the operad of framed little disks. Passing to $S^1$-coinvariants you obtain a gravity algebra (https://arxiv.org/abs/math/0605080); in particular, you get a Lie bracket on $C_\bullet^{S^1}(LM)$.

Similarly, the TFT picture seems to give a non-counital $E_2^{fr}$-coalgebra structure on $C_\bullet(LM)$. Again passing to $S^1$-coinvariants I would expect to see a Lie cobracket on $C^{S^1}_\bullet(LM)$.

However, it is known that to define the cobracket one has to pass to reduced chains $C_\bullet^{S^1}(LM, M)$. A similar construction exists in the algebraic context, e.g. in https://arxiv.org/abs/0804.4748 it is shown that Hochschild homology is a BV algebra and cyclic homology is a gravity algebra (in particular, has a Lie bracket) while only *reduced* Hochschild homology is a BV coalgebra and *reduced* cyclic homology is a gravity coalgebra (in particular, has a Lie cobracket). (Note that the paper deals with the Koszul dual situation of a cyclic coalgebra.)

**Question**: how can one see that the string cobracket exists only on reduced cyclic homology and not on the unreduced one from the TFT perspective? How does reduced Hochschild homology appear in the TFT?