String cobracket from TFT

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?

1 Answer

The string cobracket you are referring to is not part of the TQFT structure of string topology given by the smooth Calabi Yau algebra structure on $C_*(\Omega M)$ but rather associated to an action of the chains of certain compactification of the moduli space of Riemann surfaces as explained in Sullivan's survey "String Topology: background and present state". It is an operation arising from a homotopy at the chain level.

In fact, this cobracket arises from a secondary coproduct on $C_*(LM)$. Given a closed manifold $M$ of degree $d$ we may define two coproducts of degree $-d$ in ordinary singular chains $C_*(LM)$ that pass to homology: namely we may consider self intersections at $t=0$ and split into two loops or at $t=1$ and split into two loops. Note that these are really coproducts factoring through maps $C_*(LM) \to C_{*-d}(M \times LM)$ and $C_*(LM) \to C_{*-d}(LM\times M)$, respectively. These two coproducts are chain maps and moreover they are chain homotopic, so they define the same coproduct of degre $-d$ in homology, which is in fact part of the TQFT structure of string topology on $H_*(LM)$ (the upside down pair of pants coproduct). However, there is a canonical chain homotopy between the two coproducts at the chain level given by a map $\vee: C_*(LM) \to C_{*+1-d}(LM \times LM)$ which may be defined as a one parameter family of self intersections (the transversality assumptions are rather subtle). The map $\vee$ of course is not a chain map, however, it is a chain map if we work modulo constant loops so it defines a coproduct in relative homology! It is in this sense that $\vee$ is a secondary operation. Then the string cobracket in $S^1$-equivariant homology modulo constant loops is induced by $\vee$ and the maps relating ordinary and equivariant homology in the long exact sequence.

Thus, there are two coproducts, one of them of degree $-d$ defined on $H_*(LM)$ and is trivial if we work modulo constant loops (also trivial if Euler characteristic is zero, actually this coproduct is rather boring since it is essentially multiplication by the Euler characteristic) and the other one of degree $1-d$ defined on $H_*(LM,M)$ which shows a more interesting behavior. In particular, the latter induces a cobracket of degree $2-d$ equivariant homology modulo constant loops and it defines a involutive Lie bialgebra structure together with the string bracket.

The Hochschild story is completely analog to what I have described above. An interesting subtlety on the algebraic Hochschild theory of Frobenius algebras (Koszul dual to the one with $C_*(\Omega M)$) is that "reduced" Hochschild complex only makes sense for commutative dg algebras (or cocommutative dg coalgebras, if you are working with coalgebras) while the algebraic versions of the loop product make sense in the associative case. In a recent paper of Zhengfeng Wang and myself (https://arxiv.org/abs/1703.03899) we actually describe a way to combine the algebraic loop product and the algebraic coproduct (the one of degree $1-d$) in an extended version of the Hochschild complex without having to assume commutativity. Essentially when there is a problem at the algebraic version of the "constant loops" we change the differential of the underlying complex instead of killing these "constant loops".