Nikita Markarian’s mathblog

Koszul duality for n-algebras

Posted in Mathematics by nikitamarkarian on February 19, 2011

Below there is a construction that from any augmented  dg algebra over the operad of chains on framed little $n$-discs produces a coalgebra over the same operad.

This construction generalizes the Koszul duality, for $n=1$ this is the usual Koszul duality for algebras.

Another case of this duality is from topology. Take a topological space $X$. Then $\Omega^n X$  is a module over framed little $n$-discs.  The union of symmetric powers of $X$ is a comodule over the (trivial) operad $Com$ and thus over framed discs (I am grateful to Victor Turchin for this observation). (Co)chains of this (co)modules over operads are dual to each other in our sence. This explains why one may calculate homology of mapping spaces from a $n$-manifold to $X$ either by means of the generalization of  the higher Hochschild homology or the spherical homology.

It seems that it is connected with work in progress of M. Ching and P. Salvatore “Geometric self-duality for the little discs operads” (I found it on the homepage of Michael Ching).

UPD: This Koszul duality is mentioned in the draft of J. Lurie’s “Higher algebra”(see e.g. example 7.3.6.7). Unfortunately, details are postponed for future work. See also “Moduli Problems for Ring Spectra”.

Suppose $n=2$, the generalization is straightforward.  Let $A$ be a dg $f2$(=chains on framed little 2-discs)-algebra and $e$ be an augmentation that is a one-dimensional module over it. “Module” means that $\ker e$ is closed under all operations.  Let $HH(A)$ be the Hochschild complex of $A$ considered as an algebra, see this post.Then $A$ is a left and right module over $HH(A)$ (ibid.) and the composition gives a morphism $HH(A) \to e$ that is also an augmentation. Define $A^!=Tor^\bullet_{HH(A)}(A,e)$.

Let us show that this is a $f2$-coalgebra. The reasoning is similar to the one about the action of the operad of framed little discs on a multiloop space. Let $D$ be the closed disc. One may present $Tor^\bullet_{HH(A)}(A,e)$ as the complex $\oplus_k (C_*(fD^2(k))\otimes A^{\otimes k})$ (we use notations from here) modulo relations given by the operad action and modulo the augmentation ideal for copies of $A$ lying on the border:

Instead of placing the augmentation on the border on can put it on a closed subset of the disc that is the supplement to the union of non-intersecting discs:

The corresponding homology is the tensor product of $A^!$‘s by the number of white discs. The augmentation map gives the map from $A^!$ to the tensor product:

This defines the map $C_{*}(fD(n))\otimes A^!\to \underbrace{A^!\otimes\cdots\otimes A^!}_n$, where $C_*(fD(n))$ is the complex of chains on the space of framed discs. Dually this defines a $f2$-structure on $A^{!\vee}$. In fact it gives only a $2$-algerbra, but it is natural to propose that it be lifted to $f2$. In presence of an augmentation the difference between $2$ and $f2$ is not so big, I hope to discuss it somewhere.

Natural constructions of dual $f2$-algebras are dual to each others: S-homology for a compact surface $S$ are dual (call it the Poincare-Koszul duality), Hochschild homology are Koszul dual and so on. As an example consider a Lie bialgebra. Its Chevalley complex is (nearly) a BV algebra, the formality theorem gives a $f2$-algebra. The spherical homology of this $f2$-algebra equals to the tensor product of the Chevalley complex and the complex dual to the Chevalley complex of the dual bialgebra.The dual $f2$-algebra is the one corresponding to the dual bialgebra. Thus spherical homologies of dual algebras are dual, that fits in with the statement above.

I guess that this duality plays an important role in something like string theory. For example, one may propose, that BV algebras coming from A and B models for a given Calabi-Yau manifold are dual to each other.

An interesting question is to describe an algebra Koszul dual to the one coming from the string topology.