Let be a -algebra. Then is equipped with a structure of -algebra. Denote this -algebra by . In the preprint I explicitly write down the morphism from homology of to the manifoldic homology of on a parallelized n-manifold without borders. This morphism is the composition , where the first arrow is the tautological morphism and the second arrow is induced by the canonical morphism.

How is it connected with the Chern-Simons perturbative invariants? The point is that for some algebras the space is one-dimensional for any compact without borders, thus we get an invariant of lying in the cohomology of . To get an example of such an algebra consider a commutative algebra , then consider it as -algebra and take its deformation complex . It has a structure of -algebra.

**Conjecture.** For , a compact -manifold without borders and a commutative algebra the manifoldic homology is the shifted derived de Rham complex of . In particular, for the polynomial algebra the manifoldic homology is one-dimensional.

Note that this conjecture is consistent with the rigidity of the manifoldic homology that was discussed here.

For the statement is that Hochschild homology of the ring of differential operators is one-dimensional (it sits in the degree , where is the dimension of the affine space). This fact plays a crucial role the Bressler-Tsygan-Nest proof of the Riemann-Roch theorem cited in the Costello’s paper. Note that the morphism from the homology of the Lie algebra of differential operators to Hochschild homology of differential operators is very important there.

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Given a homotopy Lie algebra , the Quillen functor produces from it a homotopy cocomutative algebra, that is a comodule over the operad . This coalgebra is the homological Chevalley complex of , denote it by . Due to the embedding one may treat as a co--algebra and further as a co--algebra in the category of homotopy cocommutative algebras. Applying the Koszul duality for -algebras to this co--algebra we get a -algebra in the category of homotopy cocommutative algebras. In other words, we get a -algebra with compatible homotopy cocommutative coalgebra structure. Call it the *universal enveloping -algebra* of and denote it by .

List some important properties of the universal enveloping -algebra.

**1**. As a commutative coalgebra is freely generated by .

**2.** For a compact parallelized -manifold without borders consider the manifoldic chain complex . By the Koszul-Poincare duality the latter complex is isomorphic to the manifoldic chain complex of being considered as a co--algebra (one may define manifoldic homology of a co--algebra in same way as for an -algebra). There is a tautological morphism from to the latter complex. Thus we get a morphism , call it the *tautological morphism*. If and manifold is a circle for a Lie algebra the tautological morphism is the morphism from the Chevalley chain complex of to the Hochschild chain complex of the universal enveloping algebra.

**3.** Let be the (super) Lie algebra of homotopy groups of a topological pointed -connected space with respect to the Whitehead product: . Then is the chain complex of and is the chain complex of the n-fold based loop space of . For a parallelizable compact -manifold the manifoldic chain complex is the chain complex of the space of maps from to that sends the border to the marked point. For a manifold without borders the tautological morphism is the map induced by the embedding of to as the subspace of maps that factors through a point.

**4.** Denote by the operad that manages homotopy Lie algebras with the bracket of degree . There is a morphism that is Koszul dual to the embedding . Thus for any -algebra the complex is a -algebra. Denote it by . The functor is left adjoint to . It follows that for any -algebra there is the *canonical morphism* of -algebras .

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Equivariant Hochschild homology of an algebra with an algebraic group action is homology of a mixed complex, that is a complex with an extra differential of degree -1 commuting with the main differential. For a trivial group this is the usual Hochschild complex. I believe that if the group is finite, then the equivariant Hochschild homology is the Hochschild homology of the semidirect product of the algebra and the group algebra. If the algebra is trivial, then the equivariant Hochschild homology is the complex of functions on the cyclic nerve of the group, that is the cohomology of the group with coefficients in the functions on the group under the adjoint action. The aim is to show that the operator is up to a coboundary equal to the standard operator that is induced by the cyclic structure on the cyclic nerve. This is a routine consequence of routine facts about cyclic sets. I did not find exact references that is why I devoted first three subsections to a reminder about cyclic stuff.

The only new concept in these three parts is functor that produces a cyclic set from a cocylcic simplicial set. The point is this functor respects “circle action”. To be more precise, when passing to the chain complex of a cyclic set and to the total chain-cochain complex of a cocylcic simplicial set functor respects (up to a boundary) operators that are naturally defined on both sides.

This concept implicitly appeared in the classical papers on cyclic homology as follows. Given a cyclic set, on its chain complex an operator of degree -1 naturally acts and it gives an operator of degree -1 on homology. On the other hand, the geometric realization of the cyclic set is acted by circle. This also gives an operator of degree -1 on homology. The problem is to prove that this is the same operator. Informally the question is as follows. Consider the set of s of the cyclic category . It is naturally a cyclic and a cocyclic set. Roughly speaking one needs to prove that these two “circle actions” (“from the right” and “from the left”) are homotopy equivalent to each other.

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This construction generalizes the Koszul duality, for this is the usual Koszul duality for algebras.

Another case of this duality is from topology. Take a topological space . Then is a module over framed little -discs. The union of symmetric powers of is a comodule over the (trivial) operad 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 -manifold to 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 , the generalization is straightforward. Let be a dg (=chains on framed little 2-discs)-algebra and be an augmentation that is a one-dimensional module over it. “Module” means that is closed under all operations. Let be the Hochschild complex of considered as an algebra, see this post.Then is a left and right module over (ibid.) and the composition gives a morphism that is also an augmentation. Define .

Let us show that this is a -coalgebra. The reasoning is similar to the one about the action of the operad of framed little discs on a multiloop space. Let be the closed disc. One may present as the complex (we use notations from here) modulo relations given by the operad action and modulo the augmentation ideal for copies of 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 ‘s by the number of white discs. The augmentation map gives the map from to the tensor product:

This defines the map , where is the complex of chains on the space of framed discs. Dually this defines a -structure on . In fact it gives only a -algerbra, but it is natural to propose that it be lifted to . In presence of an augmentation the difference between and is not so big, I hope to discuss it somewhere.

Natural constructions of dual -algebras are dual to each others: S-homology for a compact surface 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 -algebra. The spherical homology of this -algebra equals to the tensor product of the Chevalley complex and the complex dual to the Chevalley complex of the dual bialgebra.The dual -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.

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I will follow the point of view of this post. Denote by a homotopy BV algebra and by the -algebra corresponding to it under the formality quasiisomorphism between BV and f2 operads. Recall that Hochschild homology of being considered as a (homotopy) algebra is equipped with a multiplication and is equipped with a Poisson structure. The former may be considered as a quantization of the latter. Algebra is a homotopy module over , this module structure may be considered as a quantization of the trivial action of on that sends image of Reinhart-Connes-Feigin-Tsygan differential to zero.

Now suppose that comes from a Hycomm algebra. In other words, is homotopically (but not coherently) trivial. This follows that the bracket of the Poisson structure on is trivial. Thus algebra is commutative as it is a quantization of . And moreover it is isomorphic to equipped with the standard commutative multiplication, at least formally by the quantization parameter. So the algebra structure on does not change under the quantization, but the structure of module on does. The element corresponding to this deformation is given by , where is Reinhart-Connes-Feigin-Tsygan differential and is the standart generator of the operad. One may see that the Maurer-Cartan equation follows from relations in .

It seems that in a sense an inverse statement is true: for a commutative Frobenius algebra (that is a finite-dimensional algebra with a non-degenerate pairing) a deformation of it as a module over its Hochschild homology that respects the pairing is equivalent to a deformation of the algebra as a -algebra. Or better one may consider a module with filtration over the Hochschild homology such that the associated graded module is isomorphic to the algebra itself as a module over Hochscild homology. Anyway, given a $Hycomm$-algebra it is worth looking at the pair .

This approach well fits the theory of quantum cohomology as it was discussed in the previous post. There for a compact complex manifold twisted K-theory of the loop space was introduced. Besides there is the usual K-theory of the loop space . is equipped by a product à la Chas-Sullivan. Conjecturally there is an action of on . Algebra equals to the cohomology ring and equals to the Hochschild homology of . In some cases is isomorphic to as a vector space, for example for . I do not know how to define a filtration on such that the associated graded module is , it seems that one have to take into account the circle action. Anyway, as soon as this is done, we have the pair that gives the desired -algebra.

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If one take as a starting point the paper of Witten and two papers of Givental (this and this), one may develop a theory of a different flavour sketched below. Conjecturally it coincides with the usual one, at least in some cases.

Let be a Kahler (it may be relaxed) compact manifold with and . Let be the fiber bundle over , that is the fiber product of a set of line bundles that form basis of . This space is acted by and the factor is . Let be the space of free loops (in the usual sence) of . This space is acted by . The latter group contains .

Consider the -equivariant twisted K-theory (M. Atiyah, G. Segal, “Twisted $K$-theory”) of . The twisting is given by the semi-infinite determinant of the tangent bundle. For a loop that bounds a holomorphic curve this is the determinant of the subbundle of the tangent bundle that may be continued on the whole curve. Different bounding curves give subspaces that differs by a finite-dimensional vector spaces. Thus the semi-infinite determinant of the tangent bundle gives a -gerbe. I would like to have a sharper definition of this twisting gerbe, which is the only non-topological ingredient of the construction.

Firstly, the twisted equivariant K-theory is acted by , because the group acts on the loop space and the gerbe is invariant under it. So is a module over the group algebra of . Then, due to the seminal paper of Chas and Sullivan, the twisted K-theory is a BV algebra. In fact a lot of work should be done here: one should replace cohomology in Chas-Sullivan theory with K-theory and show that the gerbe is multiplicative in a proper sense (**UPD:** this point is very weak, it is not clear for me how the gerbe behaves under compositions of loops; it seems much more plausible that there is an action of the non-twisted K-theory on the twisted one, see the end of the next post). Then the this BV algebra is a module over and moreover it is an algebra over this ring. Thus we have a family of BV algebras. Finally, one should show that this family is curvy trivial and trivialized in the sense of this post.

Now we may apply methods of the mentioned post. Spherical homology of BV algebras form a D-module over , and this is exactly D-module that play the major role in the mentioned papers of A. Givental: “Homological geometry and mirror symmetry” and “Homological geometry I. Projective hypersurfaces” . Moreover, it seems that my definition is (a part of) what Givental means, but written in a slightly different language. It is natural to propose that -homology is responsible for higher genera. Note that the spherical homology if our BV algebra is not naturally isomorphic to cohomology of , it is not even clear that dimensions are equal.

In the paper E. Witten, “The Verlinde Algebra And The Cohomology Of The Grassmannian” it is shown that in some cases quantum cohomology of a Grassmanian equals to some Verlinde algebra; by quantum cohomlogy Witten means the fiber of the Givental D-module at . In the light of our definition it is not so suprising. By D. S. Freed, M. J. Hopkins, C. Teleman, “Loop groups and twisted K-theory I”, the Verlinde algebra of a compact Lie group equals to the twisted K-theory of free loop space of the classifying space of with some special twisting (level). The quantum cohomology of a Grassmanian at in our definition is twisted K-theory of free loop space of the Grassmanian. It is natural to propose that under the natural embedding of our Grassmanian into the classifying space, is induced from some level and the inverse image gives the isomorphism stated by Witten.

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Given a 3-manifold and a (say) connection on it with zero cohomology, the perturbative Chern-Simons theory gives an invariant of it, that is a power series of the quantization parameter. It would be plausible to make this invariant two-step: on get from the initial data some structure (like a homotopy algebra over a operad) and then take an invariant of it.

The structure involved is a 1-parametric family of homotopy BV algebras that is curvy trivial and trivialized . The spherical homology gives invariants of this structure.

Let me explain and comment last two phrases.

Begin with the “curvy trivial” (horrible term) deformation. For any algebra over a operad there exists the standard notion of the deformation complex : differentiations of the free cooperad generated by the complex dual to the algebra (for a BV algebra see Imma Galvez-Carrillo, Andy Tonks, Bruno Vallette “Homotopy Batalin-Vilkovisky algebras”). Honest 1st-order deformations form its subcomplex and the rest is responsible for “gerby” or “curvy” deformations. Let us consider a familiar example: (homotopy) associative algebra . The deformation complex is the Hochschild cohomological complex and honest 1-order deformations are described by the subcomplex . For a non-central element of degree 1 the 1-cocycle is a coboundary in , but it gives a non trivial 1st-order deformation of . This is an example of what I call a curvy trivial deformation.

**Definition. **Call a 1st-order deformation of an algebra over a operad *curvy trivial* if the corresponding class lies in the kernel of the embedding morphism and call it trivialized if the class is lifted to the cohomology of . A smooth family of algebras such that the tangent 1st-order deformation at any point is curvy trivial (and trivialized) we will call curvy trivial (and trivialized).

An example of a curvy trivial deformation of a dg BV algebra is given by the cocycle for a -closed and choice of is the trivialization.

A classical example of a curvy trivial and trivialized family is the following. Consider a family of Calabi-Yau manifolds. Comlexes of polyvector fields form a family of dg BV algebras that is curvy trivial and trivialized. Indeed, deformation of the BV algebra corresponding to a deformation of Calabi-Yau is given by cocycle , where is the tangent vector to the moduli space of Calabi-Yau manifolds.

Now the spherical homology come into play. The spherical homology (and any -homology for a compact oriented ) is obviously acted by the deformation complex of BV algebras (or f2-algebras) . It follows that the spherical homology of a curvy trivial and trivialized family is equipped with a connection. A section of this local system obey some differential equation that is an invariant of the family. For example, the unit of the algebra gives such a section. For the example above about a family of Calabi-Yau manifolds this section is the integral of the volume form.

Here I associated a BV algebra with an oscillating integral. On may easily see that for a family of functions corresponding family of BV algebras is curvy trivial and trivialized. And (as it follows from the mentioned post) the perturbation theory deals with the local system of spherical homologies. From this perspective phrase “a curvy trivial and trivialized family of BV algebras gives the perturbative Chern-Simons invariant” sounds like a synonym for “the perturbative Chern-Simons invariant is given by an oscillating integral”.

Suppose the 3-manifold is triangulated. What one should place in a simplex? That is what is a perturbative analog of 6j-symbol (this question is posed as problem 7.5 in J. D. Roberts. Quantum invariants via skein theory**, **Phd. Thesis)? I guess the answer should come from the quasiclassical quantum group theory (connection between BV and quantum group was already mentioned here). Is it classical -matrix that responsible for trivialization of the curvy trivial family of BV algebras?

Finally, let be a 3d manifold that is a product of a surface and a circle. The usual Chern-Simons theory is not applicable in this case, because it is not a rational homological sphere. But the BV algebra in hand may be easily described. Consider the moduli space of connections on . It equals to the space of pairs (connection on , its automorphism). At a general point this is simply moduli space of connections on . It is equipped with a symplectic form (note that it may be produced (for a surface with holes) from the classical -matrix, see V.V.Fock, A.A.Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix”). The BV algebra is the de Rham complex on this moduli space and is the Brylinski differential associated with plus the de Rham differential.

In the paper of Hitchin, papers of Simpson and others a hyperkahler structure on the space of local systems on a complex curve was introduced. I wonder if this structure is connected with the family of BV algebras mentioned in the previous paragraph?

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**UPD:** Things described here and in some previous posts are particular cases of the topological chiral homology introduced in J. Lurie, “Derived Algebraic Geometry VI: E_k Algebras”. The only difference is that I need framed discs. I thank Damien for this comment.

**UPD:** Following M. Kapranov, I will use term *manifoldic homology* for the higher dimensional generalization of -homology.

Let be an oriented compact 2-manifold possibly with a border. Let be the complex of rational chains of the space of framed discs and on and semi-discs on the border. -homology of a dg f2-algebra (that is an algebra over rational chains of framed little discs operad) is the product of on over the operad of chains of framed little discs, that is modulo relations like in this definition.

Consider some examples.

**1. ** is a disc. (to be more precise, the underlying complex). It is not suprising, because -homology is homotopical invariant with respect to .

**2.** is a annulus. , the Hochschild homology. Picture

proves that is a (homotopy) algebra (this fact is true for any 2-algebra, structure of f2-algebra is not necessary). -homology of of a surface with holes is a module over . Glueing of surfaces gives maps between derived tensor products over .

Picture

proves that is a (homotopy) module over , see also this statement (in contrast with the previous picture, here is necessarily a f2-algebra).

**3.** is a torus. , more exactly the total homology of the corresponding bicomplex. Note that this complex is acted by automorphisms of the torus, that is , what is not so obviuos if one define it as the double Hochschild complex.

**4.** is a sphere. is the spherical homolgy that was discussed here, here and here.

**5.** is a pair of pants or surface of higher genus. The meaning of the corresponding -homology is not clear.

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Let us start with Hochschild homology of a DG category . The standard complex calculating it is built of , where are objects of the category (the definition may be easily given for a homotopy [i.e. -] category as well). If the category has only one object, then we get the usual Hochschild complex.

This may be rephrased as follows. Take a circle and consider all triangulations of it, that is all sets of finite number of circle points. Then put objects of the category in the 0-cells, that is points, and Hom’s – in 1-cells. It means, take moduli space of different points of the circle times and consider the complex of chains on it with coefficients in the tensor product of Hom’s, corresponding to intervals:

Now glue equivalent triangulations, that is if we remove a 0-cell, then tensor product of Hom’s we replace with the composition:

The resulting complex obviously calculates Hochschild homology of the category. (Here is some cheating, we did not consider *all* triangulations, it is not clear, what to do when two arrows go towards. But I am sure that the construction may be tuned in a suitable way.)

Let be a connected topological space. Consider the following DG category. Objects are points of . The complex of morphisms from to is the chain complex of (Moore) paths from to . As it follows e.g. from the Goodwillie’s result that we discussed many times, Hochschild homology of this category is the chain complex of the free loop space .

Now let us turn to 2-categories.

At the beginning, the following was not explicitly stated: in the definition of the spherical homology on may replace 2-sphere with any oriented 2-manifold . Let us call the resulting object by *-homology*. For a space , -homology of the chain complex of is without a doubt the chain complex of the space of maps .

By the previous post, a categorical analog of a f2-algebra is a homotopy cyclic 2-category. This observation leads to the following conjecture.

**Conjecture:** For any compact oriented 2-manifold there exist a functor from (homotopy) cyclic 2-categories to complexes such that for a category with one object and one 1-morphism this is -homology.

Call this functor by *-homology of a category*.

The category of 2-paths was considered in the end of the previous post: for a connected and simply connected topological space objects are points of , 1-morphisms between and are paths from to and complex of 2-morphisms is the chain complex of homotopies between paths. -homology of this category must be the chain complex of the space of maps .

Another example is given by the following (linear) 2-category (see e.g. section 8.3 of J.C. Baez, A.D. Lauda “Higher-Dimensional Algebra V: 2-Groups” and references therein). It has one object, 1-morphisms form a group and all 2-morphisms are only from a 1-morphism to itself and one-dimensional. This category is defined by the 3rd cohomology of in the multiplicative group of the base field. For a finite group this example is connected with Cher-Simons theory for finite groups, see R. Dijkgraaf, E. Witten, “Topological gauge theories and group cohomology”. If is an algebraic group (e.g. ) and the cocylce is the standard one, that is the second Chern character, this example is connected with the Chern-Simons TQFT, see the mentioned paper of Dijkgraaf-Witten and also series of papers D.S. Freed, M.J. Hopkins, C. Teleman, “Loop groups and twisted K-theory”.

How does a definition of -homology of a category could look like? The only idea I have is to consider the space (more exactly, the limit of spaces) of all triangulations of with vertices marked by objects of the category and edges marked by 1-morphisms and take cohomology of the local system (limit of local systems) formed by 2-morphisms. More manageable definition could include surfaces with boundaries, as in TQFT. For a surface with border circles the functor should give a complex of (twisted) sheaves over , where is Hochschild homology of the 2-category considered as a (1-)category and by “twisted” I mean twisting by an Abelian gerbe. If 1-morphisms form a set, then is a simplicial set, but if 1-morphisms are equipped with some additional structure (e.g. it is a scheme), then this set inherits it, and twisted sheaves goes with an additional stricture (e.g. they are twisted sheaves of -modules). The picture should be supplemented by gluing rules.

Continuing the flight of fancy, one may associate a metric with any triangulation, then consider the projection from the space of all triangulations to the moduli space of conformal classes of metrics and take the direct image. In this way one get a complex over this moduli space.

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DG 2-category is defined by the following data

- a set of objects
- a set of 1-morphisms for
- a complex of 2-morphisms for any for , where .

For any set of objects and set of 1-morphisms a morphism (composition)

is given, where and . These morphisms obey natural relations, for details see the mentioned paper.

A homotopy 2-category is defined by the same data, but the composition is replaced by the morphism

where is a complex. Essentially, this is chains of little discs. For details see ibid.

**Definition:** A cyclic (homotopy) 2-category is a (homotopy) 2-category with the following additional structure: for any pair of cycles of 1-morphisms

isomorphisms

are given, compatible with composition morphisms.

One may easily see, that on in a homotopy cyclic 2-category the operad of chains of framed little discs acts.

As an example consider the following homotopy cyclic 2-category. For a topological space objects are points of , 1-morphisms between and are paths from to and complex of 2-morphisms is the chain complex of the space of homotopies between paths. The cyclic structure is obvious. The complex is the chain complex of the double loop space . It is acted by the operad of chains of framed little discs, that is not suprising and was already discussed.

**UPD: **It seems that the reasonable way to introduce the cyclic structure on a homotopy 2-category is to define an involution – reversing – on 1-morphisms and appropriate compatibility morphisms. One should follow the pattern given by the example above.

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