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\title{RTG Seminar\\January 29, 2006\\
Eric Harrelson}
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So I thought I would continue with last week's discussion by giving an introduction to the open closed case.  There are open closed versions of both of these moduli spaces we talked about.  I'm just going to give an intro and look at an open closed TQFT and move on to applications.

So an open-closed version of Segal's PROP.  We had Riemann surfaces without boundary and with holomorphic holes we can sew together to get another Riemann surface.  So let $\widehat{OC}$ be the moduli space of Riemann surfaces with boundary with labelled holomorphic holes just like we had last time, nonoverlapping like last week in the interior, and labelled half-holes on the boundary.  A typical point in the moduli space looks like this [Picture].  So there's some genus, and then there's boundary, and on the boundaries we have these half-holes, like this, and these are given in a similar way to the holomorphic holes, a biholomorphic map from the half-disk.  Then these are labelled.  Maybe we have one input, two output, two input, and so on.  Okay, so just like last week we can sew closed to closed, we can use the parameters of the half-holes to sew open to open.  The green things are closed and the red things are open.  This is a two-colored PROP, you can only compose when the colors match.  You can stretch this kind of thing out, and you get cobordisms from intervals and circles to intervals and circles.  So cobordisms between closed and open strings to closed and open strings.  I attempted to draw one of these guys, let me see if I can do that.

[Picture.]

In the open closed moduli space for the other moduli space I described, you get marked points on the interior and the boundary, with nodal singularities.

We want to look at the homology of this PROP, and we can restrict to the degree zero part.  We get an algebra over the degree zero part is the the same thing as an open-closed TQFT.  We can consider genus zero surfaces of this type.  We can do a lot more, but today let's just look at $H_0(\widehat{OC}).$  I should say, an algebra over a two-colored PROP or operad, maybe you can guess what it looks like, you have two vector spaces $V_C$ and $V_O$ and now elements of the PROP $P$ correspond to operations with inputs and outputs from both vector spaces, so you get $Hom(V_C^{\otimes i}\otimes V_O^{\otimes j},V_C^{\otimes k}\otimes V_O^{\otimes \ell}.$  This surface should give us an operation like this, mixed tensors.  There is a two-colored endomorphism operad.

[How do the permutations act?  Is it the same on the left and on the right?]

Well, this is not a totally general picture.

Let's give the generators for an algebra over this PROP.  We need to find enough of these surfaces so that you con build anything from them.  We have some similar stuff from last week, there is an orienteation on all of these.

So there is an open multiplication and comultiplication.  We have an open trace and unit.  So far the same pictures as last week, but note the open multiplication is not commutative, there is no homeomorphism that respects orientation and labels.

If we define the inner product $\langle a,b\rangle=tr(ab),$ now I can switch them, just rotate around, so this is symmetric.  All other relations which held in the closed case hold here as well.  So $V_O$ is a non-commutative associative finite dimensional Frobenius algebra.  So we still have $V_C$ and we have $V_O$ and now we will have maps between them.  We have a closed string breaking open, call this $\varphi_C:V_C\to V_O,$ and the opposite one, $\varphi_O:V_O\to V_C.$  So if we look at closed multiplication and then move closed to open, that will give us the same thing as doing both closed to open and then multiplying.  Also, $\varphi_C$ maps $V_C$ to the center of $V_O.$  Then there is another relation, the closed trace of $\varphi_O(a)b$ is the same as the open trace of $a\varphi_C(b)$

Define the operation $V_O\to V_O$ by permuting $\Delta$'s outputs and multiplying them.  So this is the same as $\pi=\varphi_C\varphi_O.$  That's it.  So this gives a complete description of algebras over $H_0(\widehat{OC}),$ that is, open-closed TQFTs.  These generators and relations are on Greg Moore's web page, they give all the generalizations and reference Lewellen for a proof of this using Morse theory.

I thought maybe we could stop here as far as looking at these goes.

Okay.  Part of my thesis was looking at genus zero open closed part, looking at the full homology, I have a full description of that with six generators and fourteen or fifteen relations.

[Is there an example?]

Yes.  Take a manifold, this is string topology.  You can also consider the case where you have at least one output always.  The relations change a little bit.  If $M$ is a smooth manifold, then the two spaces, if $K$ is a smooth oriented submanifold, then the homology of the free loop space and the homology of path space in $M,$ well $(H_*(LM),H_*(PM_K))$ is your $V_C, V_O.$  In string topology you don't have a trace.  

This last relation, this $\pi$ is intersecting transversally with $K$ and then intersecting the basepoints with $K.$

[Is it enough to take some subgroup of the center?]

How will you define $\varphi_C?$

I have two applications.

Let's go back to the closed case, let's just look at the closed version.  We had these cobordisms, and this is supposed to represent two strings going into one string.  There is an operation in string topology that takes two families of loops in $M$ that transversally intersects their special points, and then loop composes.  So we can say, consider maps of the pair of pants into $M.$  We can restrict to the inputs and get two loops in $M,\ LM\times LM,$ or the output and get $LM.$  We could replace the surface with its skeleton and get a figure eight, and consider maps of the figure eight into $M.$  Then this map has finite codimension in $LM\times LM.$  Then you can use the Pontrjagn-Thom construction to reverse the map and get a map to $H_*(LM)\otimes H_*(LM)\to H_*(LM)$ via the Kunneth formula.  

I just want to say what we're really doing.  We're taking two chains in $LM,$ intersecting them with the image of the map from the figure eights.  

If you have a finite codimension embedding $e:P^k\to N^{k+n},$ then let $V_e$ be a tubular neighborhood of the imago of $e(P^k).$  Then we can get a mapping from $N$ into the tubular neighborhood union one point.  Then you can identify the tubular neighborhood, the homology of $N$ goes to the homology of $V_e\cup \infty$ goes to $H_*(P)$ via the Thom isomorphism.  So here you can still do this with finite codimension, even though this is infinite dimensional.  The two mapping spaces [pair of pants and figure eight] are homotopy equivalent.  They don't have to be, I'm only using the figure eight.

We have the operad of these things, and we saw, the homology described BV algebras.  Can we get this structure on the homology of a loop space?  Think of several inputs and one ouput.  We have the cacti operad.  The cacti operad, $\Cc(n)$ is the configuration space of tree-like configurations of circles (lobes) of varying radii, and when three or more circles intersect, they are given a cyclic ordering in the plane.  There is one special point corresponding to a special marked point on one of the lobes.  Each of the circles have a special marked point.  You say where the circles intersect, and this allows you to draw it on the blackboard.  When I collapse this the output goes around all the inputs.

You can topologize this.  The composition gives you an operad.  Due to my advisor Voronov, the cacti operad is homotopy equivalent to $\hat{\mu}.$  We can realize the same structure using the cacti operad.


Finally what can we do?  We can consider the same setup but look at $L^{C(n)},$ a point in cocti and a map $C(M)\times LM^n\gets L^{C(n)}\to LM$  Now the theorem is that this is an operad action.   This implies atht the homology of $LM$ is a $BV$ algebra, in fact the one that Chas Sullivan saw.

I am talking now about the framed little disks operad, we have two dimensional disks embedded inside the two dimensional disk of radius one.  Each one comes with a frame and a direction.  This is a marking on the circle.  This is a subset of $\rR^{3n}\times (S^1)^n.$  If I compose, I take one of these, rotate it until it matches up and then glue it in.  That one is homotopy equivalent, it's the same operad, this $fD$ operad is homotopy equivalent to cacti or $\hat{M}_0.$  I put this up here because I want to look at the little disk operad.  Now you just have embedded disks, and this is, the homology of this describes Gerstenhaber algebras.  There are two operations, there is the space with two inputs, an element of $H_0(D_2)),$ this is a multiplication, and you can move one of these around the other, so you get something in degree one, and this is a Lie bracket $\{,\}.$  This is a derivation of multiplication.  This is a Gerstenhaber algebra, a Poisson algebra with a different grading.

Notice that there is a map $D\to fD$ which is an operad morphism, it respects the operad structure, it respects operad composition.  You get a map the other way which does not respect composition.  So every $BV$ algebra is a Gerstenhaber algebra, because if you have $H_*(D)\to H_*(fD)\to End_V.$  Here $\{a,b\}=\Delta(ab)-a(\Delta b)-(\Delta a)b.$

So Hochschild Cohomology, the cochain complex is the most important tool used in deforming associative algebras.  So $A$ is an associative algebra.  So $CH^*(A)$ where the $n$th space of this complex is the space of homomorphisms $A^{\otimes n}\to A,$ with operations $CH^n(A)\otimes CH^k(A)\to CH^{n+k-1}(A).$  This is the regular $\circle_i$ take $f$ and plug $g$ into the $i$ place in it.  We also have a cup product $CH^n(A)\otimes CH^k(A)\to CH^{n+k}(A)$ and so you can use the multiplication of $A,$ so you can multiply and get $fg.$  There is also $\delta$ which takes $$(\delta f) (a_1,\ldots, a_{n+1})=a_1 f(a_2,\ldots, a_{n+1})+f(a_1a_2,a_3\ldots,a_{n+1})+f(a_1,a_2a_3,\ldots, a_{n+1})+\ldots+f(a_1,a_2,\ldots, a_n)a_{n+1}$$

You sprinkle in the appropriate signs and can show that the boundary squared is zero, and you get its homology, the Hochschild cohomology.  So the multiplication is associative but not commutative on the chain level.  There is also $f\circ g,$ which is $\sum_i (-1)^{(i-1)(k+1)}f \circ_i g.$  There is also $\{f,g\}$ which is $f\circ g-(-1)^{(n-1)(k-1)}g\circ f.$

If you pass to cohomology, this is a graded commutative associative algebra, and the bracket becomes a Lie bracket and is a graded derivation of the cup product.  The individual $\circ_i$ and even the $\circ$ do not pass to cohomology.

Okay, so $\cup, \{,\}$ give operations on cohomology and give $HH^*(A)$ the structure of a Gerstenhaber algebra. This is an algebra over the homology of the little disks operad, with the direction of the grading reversed.

Deligne's conjecture says this raises to the level of chains.  The Hochschild chains are controlled by the chains of the little disk operad.  This is proven in several different ways.  I want to get into one way due to Ralph Kauffman.  I'll need to start putting stuff up.  So first of all, the chain model.  I just introduced the cacti, and there's another one which is equivalent to unframed little disks.  So first the main marked point coincides with the marked point on one of the circles.  You can attach at the marked point of a lobe.  It's always a tree.  These are called spineless, the spines are not allowed to attach except at the marked point.  So Kauffman shows that spineless cacti is homotopy equivalent to little disks, in particular the homology of this acts on Gerstenhaber algebras.  You get this by moving one lobe around another and then the second one around the first.  Corresponding to each picture you get a graph.  You start with a root, and the next one is your main marked point, and for every circle you have an open vertex, each connection gives you a closed vertex.

So we throw the root away and you get a bipartite graph.  It's a tree.  So these trees we can get a $CW$ decomposition of the space of spineless cacti indexed by these types of trees.  So we'll have a quasioperad because I need to consider that the cactus has a length of one.  These trees index the cells in a $CW$ decomposition of spineless cacti.

You take the differential of a cell by looking at a graphical representation and then collapsing angles.

[Example]

[After this I was distracted by the example.]

The constructed action of the cellular operad respects the differential and gives the standard structure on $HH^*(A)$ on the level of homology.

 
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