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\author{Gabriel C. Drummond-Cole}
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\title{RTG Seminar\\January 22, 2006\\
Eric Harrelson}
\begin{document}
\maketitle
[Leon made some proposals at the end of last term that I thought were pretty good.  He wanted to start with the Feynmann path integral in quantum mechanics (in phase and configuration spaces and holomorphic representation) and then the $S$-matrix, and then perturbation expansion and renormalization and gauge theories.  He also volunteered to start.]

Sasha and Segal talked about two dimensional TQFTS.  For every topological cobordism was assigned a homomorphism from $V^{\otimes m}$ to $V^{\otimes n}$  They mentioned that this thing was the same as a Frobenius algebra.  Now let's give these a conformal structure and look at the algebraic topology of these.  So let's just look at the homology, because the chains were overambitious.  What kinds of algebraic structures are on the homology of quantum field theories.

Let me take about fifteen minutes to introduce some language, of PROPs and operads, the objects used when the points in the space have inputs and outputs and you can compose to get other points in the space.

A PROP (named for PROduct-Permutation), your elements have multiple inputs and multiple outputs.  You should be thinking about the cobordisms.  In general you have more complicated spaces than sets.  Start with the canonical example, the endomorphism PROP.  Let $End_V,$ where $V$ is a space, topological, differential graded, vector space, whatever, and let $End_V(n,k)=Hom(V^{\otimes n},V^{\otimes k}),$ then we can take the product of two homomorphisms $End_V(n_2,k_1)\otimes End_V(n_2,k_2)\to End_V(n_1+n_2,k_1+k_2).$  As far as graphs go, this corresponds to disjoint union.  You can also compose, if the number of inputs match the number of outputs, you can go $End_V(n,k)\otimes End_V(k,m)\to End_V(n,m).$
Finally, there are actions on $End_V(n,k)$ by the symmetric groups $\Sigma_n$  and $\Sigma_k$ by permuting the outputs and the inputs.  You should think of these graphs as having labels, and then you can perform composition, and compose by permuting the labels.

To define a general PROP, you abstract these things.  A PROP is a set of spaces $P(n,k)$ with product maps from $P(n_1,k_1)\otimes P(n_2,k_2)\to P(n_1+n_2,k_1+k_2)$ and compositions $P(n,k)\otimes P(k,m)$ to $P(n,m)$ and actions of $\Sigma_n$ and $\Sigma_k$ on $P(n,k).$  This data should satisfy some properties inherent in the endomorphism PROP.  These properties are associativity properties, some equivariance of the symmetric actions, and an identity element of $P(1,1).$

Then you can define an algebra $V$ over a PROP $P$ is a PROP morphism $P\to End_V.$  You have a bunch of maps $P(n,k)\to Hom(V^{\otimes n}, V^{\otimes k})$ respecting the composition, product, action, and differential if you have that as part of your data.

We have that language to work with.  This is also referred to as an action of $P$ on $V.$  For the tree level structure, we have a similar notion called an operad, but we have only one output and several inputs.  We have the endomorphisms, in the operad $End_V,$ we have $End_V(n)=Hom(V^{\otimes n},V)$ and composition $End_V(n)\otimes End_V(k)\to End_V(n+k-1).$  So in an operad we have spaces $O(n)$ and compositions $\circ_i:O(n)\otimes O(k)\to O(n+k-1).$  These should be associative and equivariant with respect to the action.

Then we have a notion of an algebra over an operad which is an operad morphism to $End_V.$

One last thing.  Notice that if you have a PROP or an operad of topological spaces, then the homology of the PROP or operad (same with the chains) is automatically a PROP (operad) of vector spaces.  You get homology operations by applying the K\:unneth formula.

If a topological space $V$ is an algebra over a PROP or operad, then the homology of $V$ is an algebra over the homology of that PROP or operad.  We have an element of $O(n)$ which gives a map $V^n\to V$ and then we can apply homology and get a map $H(V^n)=(H(V))^n\to H(V).$

All right.  The last definition.  A PROP or operad of vector spaces with or without differential is said to describe a category of algebras if $V$ is in that category if and only if it is an algebra over the PROP or operad.  So it's not too hard, if you have a PROP of vector spaces, you can look at the structure the algebras over it have, and you have operations and relations.  The hard thing is to get the right number of operations and relations.

\ex
Before we get into the quantum field theory, let's look at the simplest example that's out there.  We'll look at a topological operad and its homology.  This is called the littel intervals operad $I.$

So define $I(n)$ is configuration space of $n$ nonoverlapping (endpoints can coincide) labelled closed intervals in the unit interval.  This can be identified with a subset of $\rR^{2n}$ by specifying the centers and radii of these intervals.  The composition is sort of obvious.  If we have another guy we shrink it down and insert it.  That's the little intervals operad.

\fact
The path components of $I(n)$ are contractible.  There is only homology in degree zero.  Furthermore, the number of path components are equal to the number of ways of permuting the lables, two things are in the same path component if and only if the intervals are labelled in the same order.

What kind of algebras are described by this operad?  You can see that all path components can be obtained from the one path component $\xymatrix{\bullet\ar@{-}[r]^1&\bullet\ar@{-}[r]^2&\bullet},$ which gives a multiplication $V\otimes V\to W$ which we will just give the notation $a\otimes b\to ab.$  It's easy to see that we have associativity, since the following are in the same path component:
$$\xymatrix{
	\bullet\ar@{-}[r]^1&\bullet\ar@{-}[r]^2&\bullet\ar@{-}[rr]^3&&\bullet\\
	\bullet\ar@{-}[rr]^1&&\bullet\ar@{-}[r]^2&\bullet\ar@{-}[r]^3&\bullet}
	$$
But if we build the same path component in two different ways, this can be explained by associativity.  So associativity is the only relation we need.  We have one generator and one relation, and this explains every relation which is satisfied in this operad.  The homology of $I$ describes associative algebras.

Let's get into the structure we're going to be looking at, or one of them.  This is the Segal PROP.  So let $\hat{M}(n,k)$ be the moduli space of Riemann surfaces, conformal isomorphism classes with $n+k$ nonoverlapping labelled holomorphic holes.  This thing has a topology.  Holomorphic holes mean, when we have the conformal structure we need a way to sew these together, boundary to boundary, it's the same kind of picture, we have cobordisms but now they have conformal structure.  These are holomorphic holes, meaning there are biholomorphic mappings from the standard unit disk in the complex plane into the surface, which extends holomorphically across the boundary.  This is so that there is a well-defined way to sew inputs to outputs and get another Riemann surface.  These don't have to be connected surfaces.

\begin{defi}So now we can define a CFT, a conformal field theory, as an algebra over $\hat{M}.$  For every point we get a multilinear operation, this is taken by physicists to be a Hilbert space or at least a topological vector space, and the morphism is taken to be continuous. 
\end{defi}\begin{defi}
a topological conformal field theory is an algebra over $C_*(\hat{M}).$
\end{defi}
No one knows what the homology of this PROP is, it's huge, so we have to look at certain substructures.  So first of all notice that the path components of this PROP are in bijection with the topological types of the underlying surfaces.  So the homeomorphism types are in bijection with the path components of this moduli space.  So a TQFT is an algebra over the degree zero homology of $\hat{M}.$  This is another PROP associated with a topological PROP, the degree zero part.

Let me look at this a little more carefully.  We're looking at topological types of surface.  We have a surface which gives us a multiplication, $V\otimes V\to V$.  We have a comultiplication $V\to V\otimes V$.  We also have the trace $V\to k$ and the unit $k\to V.$  It's associative and graded commutative (permuting inputs on the pair of pants gives a homeomorphic surface).  Multiplying with $e$ gives the unit multiplication, and if we define a bilinear product $\langle a,b\rangle=tr(ab)$ then associativity implies that this is invariant with respect to the product, $\langle ab,c\rangle=\langle a,bc\rangle.$  If you take the coproduct composed with the unit, and then compose with the bilinear product, we get the identity.  So $tr\otimes id (x\Delta e)=x,$ or $V\to^{id\otimes \Delta e} V\otimes V\otimes V\to^{\langle,\rangle}V$ is the identity.  Another way, $V\to \sum \langle v,u_i\rangle v_i.$  So this tells us, first of all, that $V$ is finite dimensional, since the $v_i$ span $V.$  Also, the bilinear product is nondegenerate because it has to be nonzero on some $u_i$ for any nonzero $v.$  This actually gives a complete description of the algebra of the degree zero part.

Let's go to higher homology.  Like I say, this space is too hard to compute the homology of the whole thing so we can just look at genus zero surfaces.

A tree level CFT can be defined as an algebra over the operad $\hat{M}_0,$ which consists of only genus zero connected Riemann surfaces with one output.  We want to look at the homology of this operad $H_*(\hat{M}).$  We still have the path component of the pair of pants in $H_0(\hat{M}_0(2)),$ which is a multiplication $a\otimes b\to ab.$  Take the cylinder and rotate around the parameterization $360^\circ,$ which is a map of the circle into this space.  This is in $H_1(\hat{M}_0(1)).$  This is a map $V\to V,$ called the BV operator, and then these two elements generate all the homology of $\hat{M}_0$ via operad composition.  We need relations on these.  I want to look at the following composition $\Delta(abc)$ (the multiplication is associative commutative.  The BV operator rotates around, and I have three inputs.  This will live in $H_1(\hat{M}_0(3)).$  It will be the sum of a bunch of homology classes.  We can break this up into three homology classes, like where three stays fixed and $1$ and $2$ switch places, so $\Delta(ab)c,$ and then the same thing with each of them, so $\Delta(ac)b$ and $a\Delta(bc).$  This doesn't quite do it, because each time we perform one of these, we're also rotating parameterizations around as we move these around.  We rotated the first one around twice, so we have to subtract off the terms from keeping two parameterizations fixed and rotating around one of them, so $\Delta(a)bc,a\Delta(b)c$ and $ab\Delta(c).$  We're in the graded commutative categary so there are actually signs here, it's the Koszul philosophy.  When you move things around that are graded you need to throw some signs in there.  

So a fancy way of saying this is that $\Delta$ is a second order derivation of the multiplication.

There's just one more thing, if you take $\Delta$ twice, what do you think you get?  If you rotate the inputs around on both of these, you end of with a two dimensional chain whose image is one dimensional, so it's degenerate and on the level of homology it vanishes.  This gives a complete description of algebras over $H_*(\hat{M}_0),$  Getzler extending the major work of Fred Cohen.  Cohen did the little disks operad and Getzler did the framed little disks operad, which is homotopy equivalent.

[Dennis: if you take the degree zero part of a conformal field theory, Rutgers people showed that this is the same as a vertex operator algebra.  Can you pass from this to a BV algebra?]

[Leon: You have a sheaf of vertex operator algebras, and then you can find the homology.]

I have more prepared than I can talk about.  I want to next look at a related moduli space, and look at the homology of that, so it's sort of the beginning of a new topic.

That's kind of the tree level story of the homology of this moduli space.  The fullest tree level structure is when you have multiple outputs, and you only compose one input and one output.  Those are called corollas.  

[What is the algebraic structure like when you have outputs as well?]

You can compose the multiplication and the comultiplication and that's the same as combining them in another order, $\Delta(ab)=a\Delta b=(\Delta a)b,$ where $\Delta$ is the comultiplication.  You don't need any new degree one generators, I always assume at least one output, and you just need to add the Frobenius relation.  You get the cocommutativity and coassociativity as well as the dual BV relations from the Frobenius relation.  Now you know what this is.

So now there's another related moduli space, there may be some confusion in the literature as to what should be called a homological field theory.  I called it this last thing.  Some other people call that an algebra over the operad (or PROP) that I am about to introduce.  Now we consider the moduli space $M_0(n)$ which is the moduli space of Riemann surfaces of genus zero with $n+1$ labelled marked points.  Now we don't have biholomorphic holes, we have points, and the last labelled point is the output.  Some people think of these as strings shrunk to a point.  There's also a moduli space where you consider a tangent direction at each point, which is homotopy equivalent to the last space.  This is a $2(n-2)$-real dimensional connected complex manifold.  We only consider $n\ge 2.$

So this by itself doesn't form an operad, but there's a compactification of this space which does form an operad.  This is called the Deligne-Mumford compactification.  Let me define it and then draw a picture that will make it clear what it is.  All right.  So this compactification is the space of isomorphism classes of stable complex curves of genus zero with $n+1$ labelled non-singular points.  This means 
\begin{enumerate}
\item There are a finite number of singularities, which are all double points.
\item Each irreducible component is a sphere which has at least three special (singular or marked) points
\item The dual graph is a tree. (a representative of the class is simply connected)
\end{enumerate}
Stable means there are only a finite number of automorphisms.  That's the second condition.  A typical point looks like this: [picture]

This is stable because each sphere has three or more markings.  Its dual graph is a tree.  You have a vertex for every sphere and an edge for every node connecting two things.  Then you have legs corresponding to the marked points.

\fact
$\bar{M}_0(n)$ is a closed manifold of dimension $2(n-2).$

[Is this related to blowing up singularities in an orbifold, like \wh?]

You can construct these spaces by blowups, but that's not quite what you were asking.

We can see that we get a nice sequence of manifolds.  We can see that these manifolds form an operad just by attaching two of these at marked points.

\begin{defi}An algebra over $H_*(\bar{M})$ is also sometimes called a homological field theory.  I've also seen it called topological gravity.
\end{defi}

All right, we want to discuss the homology of this operad.  First, just some general facts, there's a stratification of this space indexed by the dual trees.  So let $M(T)$ in $\bar{M}_0(n)$ be the subset of all points which have $T$ as dual graph.  One thing we can say is that the codimension of $M(T)$ is always twice the number of double points.  The closure $\overline{M(T)}$ contains $M(S)$ if and only if you can get $T$ from $S$ by collapsing internal edges.  In particular, the zero-dimensional strata correspond to graphs where each vertex has valence three and the top dimensional stratum is going to be the corolla, with no internal edges and codimension zero.  All right.  

\fact
The closure $\overline{M(T)}\cong \prod _{v\in \text{vert}(T)} \bar{M}(|v|).$  So in my example, the closure of the stratum corresponding to this tree is homeomorphic to $\bar{M}_0(4)\times \bar{M}_0(4)\times \bar{M}_0(3)\times\bar{M}_0(3).$

The conformal structure is given by the conformal structures of each of the pieces.  If you have $T\in T(n),$ a graph with $n$ legs, then $[\overline{M(T)}]$ denotes the homology class of in $H_{2(n-k-2)}$ where $k$ is the number of internal edges.

\begin{thm}(Sean Keel)\\
These cycles span the full homology of the manifold.  In particular, all the odd Betti numbers are zero.
\end{thm}
Since these are just cross-products of lower moduli spaces, this tells us that the fundamental classes of the $\bar{M}_0(n)$ generate $M_0(n)$ as an operad.  Then Kontsevich found some identities that are satisfied for these, and this gave a complete description of the homology of this operad.  Let $(x_1,\ldots, x_n)$ represent the operation associated to each fundamental class, from $V^{\otimes n}\to V.$  This is graded symmetric.  We're talking about $V$ being an algebra over the homology operad.  This is graded symmetric.  To every element of the homology we get such a thing.  This is a graded symmetric degree $2(n-2)$ operation.  They satisfy the following generalized associativity conditions.  If you have $a,b,c,x_1,\ldots,x_n$ then we have the following relation:
$$\sum \pm ((a,b,X_{S_1}),c,X_{S_2})=\sum \pm (a,(b,X_{S_1},c),X_{S_2})$$
where the sums are over all partitions of $\{1,\ldots, n\}$ into $S_1$ and $S_2.$  So $X_S$ is the set $x_s$ for $s\in S.$

This is an associative law with parameters.  So for $n=0$ this is $((ab)c)=(a(bc)),$ and the fundamental class corresponding to $M_0(2)$ gives you a graded commutative associative algebra.  Then $n=1$ looks like
$$(a,(bcd))+(a(bc)d)=((abd)c)+((ab)(cd).$$

Each one of these things will give us something in $M_4$ and they correspond to degree two homology classes.  I can show you pictures corresponding to these but it's not obvious that these correspond nicely.  This is due to Kontsevich and gives a complete description of algebras over $H_*(M_0).$

[Are these relations independent of one another?]

Yes.

This is called a WDVV algebra or a hypercommutative algebra.  This corresponds to Gromov-Witten or Calabi-Yau, using holomorphic Kodaira-Spencer.  That's what Manin is about.  The book discusses these algebras, together with the additional inner product that you're kind of ignoring.  There is another way of describing these with connections, and that's the description you need to get Calabi-Yau Kodaira-Spencer.  There are other examples having to do with singularities with algebraic varieties.  The whole book is about this but you're allowed gluing anywhere.

Good examples corresponding to the other one is the Chas-Sullivan loop space.  

[This thing up here, the $n$ to $1$ operation, what is it for $n=1?$]

There is no $n=1.$


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