\documentclass{amsart}
\usepackage[all]{xy}
\usepackage{pstricks}
\usepackage{amssymb}

\newcommand{\fieldk}{\mathbf{k}}
\newcommand{\cC}{\mathbb{C}}
\newcommand{\Cc}{\mathcal{C}}
\newcommand{\dD}{\mathbb{D}}
\newcommand{\Dd}{\mathcal{D}}
\newcommand{\delby}[1]{\frac{\del}{\del #1}}
\newcommand{\gGg}{\mathfrak{g}}
\newcommand{\del}{\partial}
\newcommand{\im}{\mathrm{im}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\fF}{\mathbb{F}}
\newcommand{\Ll}{\mathcal{L}}
\newcommand{\Mm}{\mathcal{M}}
\newcommand{\pP}{\mathbb{P}}
\newcommand{\qQ}{\mathbb{Q}}
\newcommand{\rR}{\mathbb{R}}
\newcommand{\sS}{\mathbb{S}}
\newcommand{\twoby}[4]{\left[\begin{array}{cc}#1 &#2\\#3&#4\end{array}\right]}
\newcommand{\zZ}{\mathbb{Z}}
\newcommand{\opf}{\bigcirc\!\!\!\!\!\!\!\pitchfork}
\newcommand{\wh}{[unintelligible]}

\newtheorem{defi}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{conj}{Conjecture}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{problem}{Problem}[section]
\newtheorem{prop}{Proposition}[section]
%TBD numbering
\title{Geometry/Physics Seminar, Spring 2012}
\author{Gabriel C. Drummond-Cole}

\begin{document}
\maketitle
\section{Dmitri Pavlov, Ordinary Differential cohomology via smooth topological field theories}

I would like to express ordinary differential cohomology of a smooth manifold $X$ as isomorphism classes of smooth $n$-dimensional field theories over $X$.  This statement involves a lot of complicated machinery. It would be best to explain the easy one dimensional case first, discrete, smooth, and homotopy being three separate flavors, which correspond to different kinds of line bundles over $X$ (flat, with connection, and arbitrary). In the second part we will discuss generalizing these three types to arbitrary dimensions.  The goal is to categorify these. The smooth one will have to do with differential cohomology.

Maybe it would make sense to start with some motivation.  The motivation is the cobordism hypothesis, which of course is now a theorem due to Hopkins and Lurie. I will only need the easiest version, with framed manifolds. If $X$ is a space and $C$ an $(\infty, n)$-category, which has morphisms at all levels and only invertible morphisms above level $n$, and I assume that all objects and morphisms and so on are dualizable, (I should have said symmetric monoidal), for instance for vector spaces it is the finite dimensional spaces, we can compute the topological field theories over $X$, $Fun^{\otimes}(Bord_n^{fr}(X),C)\cong Maps(X,k(C))$, where $k$ is the core of $C$, the infinity category you obtain by discarding non-invertible morphisms.

For example, we can take $n=1$, and if the category is one object and morphisms $U(1)$, then you'll get flat line bundles.

This statement explicitly allows us to compute the discrete and homotopy case, but not the smooth case, because that requires the smooth structure of the manifold, whereas this version of the cobordism hypothesis involves only the topological structure. Most of the talk will be about explaining how to make this smooth. If you have things varying smoothly on one side, then they should also vary smoothly on the other; that's very imprecise. I'll say more later.

I have to say something non-rigorous which will help to describe what's going on.

\begin{conj}
If $X$ is a smooth manifold and $C$ is the same as before, but now fibered over smooth manifolds $MAN$, then we would like to have some statement of the form:
\[Fun^\otimes (Bord^{fr,smooth}_n(X),C)\cong \text{``differential refinement of }Maps_{smooth}(X,k(C))\text{''}\]
\end{conj}
I cannot really define this object for you on the right hand. We can say what it is in some easy cases, for example refinements of categorifications of line bundles (bundle $n$-gerbes) should be bundle $n$-gerbes with connection.

Let's try to develop more or less rigorously some simple examples.  Before I describe differential refinements, I would like to say a few words about the familiar case, the Atiyah-Siegel style topological field theories.  So let me say some words about discrete $1$-TFTs?

\begin{defi}
If $X$ is a space, then a $1$-TFT over $X$ (I should say from the start, my target will be always very simple category, with only one morphism at every level but an Abelian group at level $1$, so $B^n(A)$ for some Abelian group $A$) is a functor $\pi_{\le 1}(X)\to BU(1)^{disc}$.  The fundamental groupoid has objects points and morphisms homotopy classes of paths.  The category on the right has one object and $U(1)$ morphisms, but with no topology, viewed discretely.
\end{defi}
There are a couple of problems. The target here is not rich enough to give flat line bundles. I will make a simplifying assumption.
\begin{remark}
All categories and functors and so on is a stack with respect to $X$, so we can assume that $X$ is diffeomorphic to $\rR^n$. Using the machinery of stacks we can get the statement for arbitrary $X$.
\end{remark}
\begin{prop}
Isomorphism classes of $1$-TFTs are the same as isomorphism classes of flat line bundles over $X$.
\end{prop}
Why is this true actually? Let's define a map from right to left. The way you turn a flat line bundle into a field theory is, you have to send all of the objects to the single object in $BU(1)^{disc}$. We have a homotopy class of paths, but since the line bundle is flat, parallel transport is invariant under homotopy, and so parallel transport gives an element of $U(1)$.

Let's end the first part here, and let's talk about removing the flat condition.  We'd expect that the definition would be pretty similar. You can still get a one dimensional $TFT$ over $X$, you can take a smooth path to its parallel trasport. How can we make this precise so that the statement is true replacing discrete with smooth?  This is well-known. The earliest reference I know is due to Dan Freed, 1992.

Let's follow the same pattern. If $X$ is a manifold, then a smooth $1$-dimensional $TFT$ over $X$ is a functor
\[1-Bord_X^{smooth}\to BU(1)^{smooth}\]
What kinds of objects are these? They are not ordinary categories. Because of the smoothness condition, we have to do something more complicated, with Grothiendieck fibrations. I will not try to give a precise description of a Grothiendieck fibration. This category, $1-Bord_X^{smooth}$, what does it mean to specify such a category? For every smooth manifold $S$ in $MAN$, we have $1-Bord_X^{smooth}(S)$ (this is an ordinary category) and for a map $S\to T$ we get a functor, a pullback functor $1-Bord_X^{smooth}(T)\to 1-Bord_X^{smooth}(S)$.  This amounts to saying you have a functor $MAN^{op}\to CAT$. You have to use the appropriate notion of functor. Let's describe what those categories and functors are.

$1-Bord_X^{smooth}(S)$ has as objects, you should think about them, over each point in $S$ you have something that looks like $\pi_{\le 1}X$, that is, $1-Bord_X^{disc}$, and similarly for morhpisms.  The objects are maps $S\to X$, smooth, think about it as, over every point of $S$ you have an object, a map from a point to $X$. The morphisms, similarly, they are slightly more difficult, but not that difficult. A morphism between two maps $f$ and $g$ from $S$ to $X$, a morphism $f\to g$ is the following data. Take a bundle over $S$ with fibers intervals, and a map $E\to X$ such that if we restrict to one endpoint we get $f$ and for the other $g$.

If you have two paths and compose them in the obvious way, the composition does not appear to be smooth.  This can be resolved in two ways.  One can allow piecewise smooth morphisms, or require that points on the boundary of the interval have a small interval which extends a little past the point, and then the neighborhoods must agree, and the neighborhood is now part of the object. You require that the little germs agree, which gives you a smooth interval.  You can also require them to be constant near the endpoints.  Again, this is all pretty much irrelevant, so I'd rather stick with the simple version, but you can make it precise if only we had more time.

If we have a map $S\stackrel{f}{\to} T$, then a functor between such categories $1-Bord_X^{smooth}(T)\to 1-Bord_X^{smooth}(S)$ is just the pullback along $f$.  One way to interpret informally is thatin this setting, the discrete setting, you can say that objects are maps from points to $X$, but you think of the objects as a manifold. Say that the objects form a smooth manifold and so do the morphisms.  When you talk about functors, they are maps on objects and morphisms, and those also have to be smooth maps.  Once we give a smooth topology, we'll get a very different answer than in the discrete case. 

I guess I have to erase something.

[Are you trying to describe conditions on the holonomy given as a functor that will determine a smooth line bundle?]

Yes.

Once we have this definition, and strictly speaking I also have to define the target category, so what is $BU(1)^{smooth}$? It's pretty easy to describe.  So $BU(1)^{smooth}$ has, fixing a smooth manifold $S$, there is only one object, and the morphisms are smooth maps $S\to U(1)$. Then we can say that a functor between such fibered categories, a functor between such categories is a natural transformation. For morphisms, for every manifold $S$, for any morphism we get a bundle of intervals over $S$, so you should get a smooth map $S\to U(1)$ in the target category. Intuitively, you think of the value of the morphism, the family of paths as a parallel transport functor, the requirement is that if you have a family of intervals varying smoothly over some base, then the parallel transport also varies smoothly over $S$.  There is no reason why this would be true in the discrete case. You'd only get this for fibered field theories.

\begin{thm}(Freed)\\
\[\pi_0(1-TFT^{smooth}(X)\cong \text{isomorphism classes of line bundles with connection}\]
These are also called first differential cohomology of $X$.
\end{thm}
Again, I suppress some technical details.  Strictly speaking, I should be taking isomorphism classes of $E$ modulo some equivalence relation, which again is not very important. 

[Is that equivalent because parallel transport depends only on the path.]

Yes. Also, on constant paths you get trivial parallel transport.

I don't have much time left. I hope the essential idea of the approach is clear. Let me very briefly describe the rest.

Let me describe how you get line bundles without connection, which is the third part of the outline.

We know how to get flat line bundles and line bundles with connection.  The remaining case is how to get line bundles without any connection.  This is given by homotopy topological field theories.  What is that? Again, I use the same structure
\begin{defi}
Let $X$ be a space. Then a one dimensional homotopy TFT over $X$ is a functor $1-Bord_X^{homotopy}\to BU(1)^{homotopy}$, so let's describe those.
\end{defi}
Those are not ordinary categories but something slightly different. If you have two objects, morphisms between them are a set, but here we want morphisms to form a topological space. I don't want to get into the formalism of infinity categories, so I'll pretend these are categories enriched in topological spaces. The objects on the left are points in $X$, and morphisms are paths between the points. There is one object of $BU(1)^{homotopy}$, and the morphisms are $U(1)$, viewed as a topological space.
\begin{prop}
\[\pi_0(1-TFT_X^{homotopy})=\text{isomorphism classes of line bundles over }X\]
\end{prop}

[Does such a thing exist at all? Holonomy on all continuous paths that composes?]

With the cobordism hypothesis, this can be computed as $[X,BU(1)]=H^1(X,U(1))=H^2(X,\zZ)$.

I don't require that parallel transport be compatible with composition, but only up to homotopy.

Let me finish the talk by erasing another blackboard and then summarizing what we have done so far. 

Summary: flat bundles are classified by $H^1(X,U(1)^{disc})$, and these give bundles with connection, and of course there is an obvious map that forgets the connection, so you get maps:
\[\xymatrix{
&H^1(X,U(1)^{disc})\ar[dr]\ar[rr]&&H^1(X,U(1)^{top})\ar[dr]
\\
H^1(X,\rR^{disc}\ar[ur]\ar[dr]&
&\hat{H}^1X\ar[ur]\ar[dr]^{curv}&&H^2(X,\rR^{disc})\\
&\Omega^1X/\Omega^1_\zZ X\ar[ur]&&\Omega^2_\zZ X\ar[ur]
}\]
These diagonals are exact and everything commutes.  A theorem due to Simons and Sullivan say that this diagram determines $\hat{H}^1$.  The diagram can be generalised to arbitrary dimension.  Set $1$ to be equal to $n$ and replace everything appropriately.

\[\xymatrix{
&H^n(X,U(1)^{disc})\ar[dr]\ar[rr]&&H^n(X,U(1)^{top})\ar[dr]
\\
H^n(X,\rR^{disc}\ar[ur]\ar[dr]&
&\hat{H}^nX\ar[ur]\ar[dr]^{curv}&&H^{n+1}(X,\rR^{disc})\\
&\Omega^nX/\Omega^n_\zZ X\ar[ur]&&\Omega^{n+1}_\zZ X\ar[ur]
}\]
\begin{thm}(Berwick-Evans, Pavlov,Stolz, Teichner)\\
This diagram, for any $n$, admits an interpretation in terms of topological field theories of those three flavors with those definitions generalized, and provides a description of differential cohomology as described in the dimension one case.
\end{thm}

[A functor in the homotopy case is in a functor?] No, it's a homotopy functor. In the smooth case it's an ordinary category.

\section{Daniel Pomerleano: Positive output two dimensional field theories}

\begin{defi} $A$ is a {\em smooth dga} over a field of characteristic zero if $A$ is perfect as an $A$-$A$ bimodule, that is, $A$ is in the smallest idempotent closed triangulated category of $A$-$A$ bimodules generated by $A\otimes A^{op}$.
\end{defi}
One thing to notice is that, if $M$ is perfect as an $A$-$A$ bimodule, then you have the following isomorphism (using $A^e$ for $A\otimes A^{op}$):
\[A\otimes_{A^e} M=HH_*(A,M)\cong Hom_{A^e}(Hom(M,A^{e}),A)\]
In particular, if $M=A$, you'll get an element in the Hom complex, and well, Hochschild homology has a circle action given by the Connes operator $B$ and the homotopy fixed points of that circle action $HH_*(A,A)\to HH_*(A,A)$ is, well, negative cyclic homology is the homotopy invariants. 
\[HC^-(A):= (HH_*(A,A)[[u]],b+uB)\]
There's a map $\pi_*$ from $HC^-(A)$ to $HH_*(A)$ (induced by $u=0$)
\begin{defi}$A$ has an $n$-Calabi-Yau structure if there exists an element in $HC^-(A)$, $\alpha$, such that $\pi_*(\alpha)$ induces a quasiisomorphism from $Hom_{A^e}(A,A^e)\to A[u]$
\end{defi}
The theorem, which is not mine, is
\begin{thm}(Lurie)\\
Given such a structure, we get a two dimensional field theory over the PROP $C_*(\Mm_{m,n})$ where the number of outputs $n$ is bigger than zero.
\end{thm}
One example, possibly a richer structure, is string topology.  If $M$ is a compact oriented manifold, then $A=C_*(\Omega M,\qQ)$, and so $HH_*(A)\cong H_*(LM)$, and you have a map $H_*(M)\to H_*(LM)$, and you want the fundamental class $\pi_*(\alpha)$.

This is all background. What I want to talk about is, can we deform, this will be formulated vaguely right now, can we deform $C_*(\Omega M)$ so that we can obtain a field theory over $C_*(\Mm_{m,n})$ with no restrictions on $n$ or $m$.  We'll make the question more precise by restricting $M$ to a nice class in terms of its rational homotopy type.
\begin{defi}
The rational homotopy type of simply connected $M$ is {\em pure} if $C^*(M,\qQ)$ has a model of the following form:\[C^*(M,\qQ)=\qQ[u_1,\ldots u_\ell]\otimes \Lambda[\epsilon_1,\ldots \epsilon_k]\]
where $d\epsilon_i=f_i(u_j)$ with degree at least two and $du_j=0$.
\end{defi}
If you do a little rational homotopy theory, you realize that the Homology of the chains on the based loop space is a Clifford algebra over $\qQ[v_1,\ldots, v_k]$, and you can apply a perturbation lemma to see that the higher operations have the property that $m_n(w,\alpha_1,\ldots, \alpha_n)=0$ if $w$ is in $\qQ[v_1,\ldots, v_k]$, which tells you that these functions $wt$ satisfy the Maurer Cartan equation in $HH^*(A,A)[[t]]$.

So you get a deformation, a curved deformation, from it.

I was going to give an example that might be familiar. Look at $\Lambda(\epsilon_1,\ldots, \epsilon_k)$, then $C_*(\Omega M)\cong \qQ[v_1,\ldots, v_k]$, and then you can consider any function, and you get a category which is the category of matrix factorizations $MF(\qQ[v_1,\ldots, v_k],w)$.  Here you have projective modules, $P_0$ and $P_1$ with maps in each direction, and composition in each direction is multiplication by $w$. 
\begin{thm}
The following are equivalent:
\begin{itemize}
\item $w$ has isolated singularities.
\item $HH^*$ is finite dimensional.
\item $MF(\qQ[v_1,\ldots, v_k],w)$ is smooth, proper, and Calabi-Yau.
\end{itemize}
\end{thm}
$HH^0(MF)$ is the Jacobian ring of $W$.  The rest of Hochschild cohomology is a module over this guy.

You can take coherent sheaves on $\qQ[v_1,\ldots, v_k]/w$ and mod out by perfect sheaves (in the sense we had earlier) on the same thing. This equivalent to matrix factorizations via the cokernel.

We will define a category $Pre MF(A,w)$ as ($A[\epsilon],d\epsilon=w$)-modules which are perfect over $A$.
\begin{defi}
Define $MF(A,w)$ as $Pre MF (A,w)$ modulo $Perf(A[\epsilon],d\epsilon=w)$.
\end{defi}
The theorem for these guys is that 
\begin{thm}
the Hochschild cohomology $HH^*(MF(A,w))$ is finite dimensional if and only if $MF(A,w)$ is smooth, proper, and Calabi-Yau.
\end{thm}
Showing the sufficiency is good because the left hand side can actually be computed.  For example, if $M=\cC\pP^n$, then your model is \[\qQ[u]\otimes \Lambda[\epsilon],d\epsilon=u^{n+1}\]
Then $C_*(\Omega M)$ is $\qQ[v]\otimes \Lambda(\gamma)$ with $m_{n+1}(\gamma,\ldots,\gamma)=v$, and $w=v^d$.  This is an example, you can check that for every $d$ you geta  TQFT which has an interpretation in terms of Floer theory and the Fukaya category, but I'll stop here.

[Can you explain how the deformation gives a TQFT?]

It satisfies all the conditions to be fully dualizable.

[Is there something to do with the fundamental chain of the moduli of holomorphic disks?]

The relationship with Floer theory that I have in mind is that a compactification of the cotangent bundle gives something, and there are reasonable compactifications for the cotangent of $\pP^n$.
\section{Dennis Sullivan: Solved and unsolved puzzles in the FRG's focus on 2D field theories}
I want to state four problems which are precise, although my presentation may not be. We may have to discuss it more. This meeting is using a lot the language of field theories. I wanted to concentrate on two dimensional field theories.  There's always been this mixing of language that has confused and puzzled me. I'll start by making a comment about that. 

Two dimensional field theories, I'm concentrating on those, can be caricatured, this talk will be a caricature of a math talk anways, by interactions of strings, if you have such a thing, even in its extended form, the boundaries are circles or arcs, and then you have algebraic structures on the surfaces with these boundaries. You can think of the surface as circles and arcs being swept out, and they interact and come out the other end. I remember fifteen years ago, physicists would draw pictures of string theory, where a string moves around in space, in this language, the vector space or thing that you attach to the circle, an element in that thing attaches a state to the circle. Think of the state as position in a manifold, it's moving in time according to some unknown evolution, you have more than one, and two come close together and maybe interact. Call that A. Or maybe you have a string that comes close to itself and breaks in two. Call that B.

Then you could also have non-closed strings, then I might change to open strings, and you could have them touch at the endpoints and come together, or they could break apart in a similar way. Call these C and D.

There are many more. You can mix these, have a non-closed [sic] string interacting with an open string and coming to an open string, or an open string breaking off a little closed string.  These are E and F.

For example, let's talk about A. This one is very homogeneous. No matter where the interaction took place, there's no boundary.

Oh, I wanted to say, ``Ho'' can be completed to homotopic, homologous, or holomorphic. One can imagine discussions where these words are of primary importance. I don't want to say much about this, except to say the strings, in algebraic geometry, there's a version of talking about closed strings in a holomorphic context. Drinfel'd and Beilenson talk about it in dimension two and I think Kevin's extending it to higher dimensions. I"ll be mainly thinking in homotopic and homologous, and mostly in homologous.

So comments on A. It has no boundary. This is kind of an algebra, two things combining. This is not associative, it satisfies Jacobi, it's related to a Lie algebra structure. You might think, how does this satisfy Jacobi? Three pieces giving zero? You have to interact three strings to satisfy Jacobi. The third string interacts, and you get six possible terms which cancel in pairs. You should draw this picture and then try to make it concrete.

Okay, this structure is not literally in Daniel's statement of Lurie's theorem, but this is an equivariant version. By the nature of Lurie's theorem, if these strings are moving in a manifold---

\begin{defi}
String topology is all conceivable operations where you interpret states as families of strings and perform classical intersection procedures to make things make sense, interpreted in algebraic topology.
\end{defi}

There are a couple of general things I forgot to say. For F, a non-closed theory gives you closed strings.

According to Lurie's machine, this A is homotopy invariant. That's good news in low dimensions, that's powerful in two dimensions and three dimensions, since most things in those dimensions is homotopy invariant. For example, Chas and Krongold showed that this Lie algebra solves the disjunction problem. Suppose you have two complicated curves on a surface. You can make them disjoint if the brackets of all powers of the first with all powers of the second are zero.

On a surface, curves automatically interact or not. This is a non-trivial theorem.

In three dimensions, you can use the structure of this Lie algebra (this is Chas-Gadgil), interacting with geometrization, you can read the number of pieces and decomposition sites using this Lie algebra.

B has a boundary.  You can imagine tiny interactions, and then you get no interaction. One piece gets smaller and then disappears. This is a kind of boundary in the middle of the process. We'll call this boundary an ``anomaly.''

Let's remember it has a boundary. The operation $B$ has a boundary. It satisfies a coJacobi identity dual to the other thing. There's a way to get rid of the boundary by modding out by the small loops. Then algebraically you repress the boundary and get something defined modulo fixed points of the circle action. The two operations satisfy the Drinfel'd Lie bialgebra identity and another, called the involutive identity. These identities correspond to studying the four surfaces of Euler characteristic $-2$. The point I'm dealing now is irrelevant, but when you deal with the anomaly, you can add pieces at infinity, and you get these four relations.

The first problem can be stated now.
\begin{problem}Is this combined structure {\em not} a homotopy invariant?
\end{problem}
This is an involutive Lie bialgebra. For surfaces, this was done (without involutive) by Goldman and Turaev.

One piece of this is described carefully in Sullivan, more is in Kate Poirier's thesis, and there's a natural compactification germane to this problem, and joint work of Kate Poirier with Nathaniel Rounds and Gabriel C. Drummond-Cole is describing this.
\begin{problem}Describe the relevant compactification of moduli spaces that describes (universal) string interactions.
\end{problem}
You'd like to hide as much boundary as you can. I'm stating this for closed strings, you could also do it for open strings. What can we say about that? For C I didn't say enough before, it's natural to think of the open strings as running between two things.

The compactification will have limit points where you go toward Deligne-Mumford points while still keeping a harmonic function. You can intuitively see that it acts in string topology.

So C. C has no boundary, no new boundary. It's associative. If you get technical, you might have to do an $\infty$ version of that, and maybe you have to do a category version, let's call it associative.  There have been many papers written, Graeme Segal had this idea that non-closed strings should be morphisms in a category, this leads to categorical language, then many people have done many things about this category. In particular, Kevin (Costello) explained how when you have non-closed strings, where does this operation come from, if you take a Hochschild construction, you get the closed strings. If you take the associative algebra and take a Hochschild, you get a Lie algebra, and so $A$ comes in directly out of $C$, thanks to Kevin. Did you think of that yourself?

[Kevin: I learned a lot from Kontsevich.]

[The cobracket has an anomaly? Does that have an interpretation?] I think that's work to be done. Let's say I don't know, I have some things I could say, but, [unintelligible].

Kevin says that the category, if you take the non-closed strings, that forms a category, and [missed].

I have another list of names here. There's a solved problem, if you put yourself in the appropriate setting, and assume that the equivariance of the circle action can be deformed to the identity, then you can see that the theory extends to the full, Deligne-Mumford compactification. So this leads to Hochschild, et cetera.

Now, D, let's discuss D. I've always wondered, you get a category here. Do you guys talk about cocategories? D is a coalgebra, maybe sometimes you don't have to say the co word if you use duality.  So D is coassociative.  It's like a coalgebra, this was a Lie algebra, Lie coalgebra, associative algebra, and now we have the associative coalgebra, but now it has boundary. Whatever made this string cut, it could be the end of your surface was like this, your cut may happen very near the boundary. There's some special circumstance you have a cut too near the boundary. I've been puzzling about this for years in the very simple setting, where the based loop space is a non-closed string where the two ends agree. So you should think of this as part of non-closed theory. Just this case is interesting, it would be a free loop that intersects itself near the ends. You can't do the trick of modding out by short things, because they are the unit.

Now the anomaly of D has been treated, after a long interlude, using transversal boundary conditions. You say, if they touch the boundary condition again, they're transversal. So I do these operations in the new space of strings, cutting out the infinite but finite codimension subset that you don't want. The algebraic topology changes, and you can still do operations. I'm hoping that when you do these operations on geometrically restricted strings, you get invariants, the dream is to get four-manifold invariants.

Basically, three manifolds are determined by their homotopy types, except for lens spaces and formal things made up out of them. These have played an interesting role in topology, so for example $L(7,1)$ and $L(7,2)$ have the same homotopy type ($L(p,q)$ and $L(p',q')$, the fundamental group gives you $p$ and if $q$ and $q'$ differ by something with a square root, you get isomorphic cohomology rings). So these are homotopy equivalent but not homeomorphic. 

This was a problem we posed, is string topology a homotopy invariant? This modified version is not. What Somnath does, you take the space of all these transversal strings, several intersections, you can't remove a transversal intersection.

You want to get this coalgebra structure. You take the chains on this space, you have the coalgebra $\Delta$, and on a chain, it's the alternating sum of its splittings. You make a differential which is the sum of the internal differential on the chains, plus the alternating sum of resolution, and you take the sum over all ways of avoiding each bad point. You get a differential coalgebra out of the open strings, and then step two, you have to take a fiber construction and you take the cobar construction on this, and it turns out you kind of find the Pontrjagn ring of the based loop space of the compliment, twisted by $\alpha$, the element of $H_1$ in the loop space. You use rational homotopy theory ideas here to find Massey product structures in covers of $L(7,1)$ and $L(7,2)$, and one of these has trivial Massey products and the other doesn't. This is based on the work of Salvatore and Longoni showing that these have different Massey product structures.

There's an interesting codimension two example, I'll do this fast so I can state the other two problems, this is joint with Michael Sullivan, using Somnath and Jason McGibbon. If you look at all the spheres, the top homology of the zero sphere is $\zZ\oplus\zZ$. You'll add these two ways of resolving a codimension two singularity. You take the cobar as Somnath does, but with this notion of resolution, getting a differential graded coalgebra, and then you get an algebra, it doesn't have a unit because of the twisting, and you get the twisted group ring of the complement, applying this to knots in $S^3$, and then (that was paper one) you add a unit and put the meridian in the center, adding a relation, an algebraic construction out of D, and this is isomorphic to the Lenny Ng coalgebra, produced out of the knot complement, and we get this chord algebra out of a functorial operation in the string theory. He was able to distinguish all knots up to 11 crossings. It's also isomorphic to degree zero Legendrian knot homology, [lost]. So that's a powerful knot invariant.

For the second paper, the Costello idoa is that somehow the open strings determine the Lie algebra of closed strings, and applying that to this type of algebra, using C, resolving, and one finds the Lie algebra of the complement, and then by Chas-Gadgil argument, you can read off the geometric decomposition of the complement. One can read that off by applying Hochschild to the combined structure.  That leads me to:
\begin{problem}
Describe the algebraic structure of this transversal open string theory. You can get rid of the anomaly and the answer is different.
\end{problem}
When you state a problem like this, the problem is to formulate the problem. What Jacob Lurie did with this category theory, he found the structure of fully dualizable objects. To describe what these structures might be, as a conjecture, there are many instances of this thing I just mentioned, the Ng chord algebra which is something you get out of looking at the cotangent to $S^3$ and $J$-holomorphic curves, you have a general situation where taking all the possible constructions, this is some sort of real analogue of $J$-holomorphic curves. When you study open sypmlectic manifolds with contact boundary, and look at the $J$-holomorphic curves there, and the compactification is almost the same, here there is a function which is superharmonic. There are minima but no maxima. You use the symplectic filling of the contact manifold. I don't know the analogue of the transversality.
\begin{problem}
Formulate conjectures relating the naive real analogue string topology corresponding to $J$-holomorphic curves.
\end{problem}

\end{document}