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\author{Gabriel C. Drummond-Cole}
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\title{Graeme Segal\\December 7, 2006}
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It's very nice to be here, and I'm grateful for the invitation.  I'll be giving a variant of a talk I've given elsewhere.  I'd like to express my sympathy to those who have heard this before.

Quantum field theory is part of quantum mechanics, which I never found appealing.

Classical physics tells you what the world is made of, particles situated in space, which interact with fields, functions or whatever defined over space, and these things change over time, and you assume you know what time is.

This tells you that you have an infinite dimensional manifold of possible states, with its tangent bundle which tells you how this can change.  Then physics says this is a symplectic manifold which evolves by means of Hamiltonian mechanics.  Quantum mechanics doesn't discuss what the world is made of.  It starts with a configuration space on which you are doing Hamiltonian mechanics.  

You have a noncommutative algebra $A$ which maps by $\theta_t$ to $\cC.$ You start by recognizing this as being a deformation of a commutative algebra, which gives you a symplectic structure on tho state space which is a manifold.

Quantum field theory tells you this is not just an abstract algebra but something organized by space time.  So you start with a space-time manifold $X.$  We assume that for each point in here we have a vector space $O_x$ contained in the algebra $A$ of all observables.  These are things that can be observed at this point of space time.  Then quantum field theory is the idea that this is local.  The output is this bundle, as well as that they are meant to be contained in this $(A,\theta_t)$ system.  Then there is a $\theta$ which is a multilinear map $O_{x_1}\times\cdots\times O_{x_k}\to \cC.$  The traditional way of looking at this is to have these vector spaces and the multilinear functionals.  Presumably there will be some kind of homogeneity; as $x$ changes we imagine this changes only a little, if at all.  The world is then made up of things you observe.  This is an oversimplification.

So this is too flabby.  What kinds of properties should this have?  Let me mention one of the guiding ideas to axiomatize this.  You associate to the space-time $X$ something $\Phi_X,$ fields on $X.$  An example would be if $\Phi_X$ were smooth maps $(X:M)$ for some auxillary manifold $M.$  This could be some set of bundles instead or whatever.  The idea is that $O_x$ should be functions $\Phi_X\to \cC$ which depend only on the germ of $x.$  So for example, if $\Phi_X=\text{Maps}(X\to M)$ and $\varphi\in \Phi_X,$ with $f:M\to\cC.$  Then an example in $O_x$ would be $\varphi\mapsto f(\varphi(x)).$  

Now the mythology is that there is a probability measure on $\Phi,$ and $\theta(F_1,\ldots, F_k)$ should be $\int F_1(\varphi)\ldots F_k(\varphi)\Dd\mu(\varphi).$  What properties should this measure have?  That will put properties on the $\theta$s.  Look at the maps from $X$ to $M$ for simplicity.  If $M$ is Riemannian, we can evaluate $\varphi$ at each point and evaluate $d\varphi$ on it.  Then we could take a product.  This would be a white noise measure.  You want to concentrate this on a geometrically interesting part by multiplying by $e^{-S(\varphi)}$ for some geometrically interesting $S.$

You might take $$S(\varphi)=\frac{1}{2}\int \langle d\varphi\wedge*d\varphi\rangle=\frac{1}{2}\int_X ||D\varphi||^2 d\mu(x).$$
If this were just the integral of like $||\varphi||^2,$ this would just change the measure on some parts of $M.$  It would still be white noise.  But in quantum field theory, the basic point is, whether or not you expect to construct such a theory depends on the geometric behaviour of $S.$  If you can go down a gradient trajectory you expect to be able to construct this measure.

If $X$ is one-dimensional there is a good such measure, Weiner measure.  Without bubbling, in two dimensions, you can do things and get two dimensional sigma models.  The map breaks up like pieces of fishing net if you try to do things in three dimensions.  You will see the same kind of white noise as if you didn't have $e^S$ there at all.

Now, I, a long time ago, tried to give an axiomitization of this a long time ago.  I want to discuss the pros and cons of it.  So $d$ is the space time dimension.  So to each compact oriented closed $d-1$ dimensional manifold $Y$ we assigned in a functorial way a topological vector space $\Hh_Y.$  To each cobordism $X^d$ you assign a linear operator $U_X:\Hh_{Y_0}\to \Hh_{Y_1}$ which should be trace class (don't worry about it).

This should have only two properties.  One is a concatenation property, if you concatenate two cobordisms, then $U_{X'\cup X}=U_{X'}\circ U_X.$  The other is tensoring, $\Hh_{Y\amalg Y'}\cong \Hh_Y\otimes \Hh_{Y'}.$

Why does that give us anything like what we wanted.  From the tensoring property, $\Hh_\emptyset=\cC.$  Now suppose we have a closed $d$-dimensional space time, with some points, and let $X_0$ be the $X$ disks about those points.  Then $X$ minus these disks is a cobordism from the boundary of the disks to the empty set.

The natural thing to consider for $\Oo_x$ is the inverse limit of $\Hh_{\boundary D}$ for the inverse system of disks containing $x.$  The cobordism $U_{X_0}$ gives you maps $\Hh_{\boundary D_1}\otimes \cdots\otimes \Hh_{\boundary D_k}\to \cC.$

Is this good enough?  What does it help to answer?  In its first form, the quantum theory was for answering questions about nature.  How unique is this theory?  Can you define a moduli space of possible theories, how many parameters does it have?  These were thought of as rare.  It's a recent discovery that these even exist in dimension six.  If you're interested in string theory, you have to be interested in two dimensional quantum field theory (these are like Riemmanian geometry.  To put it crudely, the phase space of [\wh] should be a moduli space of two-dimensional quantum field theories.

One of the first questions is then about, if you are trying to introduce locality to quantum mechanics, have we done that?  When $x$ is contained in a disk, then $\Hh_{\boundary D}$ gives you $O_x,$ but you think that the $O_X$ will live inside $\Hh_{\boundary D}$  Think of this as being a Sobolev space, with $O_x=C^\infty(M).$  A better example would be to expand by propagating with a heat kernel.  This is metaphorical, it's a bit more complicated than that.

If we now maginify the picture and take a bouquet of points, because we could have put a smaller disk around those, when we have points in the interior of $D,$ we will get a map $O_{x_1}\otimes\cdots O_{x_k}\to \hat{O}_x$  This is into a sort of completion, defined for each point, this is a sort of operad composition law.  This is given by the cobordism from smaller disks out to the boundary.

The point is that $O_x$ is an algebra structure, but it's not an algebra, it has a bunch of operations parameterized by a bunch of points close together.

Of course in dimension two this is a vertex operator algebra, there just isn't a direct algebraic theory except in dimension two.

Kontsevich gave an algebraically-oriented talk about $D$-algebras and a bunch of stuff about their Hochschild homology.

In the physical literature, the operadic structure is called the operator product expansion.  Does this tell you enough?  I started with a whole manifold.  That picure was motivated by seeing what properties of the path integral, well, by what happens when you cut.  You can ask whether you can link these two together.  I just realized a philosophical point that I hadn't realized.  Is the electromagnetic field local?  In the 19th century, it was thought that this was a force exerted by electrically charged particles.  Already people had a field, and it was thought to be closed, and since everything was simply connected you got $F=dA.$  You had to check gauge invariance and $A$ was not supposed to be physical.   Then they did an experiment where an electron was made to go in a circle and was affected by the field.  The field was then instead a one-dimensional Hermitian line bundle with a connection.  Mathematicians know that this is local.  But in some sense it's not local because you have nothing to measure.  In a symplectic manifold you have two tangent vectors, you can measure their form.  If you think the vector bundle is really there, well, \wh

There are a few things I want to say.  The first is about local global things.  This pertains to work I did with Dusa McDuff long ago.  I mentioned the word operad.  I belong to the generation who was interested in these things topologically.

[At this point I was fatigued, my notes are spotty.]

We have the space $K.$  It's a $d$-fold loop space if whenever we have a $d$-dimensional manifald with points labeled by $k_i$ in $K,$ now we want to put a topology on this.  If we let the points move around, that's easy, but if they come together we would like to bubble out.  When points come together, there should be some way to put things together.  The union of the parts supported in the disks contains $K_k.$  Dusa and I proved that when you consider this configuration space, then it has the homotopy type of the space of maps of $S,$ a $d$-fold iterated classifying space, This is a local global theorem.  Suppose we wanted to describe maps to an arbitrary $M$ this way.  This would be $\text{Map}(X,M)\cong C(X,\Omega^d M)$  This is possible onl if it kills all data belaw $m.$
From th local description, when you want to look at it on the large, you have to put in $D$-branes.

I think that this is exactly, if you think about this it's like something that comes along in operator field theory.

I'd like to change to the top-down point of view, because my only contribution has been concerned with the question, we have $\Hh_{Y^{d-1}}.$  Then we ought to be able to answer, when we cut $Y_1,Y_2$ out of a bigger space, we would like to be able to amalgamate and reconstruct the space.  What happens quite frequently is that you can associate to the cut thing a ring $R_Z,$ and then $\Hh_{Y_i}$ is a right, left $R_Z$-module  It's an important point even if you're not interested in the subtleties of the formalism.  If you're doing physics, if things are a long way away you notice that you can forget about them.  If you think of this as modelling space time, then this would be a slice across all of time.  The idea that a basic Hilbert space should be associated to this slice is naive.  You have a small ball around you.  If you can't define $\Hh_{Y_1}$ then you would never be able to do quantum field theory.

We have fields on the boundary and we can look at $L^2$ functions on that, and we want to factorize this into two pieces.  I don't have time to tell you too much about it.  The picture that works is that usually you define $\Hh_Y$ as an irreducible representations of a certain Heisenberg group made up by translation and multiples of the dual.  You can consider the group of fields supported in a tubular neighborhood of the boundary.  You have to have two groups around it.  You have two, $g_\ell$ and $g_r,$ and you make a represeentation of their respctive groups.  We combine these, well, $\Hh_0$ would be for the tubular neighborhood.  This has an action of these three groups, and you can make $\Hh_{Y_1}\otimes_{\Hh_{Y_0}}\Hh_{Y_2}.$  Then you have \wh.  This you can look at in terms of a $d$ dimensional manifold.  Look at wings of a hinge that rotated around in a circle.

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