2 dimensional (T)(C)FT choose at least 1

Theo, Zach and Dan B-E at tea: February 2013

What does a TFT do to every topological surface?

It gives an invariant. Let us say that invariant lives in V.

What does a CFT do to every topological surface?

It gives a function on the moduli space of Riemann surfaces with that genus. The function will take value in some V.

What about a TCFT?

Well now we want a locally constant function on the moduli. In this way a lot of the dependence on conformal structure is lost. But large transformations, namely the mapping class group, can still do stuff. Another way to say that is it is a closed degree 0 form. But V might have a Q on it making it differential graded vector space.
So what we are really looking at is  

There are maps to TFT and CFT.

If you have a choice of volume form on the moduli stack you can integrate the CFT and TCFT results. This is what you do in perturbative string theory where you need to integrate out all the choices you made about conformal structure on the worldsheet.

Koszul Duality

I have a f.d. semisimple Lie algebra L. What is it's structure? I could give you a basis and structure constants and then you could check that that works.



Or I could give you this chain complex where the differential is as follows






This chain complex is generated in degree 1 so that is all I had to give you. If you compute  , you get Jacobi Identity. This is Chevalley-Eilenberg.

So the data of a Lie algebra is the same as a semifree differential graded-commutative algebra generated in degree 1.

Aside:
You can change 1 to 1-n to get a Lie n-algebra, but then you would need to specify the higher brackets because of the other generators. If you replace it with just n, you get a n-Lie algebra. Removing all restrictions on generators whatsover lands you in   .

Now let's flip the arrows and go the other way.

I have a free graded Lie algebra structure on something called SC^*. If you see the condition for it to be a complex, you see it forces C to be a commutative algebra.

Again we could play with where it is generated to get the rest of the E_n type structures instead of just Comm.

AGT back in 1996

http://arxiv.org/pdf/hep-th/9605193.pdf

Atiyah Singer III: The Witten Genus

The two maps on top are analytic and topological shriek maps.

If you mix up which of the analytic or topological maps you do, you get the index theorem in all it's forms.

Replace K theory with TMF and the A-hat with the Witten genus should give you a beefed up version.
I don't know which of the maps have been constructed, but this is where a punt to Stolz-Teichner and crew goes.

The idea is we know the Spectra which represent K theory in all it's flavors, BU and all that. If that TMF is also a generalized cohomology theory, what is it's spectrum. Insert the suspected one here.

credit talks given by Dan Berwick-Evans

Doplicher Roberts

The statement in Chari and Pressley Guide to Quantum Groups

Associated to every local quantum field theory in four or more spacetime dimensions is a compact group G whose representation ring is isomorphic to the fusion ring of the theory.

Also the appendix of
http://arxiv.org/pdf/math-ph/0602036v1.pdf

Need to unpack this into a post.

Ishibashi (as in without the k)

Consider the annulus.

You can view it as a segment rotated around to meet up with itself. Supposing boundary condition a and b on the outside and inside, what you are doing is calculating the effect of two boundary changing operators at 0 and at infinity.

If you calculate the partition function you get some linear combination of characters over all levels h showing up with some coefficients n.



where  is the

But you could also thing of this same situation as a closed loop moving from the a side to the b side in imaginary time . In this picture we need to calculate the matrix element  

But what a and b make sense to put there? We have the condition     in the weak sense on this space.
Look at n=0, this says that the state lives in  with no cross terms.

Keep looking at the others and you see an explicit expression



equal combinations over everything allowed with that h.

a and b have to be a linear combination of these.



Now let's get back to that matrix element.



That last two pieces is the matrix element between the Ishibashi states. But we see the argument of the character is the victim of an S transformation, so we can rewrite it in terms of the character we had in the first picture.



These are called the Cardy conditions. It is an positive integer condition so it is restrictive. The operator content at the boundary is constrained to fit this.

nKdV Hierarchies

I haven't had a new post in a while, so let's finish up some of the drafts.

http://math.berkeley.edu/~ilya/software2/tmp/Dickey-classical-W.pdf

Write an nth order differential operator L.



Consider the algebra generated by the u symbols and their derivatives A.

Everything in A can be differentiated. Take the quotient by the subspace of closed elements.

Also consider a ring of Laurent polynomials in the derivative symbol but the other way around. Meaning they can go down as far as they need but have to cut out at the top.
This ring splits up into the positive part and negative part.

You can also take residues in the same manner with taking the term that looks like  .

Because the top coefficient of L is 1 we can actually take it to a fractional power m/n.   which commutes with L.

But let's mess that up by a violent projection to the positive part. Now we have a n-2 order differential operator which we can call the time derivative of L which describes how the u's are changing. Since the first u that shows up is at the right place, we are OK.

If you take different fractions, you see lots of commuting times, so we get lots of conserved quantities which are written via residues of all those     for all m.

Let n=2. Then we only have one u and this becomes KdV classic.