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.

Clavelli-Shapiro Trick

Computing the expected energy for a harmonic oscillator in a heat bath isn't bad. Other observables aren't too bad either, but here is a trick that will generalize.

First introduce new b operators obeying the same commutation relations.

To compute the thermal expectation value with parameter y ( fugacity goes with chemical potential, I mean the one that corresponds to temperature), you take the regular expectation value of your operator, but replace all the a's and their adjoints by

respectively.

3 Proofs in A.6 at least in a useful special case.

This seems insanely useful depending on how far it is pushed.

Relative QFT

http://arxiv.org/pdf/1212.1692.pdf

Expectation 5.3: Given a real Lie algebra, invariant inner product such that all coroots have length square 2, and a full lattice in the Cartan such that on the lattice the inner product is integral and even, there should exist a 7 dimensional topological quantum field theory and a 6 dimensional theory relative to it.

This is supposed to be the big goal of the (2,0) siX theory.

A relative QFT is an extended field theory in one higher dimension. To be continued...

Decay

MOA2DIS mother of integrability

Why self dual yang mills in 4 dimensions is the Mother Of All 2d Integrable Systems. At least that is what Ketov says in his last chapter.

Found the sources
http://arxiv.org/pdf/hep-th/9307021v2.pdf
http://ac.els-cdn.com/037596018990964X/1-s2.0-037596018990964X-main.pdf?_tid=410488d6-3295-11e2-8bd0-00000aab0f6b&acdnat=1353362920_03278f52f496a8df2a191a8dae74b67f

Let's work on (2,2) signature  with metric  . Look at the self-dual Yang-Mills equations on it. In these coordinates they read:




Calling the components of the gauge field A,B,C,D respectively. We can ask that they don't depend on u or y. We can also ask that A=D, but I don't know why you would ask such a thing.




We have put three equations into 2 by inserting a  and asking to read of the coefficients of the commutator of the above operators.

The quadratic piece demands that B not depend on x. There are three possibilities for B after gauge transforming. Let us pick the one where B is proportional to the generator for the Cartan in it's usual form multiplied by some t dependent function.

The equations are solved if we write the following for A and C.




and



The second choice above gives the nonlinear Schrodinger equation with attractive self interaction.

We made several choices along the way, other choices would have resulted in NLS with repulsive interaction or KdV.

If that is all there is to it, the title MOA2DIS seems to be excessive.

Enriching some categories

Start with a plain old category C. You have some objects A and B and Hom(A,B) to describe all the arrows from A to B.

What structure does Hom(A,B) have?

Not much. Ok, now I'm stuck. I can't say any more.

If the Hom(A,B) were just sets, it would be locally small. Not good enough.

If I somehow knew they were abelian groups, then I would be in pre-additive categories, and I would start to have some hope about doing all the abelian category chases.

What are the structures I could possibly ask Hom(A,B) to have? I need to have a well defined map Hom(B,C) x Hom(A,B) to Hom(A,C) so that tells me that it has to be monoidal in order for the left hand side to make sense.

And there should be the identity in Hom(A,A).  That means there should be a morphism from the monoidal unit to Hom(A,A) in this monoidal category.

AND coherence relations time:

Now another example, what if I enrich over chain complexes of complex vector spaces. Each hom is now differential graded. So this is a dg-category.

I can relax conditions a little bit more and give you an example of an A_\infty category. The specific one I have in mind is the Fukaya category of some symplectic manifold.

Picture time

Draw more holomorphic polygons to give the higher maps that mess up the dg-category structure.