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.

Youtube videos better than NYAN cat

Cool people saying cool things

Contact Homology

GRASP usually gives me things I can post about, and today's talk was embedded contact homology.

Start with a contact 3 manifold Y. Find it's Reeb vector field R.



Look for where this vector field makes circular orbits. Those are Reeb orbits.

Take the module generated by all unions of them.

An arbitrary element will look like

 

where R lists all the orbits and m's are the multiplicities in  .

You say the boundary of a particular union of Reeb circles is the sum over all output union of Reeb circles such that each one is counted by the size of the moduli space of holomorphic curves connecting the first set of inputs to the second set of outputs in the symplectic manifold associated to Y.

The picture looks like

The top and bottom pieces are such holomorphic curves. This therefore is showing a contribution to d^2 which we hope all cancel out.

Of course you need a condition on indices. It is so much easier to explain the index in Lagrangian Floer where it is the number of times the tangent planes turn. These indices save the day and give some control of what the end curves can be.

I don't know the index in ECH, but since he didn't say what it was in the talk, it sounds profoundly ugly. This also means I have no hope for explaining why this is a chain complex. These all have that same Morse homology flavor of trajectories breaking like shown above. In the Morse case, I can visualize all the pieces cancelling and that is where mileage can be drawn from.

Fivebrane structure

So you know how you need the second Stieffel Whitney class of spacetime to vanish for fucking fermions to make sense? Why is that?

First you have an SO(n) bundle. But you want it to be an Spin(n) bundle so that the sections on the associated vector bundle will become fermion fields.
Viewing the bundles as homotopy classes of maps instead you are looking at a lift of the map from X to BSO(n) to a map from X to BSpin(n).

There is a map from BSO(n) to BBZ_2 given by the special element 1 of [BSO(n), BBZ_2]=H^2 (BSO(n),Z_2)=Z. This is the second Stieffel-Whitney class. Pullback the universal bundle gets you BSpin(n). So in order for the map to lift, the diagram must commute. This means that the map X to BSO(n) to BBZ_2 has to vanish all the time. That only works when this class vanishes for the tangent bundle.

Now to get from Spin(n) to String(n). Repeat the construction.

There is a map from BSpin(n) to BBBU(1) given by H^3 (BSpin(n) , U(1))=Z. Pullback the universal bundle to get BString(n). The diagram commutes when the composite X to BBBU(1) vanishes. This class is half the first Pontryagin class in H^4(X,Z).

Repeat again from String(n) to Fivebrane(n). This time replace BBBU(1) with 7 applications of delooping. The obstruction class is now one sixth the second Pontryagin class in H^8(X,Z).

http://arxiv.org/pdf/0805.0564v3.pdf