Double the SUSY, Double the Drinfeld

In N=2 SUSY theories, the Drinfeld double shows up somewhere.

The Drinfeld double is  with a Hopf algebra structure built up from the Hopf algebra structure of H. Take the identity element in the above (using adjunction) and it will describe the R element that turns the double into a quasitriangular Hopf algebra. This is where exciting things happen because the representation category of quasitriangular Hopf algebras is a braided monoidal category!

http://www.impan.pl/~burgunde/WSBC09/Ddouble_Hajac.pdf

So doubling the SUSY gives you favorite type of category.

Indiana Jones and the Base of Crystals

Start with a quantum group. For each simple root  define  and similarly for the lowering operators f's. We have unique decompositions

for every u in    for that weight.

Define linear mappings M to itself by its action on  as

Let A be the rational functions of q which are regular at q=0. Now we can define the crystal base (L,B).

L is a free A-submodule of M such that the following hold:

B is a basis of L/qL
Both L and B decompose into weight spaces as you would hope.
Closure under the linear mappings said above (include 0 as a possibility on the basis B).

Using the twidled action instead of the original e's or f's allows regularity at q=0. They are off by a factor of  as you can see by comparing the formulas for e and twidle e.

A way to see all this data is with a graph. Assign a vertex to each element of the base and a colored directed edge if you can get from one vertex to the other with the a modified raising operator. The edge is colored by which i you needed to use.

You can tensor two crystal bases together. The new L and B are what you would expect, but the action of the operators is different. For example,

Even though this construction has passed to the q=0 case for the quantum group. This still retains some information about the representation theory like how reducibility, weights and multiplicities.

AGT

Lots of things going on here. I'll explain when (if) I understand it.

http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.3219v2.pdf

M5 branes wrapping Riemann surfaces. Those surfaces pinching off into pairs of pants.

Koszul-Tate

Start with a commutative ring R and an ideal M.

Treat R as degree 0 elements. Add a bunch of elements T in degree 1 such that their image under the differential produces M. Now the zeroth degree homology is R/M, but the first homology is a mess. So repeat the process with a bunch of new variables in degree 2. Lather, rinse, repeat.

Eventually you get a supercommutative differential graded ring with homology R/M concentrated in degree 0. This is exactly what you want, after all you want the number of ghosts to be 0, and nothing physical in any nonzero ghost number.

BV awesomeness

Remembering the real good way to think about manifolds and more generally schemes is as a topological space equipped with its sheaf of smooth functions. What if this sheaf locally gave the smooth functions on n dimensional ordinary space. That would give a plain old manifold.

What if it was the tensor product of smooth functions on R^p with the exterior algebra on q variables? Then we would be in superspace with p even directions and q odd directions.

Let's construct lots of examples of supermanifolds. Start with a vector bundle E over M. Now consider the supermanifold whose sheaf of smooth functions is given by sections of the exterior algebra of the dual vector bundle. If E is the tangent bundle to M, this would be the sheaf of differential forms.

Batchelor's theorem says that every supermanifold is isomorphic to the victim of the above construction for some vector bundle. But the above functor from vector bundles to supermanifolds is not fully faithful, so there actually is benefit to be gained by looking at the category of supermanifolds rather than living in vector bundle-land.

For a (n,n) supermanifold construct an odd symplectic structure by 

This is clearly odd, it has one odd variable and one even variable.

You can go ahead and repeat lots of symplectic geometry. You have from this two-form a Poisson bracket, again, you can also get a Laplacian if you provide me with a density function too.

This Laplacian is odd and squares to 0, so we can take its cohomology. This is the beginning for BV-BRST. We can properly define the integrals we need to.

More details

Toda! Toda! Toda!

(schematic for post)

Here is the Lagrangian. The m and b are parameters under your control and the n's are the coefficients of the maximal root in terms of the other roots.

The reason this is so super special awesome is in its integrability and how it reduces to lots of simpler integrable field theories. Examples include the sine-gordon model when you use a pathetically small r.

Even better it describes integrable deformation of a generalized Lioville theory.

If you analytically continue to imaginary b, sometimes you get deformations of minimal models. Magnetic deformation of an Ising model. Yay! actual physical relevance.

A quick proof of the C-theorem

Give me a quantum field theory in 2 dimensions that is rotation invariant, unitary and has stress energy tensor conservation. In return I will give you a function that is decreasing along renormalization group flow and on the fixed points, it will give the value for the central charge.







T is the component of spin 2,  is the component of 0 spin and  is the one with eigenvalue -2.

Conservation of stress energy tensor translates to the following equations:






Now define



We get



but H is positive, see the definition of it. Therefore this function decreases.

At the conformal points,  vanishes and sense so do G and H making C=2F. But by OPEs F=c/2. Therefore we get a precise match of our defined function C and the central charge at the fixed points of the renormalization flow.