Why can we solve some QM problems?

Here is some old stuff back from Paulo Bedaque's 622.

Write the two operators



They are H.C.s of each other so lets call them A and  respectively. Write two Hamiltonians




If you have an eigenstate   of the second, then you automatically get an eigenstate of the first by taking  . Therefore the two Hamiltonians have the same spectrum except possibly at the bottom rung.

Expanding out the Hamiltonians gives the two potentials as .

Playing with the ground state of the Hamiltonian gives a formula for the superpotential, or the other way around if you were given the ground state instead.

Why is this supersymmetric? Take the direct sum of the two Hilbert spaces. A and  placed in blocks of a 2 by 2 in order to switch between the two pieces. If we call those two spaces, the fermionic and bosonic pieces then we just got the supersymmetry exchanging the two. The superalgebra here is SL(1,1)

This can be done in higher dimensions after passing to the effective 1 dimensional problem after including the angular momentum barrier pushing away from the origin.

Now we can get to the title. Why are certain potentials solvable?

This happens when you have the following condition on the partner potentials.



where the a's are parameters. If we have the above condition, you can construct a ground state for the first Hamiltonian and shape invariance guarantees that the spectra are just shifts of each other. You can build up the entire bound state spectrum from this by applying raising operators and bouncing back and forth between Hamiltonians.

Your favorite potentials that you have solved like harmonic oscillators and the hydrogen atom and their partners form shape invariant pairs. So they are exactly solvable as you knew because you solved them back in your misspent youth.
Shape invariance is a sufficient but not necessary condition for solvability.

For more:
http://arxiv.org/pdf/hep-th/9405029.pdf

Algebra + Homotopy = ??

Explaining why A_\infty algebras

There is a comprehensive review article on this recently. Check out what it's all about.

http://math.unice.fr/~brunov/publications/Algebra+Homotopy=Operad.pdf

Have a dga A with a homotopy to another complex H. From the dga structure of A, you can get all the higher operations needed on H side.

For example



where is the map H to A, then use the multiplication on A and then take it back with p.

I refuse to write the formulas for any of the higher products. Instead I'll give you a pretty picture.

That is the three product on the H side.

That is the nth higher product in terms of all sorts of degenerations of planar binary rooted trees with n leafs.

There are also the horrible relations that describe all sorts of degenerations of the trees.

So we have gotten an  algebra from a dga. This construction is in fact functorial and dga's are a full subcategory of  algebras.

We could do algebraic topology with this by using the example of a chain complex being homotopic to the homology complex with 0 differential. The higher products we got by abstract nonsense become the familiar Massey products of say the Boromean rings or other such configurations where linking numbers just don't cut it.

We have gotten  , next time we continue with operads.

Equivariant Cohomology

I have a space X with an action of a group G. There is the cohomology of X and the group cohomology of G. I can do something that will degenerate to these two special cases when X or G are trivial.

First take the space   it is homotopic to X because EG is contractible. There is a G action on it, act on each factor. This action is free because it is free on EG. That means you can take a well defined quotient space which will not have any weirdness. Take the ordinary cohomology of that thing.

If G is trivial then EG is too and our quotient space just spits back X.
If G acts freely, then the constructed space will be homotopic to the plain old orbit space.
If X is a point, then the constructed space will be BG, and you remember what the cohomology of that is don't you?

Lie Algebroids

Lie algebroids are vector bundles with a bunch more data.

 called the anchor map.
and a bracket on sections.

There is also a Leibniz identity with the bracket and anchor

Some examples:
1. Take M to be a point, E to be a regular Lie algebra lying on top. The anchor map is stupid in this case, and the bracket is just the bracket of the Lie algebra.
2. E is the tangent bundle itself with the anchor map being the identity. The bracket is the Lie bracket of vector fields.
3. The cotangent bundle of a Poisson manifold. The anchor map is given as
  and the bracket is   

From a Lie algebroid you get the supermanifold E[1] by viewing the fiber coordinates in degree 1. (Remember a previous post where we caution that this is not functorial so there is actually a subtlety between vector bundles and supermanifolds even though we can go from one to another.)



is the sheaf of smooth functions.

You get a derivation on this space. Call it . It squares to 0 making this space a Q manifold.

We also get an action of   when we view  for the appropriate G groupoid. Infinetesimal generators of this are v which reads off the degree and .

If this supermanifold happens to have a symplectic structure invariant under our vector fields above, we can ask what is the degree of this symplectic form.

This is where we come to a theorem:
Symplectic Q manifolds of degree 1 are in bijection with Poisson manifolds.

The next step is natural. What about degree 2 symplectic Q manifolds? The answer needs Courant algebroids.

This comes from a talk given at GRASP on 1/27/12.

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.