Monthly Archives: July 2006


Well, I’m done with the official presentation today, so that leaves me a bit more free for some time. Considering that I’ll be offline for a week or so … once I go home, I think I should probably try to spike my blogging density for some time.


Now, last time we’d just set up the SUSY algebra for (0+1)-D systems, basically Quantum mechanics, that is actually enough for the proof of the Atiyah-Singer index theorem(The links at the end of the wikipedia are worth checking out and range from popular level to maths that is beyond me at present). So, for now, let us take a slight detour, and look at differential forms. This may be redundant but, let’s just review differential forms (horribly quickly, in a way that might appear, (because it is) pretty sloppy to a mathematician, but it’ll do for our purposes here)


So, given a manifold, M, let’s look at it’s tangent space. Basically, we can have vectors & tensors in the tangent space. Now consider all the antisymmetric tensors in the tangent space. Now, (loosely speaking) the p dimensional antisymmetric tensors is what we call p-forms. The p-forms form an abelian group under ordinary addition. Now let us define an operator, d as:

(dA)ijk… = D[i Ajkl…]

Where, basically we get a p+1-form from a p-form, by differentiating it (thus adding an extra index) and then antisymmetrising, so that the result is also a form (antisymetric tensor). This structure is what is called a cochain complex. Now, we can also define a d* operator which takes a p-form to a (p-1)form, by contracting with an additional index.


Ok, that’s enough of an interlude into this, and it’s getting to technical for a lazy guy like me to care to typeset in the horribleness, that is HTML (for mathematical typesetting that is …). So, you might say, all this is very interesting, but why should I be interested in forms? Well, there’s tonnes of uses of this language in Physics … For one, you can join the cool kids, and say:

F = dA, dF = 0, d*F = J

instead, of the boring old Maxwell’s equations … but well … it’s just pretty neat stuff, by itself … but we’ll see soon how all this can be useful, in Physics.  We’ll consider a Supersymmetric system which has as its Hilbert space, the space of all forms. As I said on the top, I’ll be offline for a week at home, so I will continue posting, only after that, if you’re impatient then you can have a look at the report I wrote up about my summer work.





It’s been nearly three weeks since I promised to write about my activities here. My apologies, in case I actually have any readers. I got swamped in last minute deadline-beating and managed to sneak in my eport just at the last second as the clock was striking 4. I’m writing this post thanks to some encouragement from Tom. So, here goes:

Day 1 : Symmetries have played a very important role as guiding principles in Theoretical Physics. A given symmetry enables us to make some statement about the behaviour of a system (namely the associated conserved charge), thus one would naturally be interested in any possible new symmetry that physical models could have. However, there is a theorem due to Coleman and Mandula, that forbids the possibility of any symmetries that transform as tensors under Lorentz transformations other than Pμ and Mμν. There is a very simple plausibility argument by Witten as to why this should be so. It basically says that, the conservation of Pμ and Mμν, leaves only the scattering angle unknown. Any new conservation law would leave us with only a discrete set of possible angles, but since the scattering amplitude is an analytic function of angle it must be zero for all angles. Thus it would seem that the Coleman-Mandula theorem, rules out all possible symmetries other than the Poincare group(translations and Lorentz transformations). This seems to spoil all our fun, by saying that no new symmetries lie in wait to be found by daring & adventurous experimentalists.


Thanfully, there’s a catch! We have only ruled out symmetries, whose generators transform as tensors under the Lorentz group. What about spinors?? That is another representation of the Lorentz group. We can have spinorial conserved charges. This is because, spinorial conserved quantities do not give us any observables, which impose constraints on scattering matrices. (For the more technically minded, the Coleman-Mandula theorem applies to Lie algebras, whereas supersymmetry charges obey a different structure, they have anticommutators instead of the Lie-bracket). What they do is to relate boson-boson and fermion-fermion scattering. Now, the anticommutator of two supersymmetry generators is going to have two spinor indices. This quantity is like a vector, if it consists of a left & a right handed spinor, but this thing being a conserved vector, has to be nothing other than Pμ. All other anticommutators have to be zero, because such a quantity does not correspond to anything that we observe. More importanty P0, the Hamiltonian, can be written as the anticommutator of a Q & its hermitian conjugate Q*. This is the simple case with just one supersymmetry, what is called, N=1 supersymmetry. Therefore, we have:

H = {Q, Q*}

Now, this simple way of writing H, as an anticommutator has several interesting properties. Next day, I think I’ll write a bit about differential forms, and slowly build up to a proof of a special case of the Atiyah-Singer index theorem(for the de Rham cohomology).


In case anybody is reading this, I’d really like to know that somebody actually reads the stuff I’m writing. And considering that my audience, would probably be a singleton, I could try to make the stuff I put here more/less technical as you might like it. It would really feel nice to know that the stuff I’m writing here, is (hopefully)useful/enjoyable to someone.