Superspace

It has been a while since I posted anything on this WordPress blog. Since I’ve just bought a domain name from WordPress, I thought I should post something, even if just to mention this update in the blog’s address. But that’s just some filler content that I post on my other blog. So here I should at least talk about some Physics.

Superspace – the greatest invention since the wheel – is where I spent most of my graduate days. After graduating, I have spent less time there but I have regularly visited it over the years. Last year, I visited it to settle a 7 (or so) years old score with my collaborators by writing a paper on quantization of supersymmetric Yang-Mills theory in 4d $\mathcal{N}=3$ harmonic superspace that is actually useful in computing loop graphs.

Now, a popular physics blog would go to great lengths to try to explain to a layperson the various technical words that appear in the above paragraph, like quantization, SYM theory, $\mathcal{N}=3$, harmonic superspace, and loop graphs. [Of course, 4d just refers to our (3+1)-dimensional spacetime: 3 space and 1 time directions.] As it turns out, 4Gravitons is just that sort of a blog and most of these terms have been explained there: Quantization, Supersymmetric, Yang-Mills theory, $\mathcal{N}=3$ (well, $\mathcal{N}=3$ is quite similar to $\mathcal{N}=4$), Loop graphs.

The most astute reader amongst you must have already realized that harmonic superspace is not explained there. Or maybe, it has been but I have deliberately not linked to that page. The reason for doing that is quite simple: I am supposed to be the (second-)last ‘expert’ on harmonic and projective superspaces in the whole world. By “(next-to-)last ‘expert’ in the whole world”, I don’t mean the pejorative phrase “world-famous in India”, I literally mean the former phrase and it does not have any connotation about fame at all in it. So, being one of the ‘experts’ in superspace, I can’t link everything to 4Gravitons (who is an expert in another subfield of theoretical high energy physics) as I believe that one should undertake the undertaking about explaining something that one is an expert on to the lay-world.

Here goes my feeble attempt then. [Disclaimer: Anyone expecting clarity and thoughtfulness in crafting the explanations, as found at 4Gravitons, will be disappointed.] First of all, we must agree with the basic fact that harmonic superspace is a kind of a superspace. Then, we need to define a superspace and later we will tag on the adjective ‘harmonic’ in a hand-waving way, which should be sufficient for the level of this post.

A superspace is an extension of the usual spacetime $x^{\mu}=(t,x,y,z)$ with extra coordinates which have no basis in reality. (There, I said it; no beating around the bush.) These coordinates are such that (let’s call them θ (theta) to exude Greek sophistication) $\theta^2=0$. Now, a layperson would say: well, that just means $\theta=0$. That is, a layperson who has never heard of matrices. Anyway, hoping for this level of mathematical sophistication from our lay-audience, we can get past the fact that the vanishing square is not a problem and we can get on with building the most basic superspace, called $\mathcal{N}=1$ superspace. This superspace has 4 θ’s [$\theta_1\theta_2=-\theta_2\theta_1$ is satisfied by multiple θ’s. Recall that ordinary numbers satisfy $x_1x_2=+x_2x_1$] and can be used to describe $\mathcal{N}=1$ (and higher) supersymmetric theories. This is useful because with this setup, supersymmetric theories can be described in a compact notation with very concise expressions. Performing calculations in superspace is also quick and easy when compared to ordinary spacetime computations. Before superspace was invented, people found “miraculous cancellations” in their calculations, i.e., they would calculate a bunch of Feynman graphs and end up with an equivalent of “+1-1=0”. The problem is that they would have to compute the equivalents of “+1” and “-1” separately (which can be tedious) and find later that they summed to 0. Whereas in superspace, there are far fewer (super)Feynman graphs to compute and one directly sees the 0’s. In other words, nothing tedious to calculate and cancel later.

Now, we come to another fact that there are $\mathcal{N}=2$ supersymmetric theories and we should probably have $\mathcal{N}=2$ superspace to describe them directly. So one may just append 4 more θ’s to $\mathcal{N}=1$ superspace and call it a day. That approach is a valid one but doesn’t go far in achieving all the successes of $\mathcal{N}=1$ superspace in simplifying computations for $\mathcal{N}=2$ theories. (Because there are certain objects that do not exist in this naïve $\mathcal{N}=2$ superspace, which turn out to be crucial for simplifying computations.) Thus, we have to invoke harmonic (or projective) variety of the $\mathcal{N}=2$ superspace. Harmonic superspace contains the usual spacetime, only 4 θ’s (which are obtained as some combinations of the earlier 8 θ’s) and a 2d sphere (an ordinary sphere, nothing exotic like the θ’s but it is again not ‘real’; though, it has a very satisfying origin story in the mathematical structure of supersymmetric theories, whose discussion is beyond the scope of this post). It might still not be clear to a layperson what has ‘harmonic’ got to do with that extra 2d sphere. The punchline being that the study of functions on spheres is referred to as harmonic analysis in the mathematical literature. (The adjective ‘projective’ is also related to the 2d sphere, which can be treated as a complex projective line. Actually, the word projective has a more satisfying origin in the way the coordinates transform in projective superspace. You guessed it right: they transform via projective transformations.) QED

Just completing the loop, $\mathcal{N}=3$ harmonic superspace consists of the usual spacetime, 8 θ’s (instead of the expected 12) and a 6d compact space (don’t really want to say anything more than that right now). Our 7-years-in-making paper studies the SYM theory in this superspace and derives Feynman rules to make computations of loop graphs easier. This problem has been unsolved for more than 3 decades. One proof-of-concept example we include in our paper slashes a 6-pages computation based on symmetry arguments to a ½-page direct computation of two very similar Feynman graphs.

So I guess that’s all for this post. Till next time, subscribe to 4Gravitons. You will learn a lot about Theoretical High Energy Physic(icist)s.