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This is, of course, a very good question. I should preface with the disclaimer that despite having worked on some aspects of integrability, I do not consider myself an expert. However I have thought …

Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware o …

With all due respect to Mikhail Gromov, Physics is not about homotoping loops in the spatial universe! That would certainly be risible. In Physics, the fundamental group reflects itself in the spect …

It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the sam …

The trace is simply a (properly normalised) ad-invariant inner product on the Lie algebra; that is, a nondegenerate symmetric bilinear form $\langle-,-\rangle$ which obeys the "associativity" conditio …

I think it would be more accurate to say that the real reason why Calabi-Yau, hyperkähler, $G_2$ and $\mathrm{Spin}(7)$ manifolds are of interest in string theory is not their Ricci-flatness, but the …

Symplectic reduction arises naturally in constrained hamiltonian systems, e.g., gauge theories. So it is not just a question of it being "interesting" as much as a fact of life. The way to deal with …

First, a historical note: the twisting procedure for $N=2$ SCFTs is due to Eguchi and Yang; although a twisting of sorts had already appeared in Witten's Topological Quantum Field Theory paper of 1988 …

The geodesics you seek are the so-called homogeneous geodesics. Not all geodesics will be of this form, but there certainly exist. In the literature, for some reason, people consider left-invariant …

Let $(M,g)$ be an orientable pseudo-riemannian manifold. Each tangent space $T_xM$ is a pseudo-euclidean space and hence has an associated Clifford algebra $CL(T_xM)$, which is the fibre at $x\in M$ …

It is difficult to give a precise definition, because there are many cohomology theories which go by the name of BRST. It might be helpful to give a couple of examples, not necessarily in chronologic …

In the context of general relativity, the universe is described as a lorentzian spacetime subject to a coupled system of PDEs konwn as the Einstein field equations. These relate the curvature of the …

The Hilbert scheme of points on a K3 surface plays an important rôle in providing a strong coupling test of S-duality by Vafa and Witten. This is the original paper on what is known as Vafa-Witten th …

I am quite late answering this question, even though I followed it when it first appeared, but it must have slipped my mind. Anyway, it's been a while now and nobody seems to have mentioned my favour …

Let me add something to what David and Urs have written already, since the way those two answers are shaping up, perhaps what I'm about to say does not get mentioned. One of the most interesting appl …