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Sure you can. In fact, this usually goes by the name of strings without strings. The basic observation is that when you quantise the sigma model when $M$ is, say, Minkowski spacetime, what you end u …

The formal delta function obeys the usual properties that the Dirac delta function does, but relative to the pairing defined by the residue. For instance, $$ \mathrm{Res}_z f(z)\delta(z,w) = f(w)$$ f …

Yes. This follows from the fact that $G_2$ acts on the octonions via automorphisms. Let $e_i$, $i=1,\dots,7$ be a choice of 7 imaginary octonion units and let $1$ denote the identity. Then by defin …

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 …

Edit The answer below is not correct. Upon further reflection, I believe that the correct causality condition is indeed global hyperbolicity and not the weaker stable causality. I believe this is …

OK, I'll bite -- although I am not convinced that this is a suitable question for MO. First of all, it depends what you mean by 'string theory'. There is mathematics which has been influenced (some …

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 …

This is not really an answer to your question, but a philosophical point. The physics of massless particles is not simply the limit $m\to 0$ of a massive particle. This is perhaps easier to see in t …

The answer to your bonus bonus question is negative, I'm afraid. The Euler-Lagrange equations only extremise the action in general, not minimise it. Nondegeneracy of the lagrangian and positive-defi …

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 …

As Mariano points out in his comment, this follows from the definition of a connection on a principle $G$-bundle $\pi: P \to M$. At every $p \in P$, the kernel of $\pi_* : T_pP \to T_{\pi(p)}M$ defin …

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 suppose that you don't really mean for both spaces to be spheres, so I will interpret your question about the (4,7) split. The context here is eleven-dimensional supergravity and the studies of suc …

I am slightly confused by this question. The fact that one can formulate the Dirac equation in either (3,1) or (1,3) signature, which have non-isomorphic Clifford algebras and hence Clifford modules …

The Virasoro minimal model $\mathcal{M}(2,5)$ (or in some conventions also $\mathcal{M}(5,2)$ is the conformal field theory which describes the critical behaviour of the Lee-Yang edge singularity. It …