TITLE: Mathematician trying to learn string theory
QUESTION [56 upvotes]: I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, Dualities, D-branes, M-branes etc. What's the easiest way to do so provided I have the necessary knowledge in algebraic geometry, algebraic topology, analysis and differential geometry?

REPLY [16 votes]: I want to understand compactifications, Dualities, D-branes, M-branes
etc.

What makes all this hard to learn in a systematic way is that the theory itself is still incomplete and proceeds in parts via educated guesswork.
In principle perturbative string theory is well defined: This says to pick a 2d super-conformal field theory of central charge 15, collect its n-point functions into one formal power series, interpret this as a loop-wise finite, hence normalized, scattering matrix known from QFT, then study this.
In principle "compactifications, Dualities, D-branes" (though not M-branes) are all well defined parts of this scheme. For instance D-branes are the algebraic data that defines the 2d SCFT at the boundary of the Riemann surfaces, Dualities are correspondences between different 2d SCFTs that miraculously yield isomorphic scattering matrices this way, and compactifications are decompositions of a 2d SCFT as a tensor product with some "finite" factor for instance a rational CFT. The target spacetime geometry that is being compactified thereby is to be read off from the CFT in a way generalizing how spectral Connes-style NCG reads off target geometry from algebraic data. (See the exposition at Spectral Standard Model and String Compactifications)
The problem at this point is that, while well defined, it is mathematically so hard. The only 2d CFTs that have mathematically been constructed as full CFTs are the "rational" ones, which only describe certain compact factors. For the others there is Segal's axiomatics, but essentially no examples. What one has is the local description in terms of vertex operator algebras. For these many examples are available, but not enough for the purposes of the physicists.
This is the first point where physics parts with a systematic mathematical development. Namely while it is hard to really construct full 2d SCFTs, the folk lore of the path integral allows to think of solutions to supergravity equations of motion as inducing 2d SCFTs via quantization of the "nonlinear sigma-model" with these spacetimes as their targets. So now instead of deducing effective target space geometry from the SCFT algebra, one prescribes classical target space geometry to which one believes SCFTs may be associated. It is from here on that much of string theory is now phrased in terms of classical geometry with some extra stringy effects sprinkled in (modular invariance, anomaly cancellation, brane instanton effects).
The uncertainty as to how well this sigma-model construction is under control is the cause of the discrepancy in the perception of how many string vacua are known: A perturbative string vacuum is equivalently that 2d SCFT which gives the scattering matrix, so in principle the "landscape of perturbative string vacua" is the moduli space of 2d SCFTs. But since this is not understood, what people instead scan is the space of target space geometries that are thought to induces 2d SCFTs. This is a subtle business, where one imagines one may incrementally approximate that 2d SCFT by adding alpha-prime corrections, cancelling anomalies, etc. Therefore one finds authors who worry that too many string vacua are known, and other authors who worry that too few string vacua are known.
If this were all there is to it, the solution would in principle be straightforward: mathematicians/mathematical physicists should simply sit down and find means to rigorously construct 2d SCFTs and to understand the moduli space that they form. That would be the mathematics of perturbative string theory, and all the answers as to "compactifications, Dualities, D-branes" would be encoded in there, under some well defined dictionary.
But now there is also the "M-branes", and that's indicative of the real problem: Since the string perturbation series is just a non-converging formal power series, as for a normalized perturbative QFT, it ought to be but the Taylor expansion of some non-perturbative theory about some points of its cofiguration space.
There are many compelling hints for this "theory formerly known as String" but so far they are just hints.

We still have no fundamental formulation of “M-theory” - the
hypothetical theory of which 11-dimensional supergravity and the five
string theories are all special limiting cases. Work on formulating
the fundamental principles underlying M-theory has noticeably waned.
[...]. If history is a good guide, then we should expect that anything
as profound and far-reaching as a fully satisfactory formulation of
M-theory is surely going to lead to new and novel mathematics.
Regrettably, it is a problem the community seems to have put aside -
temporarily.
But, ultimately, Physical Mathematics must return to this grand issue.
(G. Moore, "Physical Mathematics and the Future", talk at Strings 2014)

The way this is presently being studied is the same mix of classical target spacetime geometry with conjectured "extra effects" sprinkled in. For instance a "black M-brane" is to first approximation a solution to the equations of motion of 11d supergravity which preserves half of the global supersymmetry. By analogy with the D-branes and some other arguments, coincident such M-branes should "carry" a non-abelian gauge SCFT on their worldvolume. There is presently no way to derive this from anything, but superconformal invariance imposes enough constraints that a classical action functional could finally be guessed (the BLG/ABJM model). Now the search is on for the analogue on the M5-brane. Nothing definite is known, there are hints, and whatever one finds will justify itself by plausibility arguments. Because none of this can be derived from first principles
There are presently no first principles for full string theory, aka M-theory. There is no mathematics of M-theory to be learned. Instead, the mathematics of M-theory is waiting to be found.
This may not be as bad as it may sound. Maybe "M-theory" is easier to deduce following mathematical principles, than the historical route of the perturbtive string. I just wrote an exposition of such a point of view over at PF-Insights:
Super p-Brane Theory emerging from Super Homotopy Theory
So it seems not out of the question that, conversely, it will in the end be the physicists who will learn M-theory from the mathematicians.