Disclaimer for this section: I haven’t been studying string theory for a very long time, but I think I do have a good handle on the necessary prerequisites. This time I won’t have a “topics you can learn concurrently” subsection, since if you’ve reached this point of your own accord, you can figure out how best to devote your attention elsewhere.
This subject has a reputation of people entering unprepared, so the central advice I provide is this: don’t learn string theory via the physicist-route if your sole aim is to study compactifications, moduli spaces and the like - the mathematics route is quicker since you don’t need to trouble yourself with physical interpretations (yawn). But if you do wish to learn string theory properly, you must be prepared to essentially be doing conformal field theory half the time.
Prerequisites:
-
Quantum Field Theory: You must be friends with point-like particles before you can prove strings! Interestingly (and although I despise this terminology), string theory is technically a first-quantised system, while QFT is traditionally second quantised (second quantising strings leads to string field theory). You should be very comfortable with both canonical and path-integral formalisms, Grassman numbers, the S-matrix and Hilbert spaces, linear and non-linear sigma models (and hence pre-QCD pion mechanics), regularisation/renormalisation and the renormalisation group, non-abelian gauge theory (Yang-Mills and Higgs, preferably accompanied by the bundle formalism) Ward identities, currents, Noether’s theorem, representation theory of symmetries, the Lorentz group and $\mathrm{SU}(n)$, Young tableaux, gauge and global quantum anomalies. Some more QFT stuff that I personally recommend you learn prior (perhaps not as prerequisites, but to develop ideas and tools and to check your understanding) include solitons like instantons and monopoles and matrix models.
-
Conformal Field Theory: This forms the unstated bulk of string theory. Despite many claims that you can actually pick this up concurrently, I recommend that you are comfortable with this at an intermediate level before starting out - there are a lot of quantum mechanical and statistical mechanical outlets that you can get your hands dirty with without stepping into a stringy context. In particular, with the following topics: Virasoro representation theory/Verma Modules, state-operator correspondence, correlation functions, chiral algebra, Kac-Moody algebras, free bosons and fermions in flat space, on the cylinder, torus and orbifold, modular invariance and characters, bosonisation and a brief overview of the superconformal algebra. Conformal bootstrap and knowledge of in-depth minimal models are not required. Believe me, if you commence your string theory journey with a firm grip on CFT, it is feels ilegally easy.
-
Constrained Dynamics and Quantisation: In-depth knowledge of Gupta-Bleuler quantisation, lightcone quantisation, Poisson, Dirac brackets and constrained quantisation, gauge transformations, general covariance, BRST symmetry (antifield formalism not necessary), especially applied to non-abelian gauge theories and the point particle. Obviously a lot of experience with Lagrangian mechanics too.
-
S-Matrix Theory: It’s quite remarkable how the true foundations of string theory receive very little attention, usually compressed into a few pages in an introductory chapter. I will throw around a few names here: Regge. Mandelstam. Chew. Veneziano. Froissart. Analyticity. Dual resonance. If you can manage it, you’re best off reading their original works - understanding how string theory is an S-matrix theory is essential to understanding why it is held in such high regard - and all of this ties into pre-AdS/CFT “proto-holography” in some deep mysterious ways.
-
General Relativity: Nothing super advanced here, the equivalent of an undergraduate course in GR should suffice, even below the level of rigour of Wald, etc. I mean the Einstein field equations (obviously) from the Lagrangian formulation, black holes, basic cosmology, gravitational waves, spinors in curved spacetime, the Unruh effect (with basic black hole thermodynamics) and a mastery of all of the differential geometry that goes into it. The analytical aspects like causal structure, hyperbolicity, singularities, etc. are not necessary.
-
Supersymmetry: This isn’t strictly necessary before bosonic string theory of course, but its helpful to get some perspective by seeing supersymmetry in QM and QFT - representations, superspace, supersymmetric non-linear sigma model, SYM + Higgs, symmetry breaking and non-renormalisation theorems are a good start. In in-depth knowledge of SQCD-like theories will almost never be required, since that’s a whole rabbit hole of its own.
-
Mathematics: String theory does immediately not bring in much new mathematics to the table if you are comfortable with the physics prerequisites mentioned above, so you will be able to learn the basics rather rapidly. That said, several topics soon begin to emerge, including complex differential geometry and analysis, complex Riemann surfaces, Clifford algebras (especially the Spin groups, since an accidental isomorphism no longer exists for higher dimensions), exceptional Lie groups, cohomology, a bit of algebraic topology and algebraic geometry (particularly in compactifications). Each of these is pretty much a bottomless pit: you can probe them as deep as you want, and there will probably be some inexplicable connection to string theory that Witten thought of 30 years ago.