In my previous learning journey, I’d mentioned about baryons being topological twists in the pion field. By popular request, I will be giving a lightning review of what this construction, known as the Skyrmion model.
Before the QCD model of quarks and gluons was definitively established, the leading model for nuclear interactions was the pion model. A number of different pion fields packaged neatly into a larger, composite field (using the generators of $\mathfrak{su}(N_f)$) interacted with the nucleons (the proton and neutron). This yielded resonably good results, but was later superseded by the quark model, of which the pion model was seen to be an effective field theory. The pions themselves emerged as Goldstone bosons due to spontaneous chiral symmetry breaking - or, when factoring in the small explicit symmetry breaking due to the quark masses, they arose as pseudo-Goldstone bosons (“small” owing to the fact that $m_i\ll\Lambda_\text{QCD}$). The popular description of pions is as mesons - bound states of a quark and an antiquark, however the Goldstone bosons description, in which pions are described as vibrations in the various chiral condensates, is more appealing to me at least (the bound state presentation seems to obscure almost all of the genuinely interesting features of pions).
Consider a non-abelian $SU(3)$ gauge theory with $N_f$ flavours of massive quarks transforming in the fundamental representation. It is described by a Lagrangian
$$
\mathcal L=-\frac12\mathrm{Tr} F_{\mu\nu}F^{\mu\nu}+\sum_i^{N_f}(i\bar\psi_i\not\! D\psi_i-m_i\bar\psi_i\psi_i)
$$
where $\mathrm{Tr}$ is of course a colour trace. We can construct a low-energy approximation of this model using the so-called first-order chiral Lagrangian of the pion octet:
$$
\mathcal L_\pi=\frac{f_\pi^2}{4}\mathrm{tr}(\partial^\mu U^\dagger\partial_\mu U)
$$
where we have neglected pion masses. Here $U$ is an $SU(N_f)$ valued matrix: $U=e^{2i\pi(x)/f_\pi}$, and $\pi(x)\equiv\pi_a(x)T_a$ where $T_a$ are the generators of $\mathfrak{su}(N_f)$, and so this constitutes a non-linear sigma model. This is the simplest Lagrangian constructed out of $U$ that is symmetric under the full chiral symmetry (left and right unitary rotations, minus the axial rotations).
Now what are baryons? Everyone knows that they are once again certain bound states of quarks, but the fundamental definition is that they are objects that are charged under $U(1)_V$ and trivial under $SU(N_c)$. These are generically massive, so it is natural to expect that the Lagrangian of massless pions knows nothing them. This turns out to be wrong, baryons do in fact emerge, in a rather miraculous way!
Recall from your forays into instantons and monopoles that we can often classify field configurations on a manifold using algebraic topology. In our case, let us study static configurations of $U$ that are asymptotically the identity. This boundary condition allows us to compactify the base space from $\mathbb R^3$ to $\mathbb S^3$, and so such field configurations are maps $\mathbb S^3\to SU(N_f
)$. Definitionally, such maps are classified by the third homotopy group $\pi_3(SU(N_f))=\mathbb Z$, so every static field configuration can be assigned an integer. This corresponds to the winding number of the field, roughly because it measures the number of times a field winds around infinity. This can be computed topologically via
$$
B=\int\mathrm d^3x\ B^0
\\B^\mu=\frac{1}{24\pi^2}\epsilon^{\mu\nu\rho\sigma}\mathrm{tr}((U^\dagger\partial_\nu U)^{\otimes 3})
$$
which is a conserved current $\partial_\mu B^\mu=0$. However, trying to explicitly construct these solutions from the chiral Lagrangian above yields nonsensical energies, and this is because no such solitons exist in this framework, via Derrick’s theorem.
Tony Skyrme formulated a different Lagrangian, essentially via trial and error, but employing the general symmetry considerations described above.
$$
\mathcal L_\text{Skyrme}=\frac{f_\pi^2}{4}\mathrm{tr}(\partial^\mu U^\dagger\partial_\mu U)+\frac{1}{32}\mathrm{tr}([U^\dagger\partial^\mu U, U^\dagger\partial^\nu U]^{\otimes 2})
$$
which initially looks slightly bizzare, but not so much when you realise that $L_\mu\equiv U^\dagger\partial_\mu U$ is a nice traceless Lorentz vector that we can construct out of the $U$ field - it’s the left flavour current. Now we can calculate the energy of the static field configuration (an exercise which I leave to the reader), and finally, employing the Bogomol’nyi bound, we see that the energy is bounded by the winding number times a constant prefactor:
$$
E\ge 6\pi^2f_\pi|B|
$$
Note that $B$ is an additive charge, and we don’t have all that many $U(1)$ symmetries lying around. We have exactly one, in fact, the $U(1)_V$ vector symmetry (left-right symmetric phase rotations), since its cousin the axial symmetry is broken by instanton effects - so we should identify charge under $U(1)_V$ with baryon number. Since $U(1)_V$ holds exactly in QCD, unspoiled by anomalies, baryon number conservation is almost an exact symmetry of the Standard Model. There’s a problem when you look at the electroweak sector because there are $\mathrm{SU}(2)$ sphalerons that can violate B (but always preserve B - L, this is anomaly-free in the entire Standard Model). The good news is that sphalerons are extremely high-energy, so no-one really cares about them. On a related note, there are no global symmetries in any quantum gravity theory (it’s a simple thought experiment involving black hole evaporation - try and rediscover it), and BSM models generally only conserve $B-L$ to allow for matter-antimatter asymmetry for early-universe baryogenesis and leptogenesis.
Finally, we can construct explicit solutions that saturate the Bogomol’nyi bound by solving the equations of motion of $L_\mu$. To summarise, baryons appear in a higher-derivative pion model as field configurations with non-trivial winding at infinity. Incredible.
I recently attended the Strings 2021 conference! An annual conference, it takes place over a span of two weeks, in a different country across the globe each year, with the pre-eminent names in high-energy physics being invited to speak in front of hundreds of graduate students, postdocs and faculty. It’s always been in-person - until this year. For the first time ever, the Strings conference has been organised online, which meant that participation has shot up sixfold - I was among a handful of high schoolers attending.
I found the presentation system very engaging - it was primarily composed of research talks, where novel research in string theory was described following which the presenter fielded a few questions from the audience at the end. However, there were also longer “review” talks, where recent developments, main features and broader context of a large, active subfield of high energy physics, along with discussion sessions, which commenced with a short presentation but involved questions, comments and discussion later on. How did I decide which ones to watch? Well, it was a combination of spotting some familiar names (Clifford Johnson, Shiraz Minwalla, etc.) and finding topics which were at least slightly familiar to me (offshoots of “traditional” string theory, quantum black holes, amplitudes, supersymmetric gauge theories, etc. ). For their part, the moderators/organisers did a very good job of regulating the discussions (though let me add, the organisers are not some admin people - they included amidst their ranks Nathan Berkovits, creator of the famous pure spinor formalism, and Pedro Vieira, winner of the New Horizons Breakthrough Prize in 2020!).
The research talks were of course too heavy for me at times (I came in fully expecting this), but they were valuable since I previously had no idea what the cutting edge of string theory even looked like, and I learnt of many new interesting models and paradigms. The topics spanned several huge subfields, including “traditional” high-energy string theory, black hole microstates/information, cosmological and condensed matter applications, and it helped put into perspective what I might be working on (and with whom!) if I decide to pursue research in these fields. Another important insight that it provided was in written and spoken academic discourse online (the Slack channels for the conference were continuously active!). I will probably revisit several of the talks to watch them at my own pace with the recent literature in hand, which will probably be even more enlightening.
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.