# Mathematician trying to learn string theory

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?

• Is this a question about Mathematics research? You might get better answers on a Physics Q&A website, such as physics.stackexchange.com – Mark Grant Dec 13 '12 at 8:35
• I think it's more properly under the soft question category . I want to know the answer from a mathematician perspective. – quark Dec 13 '12 at 8:45
• I support this question being posted on MO. I think some of the users here might have a better feeling of what a mathematician needs to learn to have good intuitions in string theory-related mathematics. Also, I wanted to ask such a question myself ;) – Piotr Achinger Dec 13 '12 at 8:46
• Community Wiki, please. – Todd Trimble Dec 13 '12 at 11:54
• @David Roberts, you probably mean this - math.ias.edu/qft – Asaf Dec 13 '12 at 13:34

Many string theorists would like to know more algebraic geometry. There are a few of us who know algebraic geometry at a pretty high level (not me) but many more who would like to learn more and feel it would help with their research but find the literature very difficult. I think the optimal solution would be to find such a string theorist and agree that you will teach them algebraic geometry if they will teach you string theory.

• That's a really nice idea. On the covers of the IAS volumes mentioned by David Roberts, there are cartoons in four panels. The first two panels are set in the mid-60's; in one is a pair of physicists at a blackboard all excited about gauge theory, in the other is a pair of mathematicians at another blackboard all excited about Atiyah-Singer index theory. The second set of panels is set thirty years later; in the first the two physicists are staring with cartoon question marks at Atiyah-Singer index theory, and in the second the two mathematicians are likewise puzzling over gauge theory (cont.) – Todd Trimble Dec 13 '12 at 19:56
• Clearly the foursome should have gotten together much earlier. I'm sure there are similar blackboards with mirror symmetry on one, enumerative algebraic geometry on the other, and the two groups should somehow get together. – Todd Trimble Dec 13 '12 at 19:58
• It would be fun, but would require a lot of time and dedication from both parties. I'm tempted to ask the converse, how does a physicist who knows QM, QFT and string theory learn algebraic geometry, or at least the parts that are most relevant to string theory? The standard answer seems to be to read the first few chapters of Griffiths&Harris and lecture notes by Candelas and others but I wonder if there is a better answer that doesn't involve a willing algebraic geometer. – Jeff Harvey Dec 14 '12 at 1:26
• Candelas once told me that, when he asked Atiyah how to learn algebraic geometry, Atiyah responded: "You can't". At first Candelas thought Atiyah was making a statement about him personally, but what he was saying is that algebraic geometry is such a large subject that understanding it is a full-time occupation. I'm sure string theory is the same. Therein lies the problem, and it doesn't help that when mathematicians and physicists talk about the same object they often do so in very different ways. – anon Dec 14 '12 at 21:27

Since I am a mathematician and also spent quite efforts on learning string theory, etc., let me add some comments.

I agree with David Roberts suggesting this (published book is a little more complete, but not essentially), I partly agree with Chris Gerig "This is more just QFT, and would be a good mathematical perspective after you understand the physics of QFT / String Theory..." I would say that this more concerns lectures by mathematicians: Deligne, Kazhdan, Bernstein, which I would suggest to skip at first reading. And just look at physicits lectures: Faddeev, Gawedzki (fall semester) and Witten, D'Hoker (spring semester)

"... , in particular I don't think it help for the papers that get posted on arXiv hep-ph and hep-th." Well, yes, this volume does not cover most interesting developments made in 90-ies, but nevertheless as some basics sources, it should be Okay.

Let me also agree with Chris Gerig "I mean, if you are really trying to understand String Theory, then you're going to have to become fluent in Classical Mechanics, Quantum Mechanics, Quantum Field Theory, and General Relativity first... "

The way which many people in Russia are doing this - is via volumes by Landua Lifshitz. Let me say that volumes 1-3 (Classical mechanics, Field theory(Classical electrodynamics and General relativity), Quantum mechanics) are quite accessible for mathematicians, even for last years undergrads. But this does not contain Feynman path integral. You may look at Feynman, R.P. and Hibbs, A.R. Quantum Mechanics and Path Integrals. Also LL does not include quantum field theory. You may look at Ramond's short book. And IAS volume discussed above.

You may also look at Igor Dolgachev's (mathematician) "Introduction to physics" http://www.math.lsa.umich.edu/~idolga/lecturenotes.html

Another question it might be worth to figure out - what aim you are setting for yourself. To become a physicist? Let me tell the story - a friend of mine started as a physicist, but later turned to mathematics, I asked him why ? (cause he is really smart and surely had good perspectives). He answered: "you know in physics 1+2+3+4+.... = -1/12, can you live with this ? Me not." Another story about I. Gelfand who being at Rutgers decided to learn some physics, it is started Okay, but at some point, physicist said "here we divide by the volume of the diffeomorphism group" (you always do it in Faddeev's-Popov approach), after that Gelfand stopped this. (The story from my friend who was Gelfand's student and was personally there). I mean for a mathematician absence of proof/(clear understanding) is like a teeth pain, but true physicists will not even observe a problem :) So there is certain cultural and mental difference, and should choose what is more suitable for you.

However, my strong feeling is that mathematical community MUST somehow "learn/absorb/rework/rethink" ideas of QFT and string theory. There are certain important tools and ideas which are now hidden in some physical language and sometimes looks as trick, heuristics, etc..., but should be cleared out, polished, placed in the right position of our mathematical knowledge. We are at certain point where the part of math. community and hep-th community are quite close to each other, this will not be forever. So it is important not to loose a chance of gaining physical "intuition" and making from it mathematical theory.

Let me speculate a little on the possible place of physical ideas in math. It seems to me they are to certain extent "differential geometry of specific infinite-dimensional manifolds". I mean typically in physics we consider the space of all smooth maps from one manifold to another. We write a kind of differential form on this space and integrate it. The problem is that such integration is ill-defined business, however it somehow works. I think that the manifolds we work are not some abstract infinite-dimensional manifolds, but we should take into account that we consider the space of maps from one finite-dim manifold to another finite-dim - an this will lead to certain "semi-infinite" structures. Like vertex operator algebras more or less are loop algebras of finite-dimensional Lie algebras. To certain extent these ideas kind be made quite precise in topological quantum fields theories see e.g. this discussion: Doing geometry using Feynman Path Integral?

• In Ramanujan's second letter to Hardy he said “I told him that the sum of an infinite no. of terms of the series: 1+2+3+4+…=-1/12 under my theory...I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter….verify the results I give and if they agree with your results, got by treading on the groove in which the present day mathematics move, you should at least grant that there may be some truths in my fundamental basis.” So don't blame the physicists! – Jeff Harvey Nov 24 '15 at 21:32

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.

(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.

I mean, if you are really trying to understand String Theory, then you're going to have to become fluent in Classical Mechanics, Quantum Mechanics, Quantum Field Theory, and General Relativity first... otherwise the papers are going to be unmotivated and you won't understand the linguistics and you won't know how the results connect to the universe (i.e. they're more than just a sequence of symbols which we call math).

That being said, assuming CM/QM/QFT/GR are under the belt, the best place to start is Green-Witten-Schwarz's (GWS) Superstring Theory, followed by skimming Polchinski's String Theory. This is supported by my string theory professor when I took it a while ago, Petr Horava (discoverer of D-branes). From here you can supplement other notes and papers.

In Vol.1 of GWS, chapter 2/3 will explain the bosonic string theory (i.e. ignoring fermions) and BRST quantization, which leads to a critical dimension $D=26$. Then chapter 4 will fix this with supersymmetry (i.e. putting back in the fermions), leading to the actual critical dimension $D=10$. After this, gauge anomalies and compactification and dualities and D-branes can start being assessed.

• From what I have gathered in physics , One writes an action and lagrangian invariant under some symmetry groups and derive the equation of motion and other consquences from the action. So,I guess one important thing is to understand symmetries and lagrangians – quark Dec 13 '12 at 18:09
• Understanding Lagrangians and symmetries is important, but some theories don't have Lagrangians. In fact such theories, like the (2,0) theory in six dimensions, are the focus of much recent research. Also, while Petr Horava has done much excellent work, including work that foreshadowed the discovery of D-branes, Joe Polchinski is the person who is generally credited with discovering D-branes in string theory. – Jeff Harvey Dec 13 '12 at 18:21
• But this credit isn't fully justified, it was an independent discovery. – Chris Gerig Dec 13 '12 at 21:40

• String theory is HUGE! – David Roberts Dec 13 '12 at 10:23
• An electronic version of this book is available in DjVu format: libgen.org/…, libgen.org/… – Dmitri Pavlov Dec 13 '12 at 14:49
• This is more just QFT, and would be a good mathematical perspective after you understand the physics of QFT / String Theory, in particular I don't think it help for the papers that get posted on arXiv hep-ph and hep-th. – Chris Gerig Dec 13 '12 at 16:28
• Yeah, you could do worse. Hopefully you could do better, too, because those two volumes are really quite high-level and demanding. It's almost as if it could be titled "Quantum Fields for Fields Medalists" (as in, Fields Medalist Witten assigns "superhomework" to Fields Medalist Deligne and other high-powered friends, and their submitted solutions essentially form a book within a book). – Todd Trimble Dec 13 '12 at 16:41
• Isn't everything Witten writes superhomework for mathematicians anyway? :-P – David Roberts Dec 13 '12 at 21:46

the most basic book I know of is Enumerative Geometry and String Theory by Sheldon Katz.