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Cumrun Vafa: mathematical introduction to string theory

Placing dualities at the center

A few days ago, Cumrun Vafa of Harvard was invited to Brazil to speak about the mathematical aspects of string theory in an introductory way. You may guess that the place is in Brazil because the flag next to Cumrun resembles the 1-sigma and 2-sigma confidence bands in a colorful exclusion graph. The Brazilians must love particle physics to have chosen such colors. ;-)

The task facing Cumrun is of course tough because string theory depends on a big percentage of the intuition that people learn when they study physics, not just mathematics, and it's also hard because even though many quantities that appear in string theory are totally exact or well-defined, there is no known definition of string theory that would be both rigorous and universal – covering everything that string theorists investigate.

But it's still rather natural for Cumrun to present string theory from the mathematical vantage point because he surely belongs among the 50% of the string theorists who are excellent mathematicians at heart.

Nevertheless, Cumrun gave the 65-minute talk and you are invited to watch it. At the beginning, Cumrun says something about the history of physics and mathematics and their relationships and claims that the existing links between string theory are more tight than string theory's links to experiments which haven't proven the theory yet.

The first notion that he discusses – perhaps unexpectedly, in a way that makes Cumrun's pedagogic approach modern – is the concept of a "duality" that he wisely translates as an "isomorphism" to the spiritual jargon of mathematicians. This is of course a perfectly valid translation. A duality is some equivalence that preserves all the relevant aspects of two algebraic structures – in the case of physics, all observable features of two physical situations. The situations – like the two algebraic structures – may be described by different languages and look "different" but a closer, nontrivial scrutiny may show that "everything that really matters" (everything that isn't just an aspect of the language) about them is the same.

Closely related: You may read a 17-page transcript of a dialogue between Witten, Ooguri, Toda, and Yamazaki after they gave him the Kyoto Prize. These Japanese scholars may very well be responsible for John Horgan's having been the previous Witten's interviewer. The dialogue of the 4 scholars says a lot about the history of dualities, something about holography, and then especially on the geometric Langlands program and the Khovanov homology that Witten worked on in recent years, aside from superstring multiloop perturbation theory. A brief manual how to work with mathematicians and what a young student should do to become Witten is attached. Hat tip: V.H. Satheškumar
He maps two general calculations that depend on some parameters to each other and quickly points out that the two situations may look completely – topologically – different from each other but the invariant physics is still perfectly isomorphic. The spacetime dimension itself is therefore "ill-defined" in the purely physical sense. Dualities lead to lots of identities in mathematics that are surprising but appear to be true, anyway. Physics of string theory "guarantees" these identities, even identities that people wouldn't be able to mathematically prove before string theory entered the stage.

String/M-theory is drawn as the octopus with 6 or so tentacles and in the vicinity of these tentacles or corners, one may calculate and things agree when extrapolated to the other corners. The extrapolation of the most general quantities to the deepest bulk of string/M-theory seems hard in most cases.

The first really explicit example of the dualities is the mirror symmetry. Given the complexities of mirror symmetry, I really guess that this wouldn't be viewed as a pedagogically smooth way to start the introduction of string theory if the audience were not mathematicians. Thirteen minutes after you heard about "string theory" for the first time, they are already supposed to grasp mirror symmetry. ;-) But of course, with the mathematical background, this pedagogical approach isn't as ludicrous as it sounds here.

OK, he reverts the Hodge diamond of the Kähler manifold, says that it usually changes the topology of the space, but the physics of strings on the two manifolds is the same. The Kähler deformations and the complex structure deformations are interchanged. The counting of rational curves which is hard may be rewritten as an easy integral of a holomorphic \(n\)-form. I am not going to explain it here; there have been some blog posts in the past when I did try to explain these things but I still feel that comprehensive books do a better job in teaching the general uncontroversial material of this type.

Cumrun doesn't forget to emphasize – and I think it's very important to do so in front of mathematicians because they often make the conceptual mistake – that mirror symmetry isn't just a tool to count the rational curves. There are infinitely many different statements that are comparably nontrivial but different but that still follow from the full duality, an isomorphism between two physical situations. So the counting of the rational curves is just a tip of a Seiberg.. or iceberg.

After the mirror symmetry on complicated curved Calabi-Yau manifolds, Cumrun talks about the T-duality on tori – and links it to the Fourier-Mukai transform that mathematicians tend to know. He says that the previous mirror symmetry may be derived as an application of the toroidal T-duality on fibers at each point of the base. Physicists would probably find it easier to start with the torus and then talk about some curved complicated beasts – but mathematicians may really sort the complexity in the opposite way.

He says that the theorems and identities hiding behind other dualities are demonstrably true (many checks) and partially provable but it is not really understood well from a purely mathematical viewpoint why these things are true. He says that the ADE classification of subgroups, Lie groups, and Platonic polyhedra pretty much vindicates some ancient speculations that have been considered superstitions for centuries or millenniums. The Dynkin diagrams may really be reinterpreted as intersecting spheres, and there is a link between an ADE Lie group and a finite subgroup of \(SU(2)\), and string theory on the \(\CC^2/\Gamma\) orbifold really explains why this bizarre relationship exists (because it implies that the Lie group arises as the enhanced gauge group at the singularity). I love to say it in the same way, it's really a wonderful example of a deep insight in string theory.

Cumrun's next headline says "geometric engineering of physics". Self-dual connections or instantons/bundles on \(\RR^4\) may be embedded into stringy physics on a bundle over a resolved \(\CC^2/ \ZZ_2\) singularity times \(\RR^4\). The instantons live on the uncompactified latter \(\RR^4\) but their properties are encoded in the compactified dimensions. So we really have a triality here – instantons in \(\RR^4\) are related to enumerative geometry as well as some integrals on manifolds.

Around 38:00, Vafa starts to talk about surgeries. It's "Morse fix" of a cone over \(S^p\times S^q\), e.g. \(S^2\times S^3\), which can be resolved in two different ways. Geometric transitions have a physical interpretation. He wraps a brane over the topologically nontrivial small cycle, he switches from the symplectic to the holomorphic language. Answers computed on one side may be used for the other side. For example, you may replace \(S^3\) by a general 3-manifold and by wrapping \(N\) branes on the cycle, you get \(SU(N)\) Chern-Simons theory and you may compute the Gromov-Witten invariants related to knot theory. Lots of branches of mathematics are being connected. Vafa frames the full duality as an infinite-dimensional generalization of these "toy models".

Random Gaussian matrix models are linked, via a low-dimensional version (or example) of the mirror symmetry, to the simplified deformed conifold and to the semicircle which is the limiting distribution of eigenvalues of the (very large) random matrices. A generalization to a field-dependent exponent relates the distribution of eigenvalues to the period integrals.

His final example involves black holes on \(M^6\times \RR^4\). The clock behind Vafa shows almost the same time as my clock. ;-) Vafa constructs a black hole – you may construct yours and name it after you – by wrapping a brane on a cycle. And you may count microstates. This was probably too mathematicians-oriented presentation because if this were supposed to be something like a presentation of Strominger-Vafa, I could have recognized it only at the end when the K3 appeared. :-) Cohomologies, the Euler class, etc. for large cycles. Finally, the elliptic genus was used to solve it. The coefficients in the expansion of the elliptic genuses were linked to the Bekenstein-Hawking entropy but Strominger and Vafa could have gotten further – they had the exact, and elegant, answer.

The connections, while not always rigorous, are far-reaching and exciting.

Questions. Cumrun gave a 1-hour talk about some fascinating particular insights that string theory made for mathematics but I would have made a bet that the first question would be about experimental testability of string theory which has nothing to do with anything that Cumrun was talking about. People living everywhere are staggeringly stupid and Brazil is no exception, of course. OK, the question is whether the existence of black holes proves string theory. Cumrun says that black holes and even gravity are indeed proofs of string theory (here's a proof of string theory: drop a pen) but we want a more concrete proof that specifically confirms string theory and no other conceivable theory.

The second question was better: What happens with the duality if the black hole Hawking evaporates? Cumrun says that the black holes in his example were BPS i.e. extremal and stable. They don't evaporate. Vafa says that one may also check the law for the first deviations from the extremality. Well, there are really examples of completely non-BPS (and far-from-extremal) black holes for which the stringy counting has been successfully checked, too. The Kerr black holes in \(d=4\) which are in no way near BPS have the right microscopic entropy thanks to the AdS/CFT methods because the underlying calculation is always one of the BTZ black hole and a calculation in a \(CFT_2\).

Third question, an even more technical one. Is there any reason in the black hole description why there are modular forms? The answer is No, it is a surprise, and only string theory explains why it is relevant there. All the special features of the form that make it "modular" are about the corrections to the Bekenstein-Hawking formula which are invisible in the simple GR language. When answering the last question, Cumrun said that some \(N\) from the open Gromov-Witten theory was translated to the Kähler size of a \({\mathbb P}^1\) (with an extra string coupling factor).

This duality-based talk is a conceptually modern and unusual way to introduce string theory, one in which "strings" are almost entirely invisible, of course. They become visible if you actually try to understand where all the formulae that ultimately match come from because even though string/M-theory is no longer thought to be just a "theory of strings", it's still pretty much true that everything we know about string/M-theory (including M-theory, D-branes, and their assorted physics) may be boiled down to physics of fundamental strings plus mathematics.

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reader Tony said...


Years of downloading PDF free books on manifolds, topology, groups, etc. have paid off : I could understand some of this.

Even the term isomorphism is now more familiar than duality.

reader Luboš Motl said...

Great news, Tony.

While I usually prefer physics language over the mathematical one, "isomorphism" is still more likely to be used in non-science situations than "duality". ;-)

There's a difference here - the isomorphism is some one-to-one map of particular elements of an algebraic structure; and the map has to "commute" with some particular operations on the two structures.

In physics, the duality is mapping some "calculable objects" on both sides - which are playing the role of the elements, but are less well-defined (what is a "calculable object" in general?) and the duality has to preserve all the "truly measurable" features and relationships between the objects.

So if one tries to sell a duality as an isomorphism, it looks much less vague and well-defined, and too heuristic. However, this vagueness is just a mathematician's illusion because "what is truly physical and measurable" about a physical theory actually looks totally well-defined in any well-defined theory, like string theory or its particular compactification. It may be "derived" what is observable. ;-)

At any rate, it's great for you to have studied those manifolds etc. Have you read Nakahara's book, for example?

reader Tony said...

No. I see it can be ordered from Amazon, but after some search I found a free PDF (some student's personal site) that I'm downloading now.

Not that I'm too cheap, but I have a regular job and carrying and then reading paper books in spare time, while waiting for programs to compile, is a bit clumsy. Also, late night, before falling asleep, I prefer reading on Kindle, so PDFs are ideal.

Thanks for the suggestion and you are welcome to make many more!

reader Dilaton said...

Hi Tony,

could you give the link to the Nakahara book PDF?

I look forward to watch the talk Monday evening (rather busy at present) by applying my method of pausing after each slid to read and think about it :-).
Cumrun Vafa can explain very well, and I liked some other talks of him a lot too.

reader Tony said...

Amazon doesn't have a Kindle version, but I wouldn't buy it anyway. Kindle ebooks have horrible renderings of diagrams and mathematical formulas.

reader kneemo said...

Geometry of Differential Forms by Morita is also a good read, once you're finished with Nakahara (This is the text order at caltech, in H. Ooguri and A. Kapustin's courses). To get deeper into Hodge theory, a standard text is The Hodge Theory of Projective Manifolds (A text in the UCLA Calabi-Yau seminar).

reader Tony said...

or just search for "mikio nakahara pdf" and you will find a few. Some are only the first edition, though. The above one is the second edition.

reader Tony said...

Plus, other PDFs are grainy, fonts are not rendered well (maybe PDFs are made out of photographs), so yes, the above one is the best.

reader kashyap vasavada said...

Nice talk. But I will have to listen to it several times to
understand even part of it. However I
have a simple question. I got the feeling that mathematically ST would predict
large number of different kind of black holes. If new type of black holes are
found which cannot be explained by GR that would be a triumph of ST. Or have I
misunderstood the point? What do you think?

reader Adam Wolfram said...

I just took a huge dump and QED proved string theory. I'm a genius! Sign me up for Strings 2356 at which point Strings will have nothing to do with the theory except as a reference to its nominal origin. But seriously or not Lubos, the entropy of the Universe will go to infinity long before you guys have enough energy to do all the string conferences you're going to need to flesh this monster out. That means that you're going to need to figure out how sentient beings in another Universe can get information about Strings 2677897 so that progress can continue. Maybe do what Brian Greene did on his PBS ad and use a gravity phone! It's real expensive probably so call collect. (And if you don't like a comment, delete it.)

reader NumCracker said...

LOL, of course we love particle physics, we've discovered the pion even without accelerators ;-)

reader Don said...

Thank you for posting, Dr. Motl. I understand now when it is said "M" theory represents something but nobody knows what it is. Prof Vafa's exposition makes it quite clear that it is as if there is an underground complex of caves linking various different math theories (some of which can be interpreted in terms of physics), but nobody knows what the structure is through which the maze of connections is passing. That unknown structure is the Mystery theory.

This is a very different beast than anything I have seen on the mathematical landscape. It makes set theory and other such like category theory seem quite contrived in comparison. It looks like there is some mathematical superstructure at issue here, as opposed to "just" a physical theory.

Stupid question time: At 44:30, where he shows the link between the (excuse my lack of familiarity with the proper names) "knot" theory on the left and the "other" theory on the right, could you use this for example to study the knot structures of the magnetic field in the sun using the theory on the right and perhaps simplify what might otherwise be a difficult problem?

Thank you for pointing this video out. I feel like I've moved up one notch in my understanding of all this (from 0 to 0 + delta).

Very best,


reader Tony said...

I think he didn't delete your post in order to demonstrate how stupid and arrogant the people, who like to voice their crap opinions against the String Theory, are.

You convinced everyone here that there must be an element of truth in anything opposite to what you say.

reader Luboš Motl said...

Dear Don, it is not entirely true. Lots about the structure is known. Moreover, the term M-theory has been used for something much less ill-defined and mysterious than you suggest. It's any supersymmetric vacuum in the whole theory of quantum gravity that has the total number of spacetime dimensions equal to 11. In particular, the fully decompactified 11-dimensional spacetime of M-theory may be described by the BFSS matrix model which is as well-defined as a Hamiltonian for the nonrelativistic hydrogen atom. In principle, it is completely known.

Then there's the "full" theory allowing all compactifications, all superselection sectors, all changes of the compactification in the most general way - I use the term string/M-theory for the whole animal. Lots of it is incalculable in present, even in principle, but perhaps an even bigger part - especially the broad physical properties of objects, effects, and their transformation - is known, often accurately and sometimes with perfect accuracy.

I can't really imagine what could be so difficult in knows that appear in the Sun's magnetic field. Knot theory is only powerful once one studies really, really complicated knots and their equivalence or inequivalence and other properties. Are the knots in the Sun so difficult?

reader Peter F. said...

I agree with Tony.

I am 'adamant' in my assessment (done with a finely tuned central neural actention selection serving system) that your arrogance is somewhere at the top of a pile of the worst possible arrogances - in any Universe! ;<
Also, almost forgot to mention that I immediately associated your comment with a smell emanating from the rear of the middle portion of a Wolf's frame (or your bottom) whichever is the smelliest. ;-{

reader Leo Vuyk said...

The search for the math of knots is apart from the design IMHO challenging.
See ""Design for Sub Quantum Physics or 3D heterotic string".

reader dalpezzo said...

Lubos, I don't mean to be picky here, but by an "isomorphism" one usually means a morphism that is invertible. And what you mean by a "morphism" depends on your level of sophistication (or perversity, if you wish). E.g., it could be a morphism in the category of categories, i.e. a functor. When discussing dualities you usually need equivalences (and not isomorphisms) of categories, but nonetheless this is not as vague as it may sound :)

reader Luboš Motl said...

Dear delpezzo, partly true. The word "functor" could often be even better than "isomorphism" for "dualities".

This is a purely terminological issue but you're also right that mathematicians are using the term "equivalence of categories". As far as I can say, this interpretation of the mathematical term "equivalence" isn't a special case of the "equivalence" in mathematics that is a type of "relation". See

The article above makes the terminology especially confusing because while discussing equivalence of categories, they also often speak about opposite or "dual" categories - and therefore also about "dually equivalent categories". ;-)

The duality in physics isn't necessarily quite about being "opposite" although in some sense it is, so one can't even say whether the duality in physics is more correctly interpreted as "equivalence of categories" or "duality of categories". ;-)

One probably shouldn't get carried away because the physics dualities aren't a special example of any "specific" equivalence of categories or duality of categories that mathematicians actually work with, or am I wrong?