I've sorted the keywords in the title alphabetically.
After Peter Woit read my words about the beauty of string theory, he wrote his own essay about beauty and string theory. If even Peter Woit meditates about the beauty of superstrings, it proves that we're definitely making some sort of progress, at least in the P.R. business! :-)
Note added later: After Peter Woit looked at this page about the links between mathematics and string theory, he wrote his own version of history of topological field (and string) theories. His picture of the history has one serious problem: it ends in 1989, and no newer insights are taken into account. Why is the end of history exactly in 1989? You might think that it may be related to the collapse of communism. Well, that's not the real reason. You will understand the correct explanation why the history of maths and physics after 1989, according to Peter Woit, does not exist, if you look here.
Let me now return to Peter's first article about the beauty of string theory. He divides the problem into five main categories:
- The beauty of symmetries - Peter Woit realizes that spacetime symmetries seem to be derived concepts in string theory. He nevertheless understands why they're beautiful, and we would probably agree about this point
- Miraculous nature of cancellations of anomalies and inconsistencies - his example is the Green-Schwarz anomaly cancellation, but I was thinking about a much more general set of ideas. It's not just the usual type of "anomaly" that must cancel for a theory to make sense, and the spectrum of different details that happen to work (and have to work) in string theory is much broader
- Uniqueness of the theory as a description of the real world, including quantum phenomena and gravity - which includes the absence of adjustable non-dynamical continuous parameters
- String theory as the extension of quantum field theory - Peter realizes that string theory is the only known framework that is able to go "beyond" quantum field theory, without spoiling its essential good features
- Beautiful connections to new pure mathematics - Peter says many incorrect things which I will correct below
One can find many limits in string theory - situations, solutions, or points in the moduli space - where it naively seems that singularities and infinities start to develop. String theory always offers (and predicts) new objects or phenomena - wrapped strings; wrapped branes; infinite tower of new states that "smear out" the spacetime at short distances; enhanced gauge symmetries; new light states in general; non-commutative geometry; worldsheet instantons, and so forth - and these new objects or phenomena imply that physics continues to be completely controllable, predictable, and smooth. That's a beautiful interplay of all the players - something that is necessary to avoid many otherwise conceivable disasters.
But let's say a couple of words about the last point - the relations to mathematics. Peter Woit speculates that string theory "has been an utter disaster for theoretical physics". Well, most of us know that it is nonsense and we know why Peter Woit likes to say this nonsense.
String theory and its links with mathematics
But he also writes many more specific statements that are not true. He admits that many interesting things come from two-dimensional conformal field theory. Well, perturbative string theory is nothing else than a sophisticated application of the important tools and theorems in two-dimensional conformal field theory. Consequently, it would be very hard for Peter Woit or anyone else to argue that the developments in two-dimensional conformal field theory have nothing to do with string theory.
But the last two sentences of his text may be even more problematic. He claims that it is quantum field theory, not string theory, that has had this huge impact on mathematics. He even says that Witten's Fields medal was "not for anything he has done using string theory".
Well, Peter Woit is not right. Let's start with topological string theory and the Fields medal because this is the most obvious way to show how much wrong Peter Woit is. A paper that Peter Woit believes is behind the Fields medal is
- Edward Witten: Quantum field theory and the Jones polynomial
The paper using the three-dimensional Chern-Simons theory was published roughly one year after another important paper in which Witten established topological string theory:
- Edward Witten: Topological sigma models
Later, another 1+1-dimensional conformal field theory was found to be closely related to this three-dimensional Chern-Simons theory, namely the worldsheet theory of the topological string.
Note added later: Incidentally, I said that "Peter Woit believes that Witten has received the Fields medal for that paper" because the correct answer is different: Witten received his Fields medal mainly for
- his proof of the positive mass theorem, based on supersymmetry (certainly something rooted in reasoning related to string theory; finally, Peter Woit hates supersymmetry nearly as much as he hates string theory, so it does not make much difference)
- his paper on supersymmetry and Morse theory (the same comment applies)
- his work on the Jones polynomial from the Chern-Simons theory
You know, a similar story can be said about most of other ways how theoretical physics influenced mathematics in the last 20 years. If you say that these influences were really influences of quantum field theory only, you really misunderstand the situation; you misunderstand what "string theory" means.
Today, we still don't know the universal principle behind string theory. Nevertheless, we're able to study string theory using many different techniques. But it is important that all these known techniques are based on quantum field theories in various spaces. It's important that these spaces are not necessarily the spacetime itself - the relevant quantum field theories may describe the worldsheet or worldvolume of strings, branes, and other objects. These spaces may become boundaries of the spacetime, and so forth.
The whole perturbative string theory is usually computed using a sophisticated and unusual treatment of two-dimensional conformal field theory. The S-matrix in spacetime is calculated as the correlator of "vertex operators" - operators associated with the external string states - integrated over all possible Riemann surfaces (worldsheets) into which these vertex operators can be inserted. Two-dimensional conformal field theory is a quantum field theory.
The AdS/CFT correspondence describes the exact stringy physics (a stringy extension of quantum gravity) in the (d+1)-dimensional anti de Sitter space as a conformal field theory in d dimensions. Again, this conformal field theory is a quantum field theory.
Also, Matrix theory describes physics at some backgrounds of string/M-theory in terms of a quantum field theory (or quantum mechanics, which is nothing else than a 0+1-dimensional quantum field theory) on some auxilliary space.
An off-shell approach to string theory, called string field theory, is a generalized quantum field theory in spacetime with infinitely many (component) fields. Moreover, string theorists systematically use effective field theories in spacetime, and effective field theories on the branes (as low-energy approximations). Even more exotic theories, such as the (2,0) theory or non-commutative Yang-Mills theory, can be viewed as generalized quantum field theories.
What I want to say is that all approaches to string theory that we know of today are based on quantum field theories in various spaces. What's "stringy" about them is the way how you use these quantum field theories, how you relate them, and how you interpret them physically. This also determines what the relevant and interesting generalizations are, and so forth. The "real" string theory is a set of ideas that tell us which quantum field theories are interesting, and what new, physically interesting behavior these theories give us. String theory also teaches us about hundreds of non-trivial connections between these quantum theories. These are theories whose properties are always described by a sophisticated application of quantum field theory.
String theory implies new ways to think about quantum field theories, and it has allowed the physicists to derive many unexpected properties of quantum field theories.
Once again, string theory, as we know it today, is the "spirit" that organizes the insights about different quantum field theories. We definitely think about string theory as something that is more general than quantum field theory, but every time we want to describe this more general structure quantitatively, we use the tools of quantum field theory (usually QFT in other spaces than you would expect).But we still have the full string theory and its logic in mind.
It is this "spirit" that was important for Witten to make most of his important insights that affected mathematics. It is this "spirit" that Seiberg and Witten had in mind when they were solving the N=2 supersymmetric theories in four dimensions. They did not write down that they were solving a problem rooted in string theory - because they did not want to lose the citations from the colleagues who would never cite a paper with the words "string theory" in it (except for your papers, Peter).
Nevertheless, it was string theory and its "spirit" that led them to this solution, too - and the connections of the followups with string theory are much more obvious and much more difficult to hide. Moreover, there are many other obvious ways how string theory influenced mathematics, and Peter Woit does not want to see them. Some of them are related to mirror symmetry - a mind-boggling relation between two apparently different six-dimensional manifolds that was first found in string theory, but it has also allowed to find many theorems in pure mathematics.
An alternative conclusion
Even though I was explaining that string theory is technically a conglomerate of "quantum field theories" in various spaces with some new ideas how to organize their observables, I believe that eventually we will find a completely new formalism that cannot be reduced to any specific quantum field theory.