String Theory's Contributions to Mathematics

[Animastryfe]

I was just reading Forbes' list of the world's most influential thinkers, and the profile on Ed Witten mentioned how research on string theory lead to entirely new mathematics. My question is, how has string theory impacted mathematics? How much original mathematics has it lead to? Do non-string theorists use this math?

P.S. I know almost nothing about string theory, so even though I know there are different types of string theory, I'm lumping them together here.

 

[petergreat]

I don't have technical knowledge about the issue, but the most lauded contribution to mathematics by string theory is mirror symmetry:
http://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory)

 

[element4]

I was just reading Forbes' list of the world's most influential thinkers, and the profile on Ed Witten mentioned how research on string theory lead to entirely new mathematics. My question is, how has string theory impacted mathematics? How much original mathematics has it lead to? Do non-string theorists use this math?

I think there are many different examples. In 1989 Ed Witten showed a deep connection between knot invariants like Jones polynomials (and other topological invariants), and topological field theory (in particular Chern-Simons theory). This had huge impact on low dimensional topology and, as far as I know, was the reason why Witten received the Field Medal.

The math used (and developed) in string theory IS actually being used in other areas of physics (ie in Condensed Matter Physics), although not as much. As an example, the above mentioned result from Witten is used to study non-abelian anyons in Fractional Quantum Hall effect and to calculate link invariants among other things (see http://arxiv.org/abs/0707.1889" ).
Math such as algebraic topology and K-theory, which I think was first introduced in physics by string theorists, is also extensively used by some condensed matter physicists these days.

 

[BruceG]

The example I like is Moonshine Theory (aka Monstrous Moonshine). This carves a path all the way from number theory (the study of prime numbers) all the way to string-physics.

When I did my maths degree, I was most interested in mathematical physics, so I dropped the Number Theory course assuming it can't possibly have any relevance to physics. It seems that I was wrong.

I have yet to learn much about it yet, so I can't comment on how much the link was inspired from the physics-end: can anyone comment more on this?

 

[humanino]

The role of the Monstrous Moonshine in physics stems from conformal symmetry, which is also important in string theory, but is not restricted to string theory. Conformal symmetry would have been studied without strings, in QCD or critical phenomena in condensed matter (for instance). I do not agree that string theory per se can be counted as a contributor to the Monstrous Moonshine business.

 

[BruceG]

Conformal symmetry would have been studied without strings, in QCD or critical phenomena in condensed matter (for instance).

OK but then would not mathematicians have studied conformal symmetry without any help from physics.
What is often said about great physicists like Witten and Feynman is that they have brought physical insight to areas of mathematics. Can you say what such a claim actually means given that modern physics is so rarefied it is hard for me to see it has any intuition beyond what comes from understanding the mathematics?

 

[humanino]

This thread is about string theory's contributions to mathematics. Conformal symmetry was already known and studied before string theory came around. Many tools were already in place before 2-dimensional conformal symmetry was introduced in string theory. If you think there is a clear contribution of string theory to the Monstrous Moonshine, it is up to you to clarify it, I cannot refute something you have not provided. There exists speculations about the role of the Monstrous Moonshine in string theory, but the conformal symmetry tools relevant for the Monstrous Moonshine do not stem from string theory AFAIK.

 

[atyy]

This thread is about string theory's contributions to mathematics. Conformal symmetry was already known and studied before string theory came around. Many tools were already in place before 2-dimensional conformal symmetry was introduced in string theory. If you think there is a clear contribution of string theory to the Monstrous Moonshine, it is up to you to clarify it, I cannot refute something you have not provided. There exists speculations about the role of the Monstrous Moonshine in string theory, but the conformal symmetry tools relevant for the Monstrous Moonshine do not stem from string theory AFAIK.

A description of Borcherds work in http://www.ams.org/notices/199901/fields.pdf says it's related to string theory: "he drew on a rich variety of ideas, among them ideas from vertex algebra theory, the theory of Borcherds algebras (particularly his singularly interesting “Monster Lie algebra”), string theory (especially, critical 26-dimensional string theory and the “no-ghost theorem” of R. C. Brower, P. Goddard, and C. Thorn), and modular function theory."

 

[humanino]

The article by Belavin, Polyakov and Zamolodchikov from 1984 referred to in the reference above is cited in the book "conformal field theory" by Di Francesco, Mathieu and Senechal as belonging to statistical physics, and "by a fortunate coincidence" important progress in string theory was made the same year by Green and Schwartz.

Also, the article above says "this construction was re-interpreted by physicists during the resurgence of string theory".

So I still do not see how the contribution comes from string theory. It may be related, but it did not originate from.

 

[atyy]

How about the Brower, Goddard and Thorn "no-ghost theorem"?

 

[humanino]

How about the Brower, Goddard and Thorn "no-ghost theorem"?

If you count that as string theory, then in that case I understand the above statements.

Those are very early days of string theory, dual resonance models, when the role of the "string" itself was not so relevant, no role for a lagrangian, just operator product expansions, associative algebra and bootstrap : essentially everything given by conformal symmetry (which we know is relevant to strong interactions), without reference to a string even as an effective model. I do not know how much emphasis was given to the "string" aspect at this point in time. When I read the original articles (including those for "no-ghost theorem"), the word "string" never appears. Since I was not born yet, I am unfamiliar with the psychological setting of people at that time.

 

[atyy]

Interesting! Goddard and Thorn's paper is from 1972, which is also before I was born Their title is "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model", which doesn't mention strings explicitly. However, by 1992, Borcherds proof explicitly mentions it as string theory http://math.berkeley.edu/~reb/papers/monster/monster.pdf