## April 14, 2011

### Segal on Three roles of quantum field theory

#### Posted by Urs Schreiber

This May Graeme Segal will give the Felix Klein Lectures at the Hausdorff Center for Mathematics in Bonn.

Three roles of quantum field theory

Abstract Quantum field theory has many roles, and the lectures will be about three of them. The primary role is to provide a description of all of fundamental physics when gravity is firmly excluded. A second, at first surprising, role has emerged from string theory, which is a theory of gravitation: it turns out that a two-dimensional field theory can be regarded as a generalized manifold, and in particular can be a model for space-time. Thirdly, quite apart from physics, the concept of a field theory has taken on a new life as an organizing principle in other areas of mathematics - not only in geometry and representation theory, but even in connection with quantum computing.

The three roles can be seen together as aspects of noncommutative geometry, and that will be a central theme of the lectures. The talks will aim to show how powerful the field theory idea is by jumping between a variety of contexts, beginning from the origins of the structure in particle physics, with a little about so-called “Wick rotation”, and then moving towards more pure-mathematical applications to analysis, algebra, and the structure of manifolds.

Posted at April 14, 2011 12:20 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2386

### Re: Segal on Three roles of quantum field theory

Graeme Segal wrote:

A second, at first surprising, role has emerged from string theory, which is a theory of gravitation: it turns out that a two-dimensional field theory can be regarded as a generalized manifold, and in particular can be a model for space-time.

How important is the number 2 here? Is there a reason why 1d or 3d field theories can’t be regarded as generalized manifolds?

Posted by: John Baez on April 14, 2011 11:06 AM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

Is there a reason why 1d or 3d field theories can’t be regarded as generalized manifolds?

No, they can be.

For 1d this is well known. For 3d almost nothing is really known, but it should work analogously:

A 1-dimensional Riemannian QFT with a single binary interaction is essentially the same thing as a spectral triple (Kontsevich-Soibelman speak of graph field theory , see the $n$Café entry here):

the Hilbert space assigned to a point is the Hilbert space of the spectral triple, the Hamiltonian of the 1d QFT is the Laplace operator of the spectral triple, or rather the Dirac operator for a supersymmetric 1d QFT.

In other words, quantum mechanics is a way to talk about the Riemannian geometry of the target space in which a quantum particle propagates in terms of the energy spectrum of the particle (can you hear the shape of spacetime?). I guess this is how Connes came up with the idea of spectral geometry (usually known, misleadingly, as (Connes-style) _ noncommutative geometry_ ) in the first place, possibly inspired by Witten’s early observations on how supersymmetric quantum mechanics encodes interesting geometry on target space. I once wrote a series of $n$Café entries explaining how Connes tries to realize our observed world as the effective target space of a 1d QFT (a quantum superparticle) in this fashion: here.

When one goes from 1d to 2d, spectral triples become 2-spectral triples, usually known as 2d CFTs: there are now two Laplace operators or Dirac operators, the “left moving” and the “right moving” one, known as the Virasoro generators if we are speaking of a 2d conformal CFT. The premise of string theory is to take this seriously and investigate the hypothesis that the world around us is not actually the 1-spectral geometry that Connes considers, but an actual 2-spectral geometry. (And it is maybe amusing to note that both Connes as well as string theory find that these spectral geometries have to have KO-dimension 10, for consistency.)

Liang Kong has a nice set of notes with an exposition of this idea for a course he taught. He calls it “stringy algebraic geometry”. We are just in a middle of an email conversation where I was arguing that he should rather call it “string spectral geometry”. The plan is that his notes will make it to a contribution in our book.

One can take the “point particle limit” of a 2dQFT describing a string propagating inside such a spectral geometry, in which it goes over to look just like a point particle. Accordingly there is a limit that takes a “2-spectral triple” (a 2d CFT) to an ordinary spectral triple. This has been originally studied at a mathematical level by Roggenkamp-Wendland and by Kontsevich-Soibelman (see the above link). Yan Soibelman has now written it up all nicely in his contribution to our book. Not much longer and I can show this around.

For 3d the guess is that the story continues analogously, becoming ever richer. That’s the hypothesis, essentially, underlying the idea that “the super-membrane is the fundamental degree of freedom of M-theory”: based on the above well-understood story for particles and strings, and given a host of hints that it makes sense to consider these 2d CFTs in turn as string-limits (called “double dimensional reduction”) of 3d QFTs of membranes, the natural guess is that for suitable 3d QFTs one can understand these as describing the dynamics of quantum membranes in the effective spectral geometry encoded by the QFT.

So I think the answer is: dimension 2 here is special (only) in that it is the highest dimension where we can make progress with understanding the situation. There are hints that also d = 6 should be tractable, being the EM-dual of $d = 2$ for 10-dimensional targets

Posted by: Urs Schreiber on April 14, 2011 12:17 PM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

Urs wrote:

So I think the answer is: dimension 2 here is special (only) in that it is the highest dimension where we can make progress with understanding the situation.

That’s what I thought, mainly because I’ve been reading what you’ve written for years on this topic. Thanks for the very nice clear summary! It would be nice if Segal would mention in his talk that the focus on dimension 2 in his statement:

a two-dimensional field theory can be regarded as a generalized manifold, and in particular can be a model for space-time

is a matter of practicality rather than a fundamental principle. (Of course, the practical, computation-friendly nature of conformal field theory itself arises from fundamental principles: mainly that the complex plane is 2-dimensional.)

Posted by: John Baez on April 15, 2011 5:25 AM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

I would like to know what makes some arithmetic geometers apparently interested in conformal field theory (if that impression should be correct). E.g. it is mentioned in connection with the Langlands program, Faltings recently talked on a “Verlinde Formula” comming from it, dimensional regularization has been mentioned in connection with “fractional motives”.

Posted by: Thomas on April 14, 2011 1:04 PM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

I don’t understand this stuff nearly as well as I’d like, but my impression is that ultimately, the modern relation between arithmetic geometry and conformal field theory is just a modern outgrowth of the old analogy between number fields and function fields of Riemann surfaces. Now that string theorists have become experts at doing quantum field theory on Riemann surfaces, a bunch of their ideas should generalize to number fields.

Posted by: John Baez on April 15, 2011 5:30 AM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

I wonder if conformal field theory is “natural” in the sense that mathematicians would have found it anyway, independent from physics?

Posted by: Thomas on May 27, 2011 9:49 PM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

I wonder if conformal field theory is “natural” in the sense that mathematicians would have found it anyway, independent from physics?

One could argue that this is indeed roughly what did happen:

all the conformal field theory that pertains to phenomena seen directly in physics laboratories so far is that describing critical phenomena in thermodynamical systems, and this is all taking place on the Riemann surface $\mathbb{C}$, the complex plane, only.

The study of “full CFT” involving arbitrary Riemann surfaces is motivated so far only on formal grounds: it is mathematically compelling to consider this, and the outcome is rich enough to motivate deeper investigation. One can and does argue that there is lots of indication that this is relevant for physics at a deep level, but the main motivation for its study has not come out of the laboratory, but out of extrapolation of mathematical structures.

Posted by: Urs Schreiber on May 28, 2011 8:02 AM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

I have missed the first session today. Did anyone reading this here attend?

Posted by: Urs Schreiber on May 2, 2011 3:10 PM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

Dear Urs,

there are videos of his lectures:

http://www.mpim-bonn.mpg.de/node/3372/abstracts

http://www.mpim-bonn.mpg.de/node/3372/program

Best wishes and regards

Jenny

Posted by: Jenny Santoso on May 9, 2011 8:53 PM | Permalink | Reply to this

### Re: Segal on Three roles of quantum field theory

there are videos of his lectures:

I have collected them now on the $n$Lab at Three Roles of Quantum Field Theory .