The n-Category Café

This stuff is beautiful, but it isn’t a structural foundation for string theory.

I see what you mean, but I would say that it is actually – for the “worldvolume aspect” (as specified in the “call for papers”): for everything short of the perturbation series.

It is all too often forgotten that in string theory people talk about worldvolume CFTs and TFTs as if these were understood. Only topological TFTs and rational 2d CFTs were recently fully understood using the mathematical methods in question here. So in as far as CFT is part of the structural foundation of String theory, this stuff here is.

And much of the perturbation series is encoded in the effective background theory. Work on understanding that better is what the “call”-text is trying to indicate in its last bit. There has been progress in structural understanding there, too.

Finally it should be kept in mind that the “landscape” is really a moduli space of CFTs. Everybody in physics talks about it, nobody has the faintest idea of what it actually is, without using the structural insights that we are talking about here.

So the mathematics we are talking about here is in fact pivotal for a huge chunk of what people consider as being string theory. It is not sufficient for the full theory, yet, I agree, but certainly necessary.

For these reasons I felt the title was indeed justified. But also in light of John’s comment below I will try to think of something else. You’d have the same complaint about “mathematical founations”, I assume?

Thanks for the comment. No, I wasn’t even aware of this book and nobody pushed me to choose that phrase. It was just an idea of a phrase I had. But possibly I should try to think of something different. Any suggestions?

I liked “structural foundations” better than “mathematical foundations”, but maybe that latter term is better suited after all?

Most of my objections to the term ‘structural foundations’ vanish if you chose it, and you explain what you mean by it in the preface.

However, there may be some people who think your book is a ‘sequel’ to The Structural Foundations of Quantum Gravity. And that could be distracting.

Most of my objections to the term ‘structural foundations’ vanish if you chose it, and you explain what you mean by it in the preface.

However, there may be some people who think your book is a ‘sequel’ to The Structural Foundations of Quantum Gravity. And that could be distracting.

Okay, thanks for the input.

Inspired by this, I worked a bit more on the proposal:

first of all I did rename the thing into “Mathematical Foundations of Quantum Field and String Theory” for the moment. I like “structural foundations” better on absolute grounds, but I do see that “mathematical foundations” may communicate the idea better.

Then I tried to improve the list of topics a bit.

Mainly, your comment made me start think about what a preface for such a book might look like. So I wrote something, just a first run, here:

Preface.

I should emphasize that I am not alone on this undertaking and haven’t checked what I wrote there with anyone yet, that I am just doodling around and have no idea what of this will survive in the long run. But it should serve to help me think about what all this might be about, and help you to see what I have in mind.

The preface may also be seen as a more in-depth reply to AJ’s comment above.

No problem. Perhaps it’s a little early for proof reading. Later it might be the moment for stylistic suggestions. While I’m here though, one ‘suggested’ should be removed from

Early on it was suggested, based on the topological examples, suggested…

Later it might be the moment for stylistic suggestions.

I would much appreciate it. Thanks.

While I’m here though, one ‘suggested’ should be removed from

Early on it was suggested, based on the topological examples, suggested…

Right, thanks. Incidentally I happen to just come from re-reading the thing myeself and have fixed that one in the process.

you could explain what was accomplished by “the identification of symplectic geometry as the underlying structure of classical Hamiltonian mechanics”.

Okay, I now wrote some more about this at

Symplectic geometry and classical Hamiltonian mechanics

Perhaps we could see how mathematical underpinning of one theory helps with construction of a later theory.

I have two examples.

1. Hamilton’s equations in coordinates $\to$ Hamilton’s equation as $$dH = \omega (\Xi_H, \cdot),$$ where $\Xi_H$ denotes the Hamiltonian vector field of $H$ leads to

$n$ vortices on the sphere as finite dimensional limit of 2D Euler equations: the phase space of the system of $n$ vortices is not a cotangent bundle but $(S^2)^n$, a qualitative change of point of view of what Hamilton’s equations can describe.

2. Jacobi’s elimination of nodes $\to$ Witten’s “Two dimensional gauge theories revisited.”

Here is a telegraphic version of the story: Conservation lawas arising from symmetries have been formalized as moment maps by Kirillov, Kostant and Souriau in late 1960s. Elimination of nodes procedure has been made rigorous by Marsden and Weinstein and, independently, by Meyer, as symplectic reduction (symplectic quotient construction). In the early 1980s Mumford observed that symplectic quotients are closely related to Geometric Invariant Theory quotients and that many moduli spaces important in algebraic geometry and in mathematical physics can be realized as symplectic quotients. Atiyah and Bott used this point of view in “The moment map and equivariant cohomology” to construct cohomology classes of moduli spaces of flat connections on Riemann surfaces. In “Two dimensional gauge theories revisited” Witten conjectured a method for computing the intersection pairings of cohomology classes of symplectic quotients. The work of Atiyah and Bott and Witten’s conjecture stimulated a large research effort to understand the topology of symplectic quotients in terms of the equivariant cohomology of the original spaces. Witten’s conjecture was proved by Jeffrey and Kirwan several years later.

Thanks for your comments!

Calling the path integral folklore could result in some critisim/hostility from some parts of the physics community.

I suppose the whole book could result in such. But that line about folklore is meant to be a bit amplified, I wouldn’t like to water it down too much. But now I changed it at least to

without much more of a structural guidance than some folklore called the path integral, however useful that has proven to be.

You write:

I’m satisfied with the explanation [of conformal field theory] of e.g. Schottenloher: “A Mathematical Introduction to Conformal Field Theory”, but would you care to explain what’s amiss with that?

Right, this deserves more detailed discussion than maybe should go in a preface, but at least it should be indicated. So I added the following paragraph

Most of the literature on 2d conformal field theory describes just what is called chiral conformal field theory formalized in terms of vertex operator algebras or local conformal nets. But this only describes the holomorphic and low-genus aspect of conformal field theory and is just one half of the data required for a full CFT, the remaining piece being the full solution of the sewing constraints that makes the theory well defining on all genera.

Next you write:

With nontrivial examples you probably mean interacting theories in 4dim? That’s Ok with me, but I think the PCT and spin statistics theorems one can prove […] are interesting enough to count as useful theorems, don’t they?

Yes, these theorems are what I have in mind, among other things, with the paranthetical remark

– apart from a few isolated exceptions –

Maybe I should name them explicitly.

But notice that the next sentence says

And it is precisely this that is changing now.

I mean this both for the Schrödinger picture (FQFT) as well as for the Heisenberg picture (AQFT, factorization algebras). So far the preface just gives the details for what I regard as some major advancements on the FQFT side of life. But below that paragraph is meant to come a paragraph starting with

On the AQFT/factorization algebra side this involves

…

I have yet to write that. I am thinking here in particular of mentioning progress of connecting these formalisms to standard perturbative QFT and renormalization theory. There has been progress here both in Haag-Kastler AQFT proper as well as from the factorization algebra perspctive (okay, I break down and admit that I am thinking of stuff as from page 12 here on).

Urs, who is the intended reader of this book?

I tried to say this in the proposed Call-for-Papers, but maybe I need to work it out better:

I am thinking that there is the need for cross-communication between mathematically and theoretic-physically inolved people here:

I know that a bunch of mathematicians working on these issues whose daily work makes them throw around the terms “quantum field theory”, “string motivated” etc. would like to have a better idea about how what they are doing is a puzzle piece of a grand phyics tale.

On the other side, many theoretical-physicist want to or maybe ought-to-want-to see clearer on the magnificent advances on the formalization front recently.

I would enjoy to see a book that bridges this gap.

Noether’s theorem in the form

symmetries $\Rightarrow$ conservation laws

works for any Lie group.

It probably depends on the stage in one’s career. I publish a lot in edited volumes because they don’t usually mind if you mix a lot of exposition with the new stuff, which is what I like to do. Most journals don’t seem to want that. Edited volumes and conference proceedings count for less when it comes to promotions — but not nothing, here at the University of California. If I were feeling under pressure of some sort — looking for a job, or feeling poor — I might avoid them. Luckily I’m not.

Conference proceedings used to be harder to find than journals, since fewer libraries buy them. But luckily the arXiv cured that.

Hisham and I are busy working on pushing the introductory chapter of the book closer to final stage.

A pdf of a preliminary version is here.

I know of a bunch of insufficiencies of this text, which I will try to take care of a little later today. But there will be plenty of insufficiencies that I have not realized yet. If anyone feels like eyeballing this, I’d be most grateful for hearing comments.

On p. ix, you use “tmf” without defining it; the expansion of “tmf” as “topological modular forms” isn’t given until p. xvi. It’s slightly confusing.

On p. ix, you use “tmf” without defining it; the expansion of “tmf” as “topological modular forms” isn’t given until p. xvi. It’s slightly confusing.

You are right, thanks!

This and plenty of other things have been improved on now. Unfortunately, due to an annoying software bug, I cannot currently upload the new file version. I’ll let you know as soon as I have.

Thanks again.

The new version of the preface and introduction is now here.

This has been polished a bit further. It’s getting pretty close to the final version.

As a condensed-matter, statistical-physics kind of person, I found myself particularly intrigued by this part of the introduction:

A complete classication of rational full 2d CFTs on cobordisms of all genera has been obtained in terms of Frobenius algebra objects in modular tensor categories [FRS06]. While the rational case is still “too simple” for the most interesting applications in string theory, its full solution shows that already here considerably more interesting structure is to be found than suggested by the naive considerations in much of the physics literature.

The classification of CFTs I know about, from my stabs into the physics literature and its naive considerations, is the Dynkin-diagrammatic approach to rational CFTs. This matters to low-energy physics folks like me because, per Zamolodchikov, classifying CFTs tells you about the fixed points of renormalization-group transforms and thus gives a (partial) taxonomy of universality classes (Ising, tricritical Ising, Potts, …). It seems to me that the FRS approach subsumes the one I’m more familiar with. Am I on the right track here?

Understanding stringy stuff appears to help with understanding why the ADE classification recurs in many different subjects, and I gather that holography has been a useful tool for constructing extensions of Zamolodchikov’s $c$-theorem. (See, e.g., arXiv:1006.1263.) I keep seeing hints that all this exotic mathematics is actually relevant to aspects of nature I can reach out and play with … unfortunately, I don’t know anyone local who is as intrigued by the prospect as I am, so instead of sounding ideas against a sympathetic colleague willing to tear them apart, I’m reduced to airing speculations in tangentially-related blog comment threads.

I’m not going to be around much over the next fortnight, but I’d like to have a discussion about philosophy of geometry when I’m back. I see Liang Kong takes time in his piece to say something on this in section 2. E.g.,

…our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge.

That’s worthy of a Clifford or a Riemann.

I see Liang Kong takes time in his piece to say something on this

Yes. I like that he took the time to say these things explicitly. The observation has been around among some people for a long time, but I think it doesn’t find due attention in the literature yet.

So Liang Kong has been one of the central figures in establishing a list of theorems that rigorously classify certain full 2-d conformal field theories in terms of algebra internal to suitable categories of confomal representations. (This is the conformal analog of the more well known and much more highly developed story in the topological case, where it ultimately culminated in the full classification of extended TQFTs.) The main part of his article is a survey of these theorems.

But, as he describes at the beginning, thinking about these theorems at some point one is struck by what this means: as discussed in some detail in the contribution by Soibelman we also know that each such abstractly defined 2d CFT encodes an effective target space geometry. In a precise sense, this works by a categorified version of Connes’s theory of spectral geometry:

Connes effectively observed that it makes sense to associate to every abstractly defined system of quantum mechanics ((1+0)-dimensional QFT) an effective (possibly noncommutative) target space geometry, which is the geometry “seen” by whatever particle is described by the given system of quantum mechanics. More or less explicitly (often less explicitly) the higher dimensional analog of this statement is the basis of perturbative string theory: to every abstractly defined 2-dimensional (conformal) QFT we assign an effective (possibly non-commutative) target space geometry, which is the geometry “seen” by whatever string is described by the given CFT.

In the 1990s there have been a handful (less than 10, I think) of articles (by Chamseddine and Fröhlich) who showed interest in capturing this idea in a precise fashion. Then somehow this interest was washed away by various fashions. This is changing now. The contributions by Kong and by Soibelman in our volume present aspects of such a formalization. Liang Kong amplifies the algebraic aspect (where Yan Soibelman amplifies the spectral and hence Riemannian aspect): there is a “stringy” (higher) analog of the fundamental duality between space and algebra at work here.