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 $$\mathrm{dH}=\omega ({\Xi }_H,\cdot ),$$ 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.