## October 24, 2009

### Mathematical Foundations of Quantum Field and Perturbative String Theory

#### Posted by Urs Schreiber

Following a suggestion by some publishing company, there is the idea of creating a book that collects contributions from various authors on the topic Mathematical Foundations of Quantum Field and Perturbative String Theory .

We have an idea for a proposed “Call for Papers”. But we would like to get some comments on this, from people who have experience with such issues.

Posted at October 24, 2009 7:51 PM UTC

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### Re: Structural Foundations of Quantum Field and String Theory

I think it’s a bit dodgy to put the terms “structural foundations” and “string theory” that close together. This stuff is beautiful, but it isn’t a structural foundation for string theory.

Posted by: A.J. on October 25, 2009 12:12 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

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?

Posted by: Urs Schreiber on October 25, 2009 10:51 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Yeah, “mathematical foundations” also runs afoul of my grumpiness. :) I don’t suppose it matters that much, as it’s not likely that someone will pick up the book thinking there’s a definition of string theory inside.

My complaint, if you will, is that the worldvolume aspect of strings is just quantum field theory. That we must approach strings in this fashion seems to me to be a failure on our part, not a deep fact about string theory.

Posted by: A.J. on October 26, 2009 2:17 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

You write:

[…] it’s not likely that someone will pick up the book thinking there’s a definition of string theory inside.

My complaint, if you will, is that the worldvolume aspect of strings is just quantum field theory. That we must approach strings in this fashion seems to me to be a failure on our part, not a deep fact about string theory.

In reaction to this I have now done two things: I once more renamed the whole thing, now into Mathematical Foundations of Quantum Field and Perturbative String Theory. Then I added the following paragraph topwards the end of the supposed preface

Even in the light of all these develoments, the reader accustomed to the prevailing phyiscs literature may still complain that none of this progress in quantum field theory on cobordisms of all genera yields a definition of what string theory really is. And of course this is true if by “string theory” one understands its non-perturbative definition. But this supposed non-perturbative definition of string theory is little more than a dream of a dream for the time being. Marvelling – with a certain pride about their daringness – at how ill-understood this is has made the community forget that something much more mundane, the perturbation series over CFT correlators that defines perturbative string theory , has been ill defined all along: only the machinery of full CFT in terms of cobordism representations gives a precise meaning to what exactly it is that the string pertubation series is a series over. Maybe it causes feelings of disappointment to be thrown back from the realm of speculations about non-perturbative string theory to just the perturbation series. But at least this time one lands on solid ground. Which is the only ground that serves as a good jump-off point.

Posted by: Urs Schreiber on October 26, 2009 9:33 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Finally it should be kept in mind that the “landscape” is really a moduli space of CFTs.

No it’s not.

None of the backgrounds (F-theory with fluxes, Type-IIA/B orientifolds with fluxes, M-theory on $G_2$ manifolds with flux, …) commonly discussed in the context of the ‘landscape’ correspond to worldsheet CFTs. Indeed, in many cases (e.g. F-theory, M-theory, …), they don’t even correspond to weakly-coupled string backgrounds.

Indeed, that’s part of the point. One of the moduli you want to fix is the dilaton (whose expectation-value is the string coupling) …

Posted by: Jacques Distler on October 26, 2009 3:44 AM | Permalink | PGP Sig | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Finally it should be kept in mind that the “landscape” is really a moduli space of CFTs.

No it’s not.

It’s even bigger and less understood than that, but it contains at least the space of all weakly coupled string backgrounds coming from worldsheet CFTs. At least if we speak of the landscape as such and not as commonly discussed. It seems to me that the “as commonly discussed” perspective has lots of unjustified prejudices about which corners are worthy of attention.

Posted by: Urs Schreiber on October 26, 2009 9:42 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

It seems to me that the “as commonly discussed” perspective has lots of unjustified prejudices about which corners are worthy of attention.

Everyone understand that the corners, which are commonly discussed, are not necessarily representative of the whole. The reason they are focussed on is that those are the corners where moduli stabilization is understood.

but it contains at least the space of all weakly coupled string backgrounds coming from worldsheet CFTs.

No it doesn’t.

Those backgrounds always contain at least one unfixed modulus – the dilaton.

Moreover, a worldsheet CFT is typically only a string background at the classical level. With enough spacetime supersymmetry, it remains a string background quantum mechanically. But those backgrounds, with extended spacetime supersymmetry, are typically not what people have in mind when they talk about the landscape.

1. People, generally, have in mind backgrounds with $\mathcal{N}=1$ or $\mathcal{N}=0$ spacetime supersymmetry.
2. The backgrounds which arise as worldsheet (S)CFTs with extended spacetime supersymmetry typically have lots of unfixed moduli (in addition to the dilaton). Again when people talk about the landscape, they are generally talking about backgrounds where the moduli are stabilized.
Posted by: Jacques Distler on October 26, 2009 5:49 PM | Permalink | PGP Sig | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Just to be clear, it is perfectly respectable to study the space of classical solutions to string theory (i.e., 2d (S)CFTs). Only a madman would insist on jumping straight to an attempt to understand the full-blown quantum theory, without first thoroughly understanding the classical theory.

But the space of classical solutions to string theory is not what people typically mean when they speak of the ‘landscape’.

Blurring, or eliding, the difference will only add confusion to a topic which already the subject of enough confusion among many in your potential audience.

Posted by: Jacques Distler on October 26, 2009 7:08 PM | Permalink | PGP Sig | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Thanks, Jacques.

I see that “M-theory backgrounds” aka classical 11d supergravity solutions will in general fail to correspond to worlsheet CFTs, but why shouldn’t it be true that a classical solution to a typeII/heterotic supergravity theory defines, to the given $\alpha'$-order, a worldsheet CFT whose sum of correlators over genera computes something close to the perturbative S-matrix computed about these classical solutions?

I have a quarrel with saying “background” for something that is not what a perturbation series perturbs about, but maybe that’s my problem.

I wish though there were a reference that would spell out these issues clearly enough that even a dense guy like me would see more clearly. Maybe we can eventually create one at $n$Lab: landscape of string theory vacua.

Because, for instance I am looking at

and see, for instance on page 4, statements like

[…] we still lack an understanding of whether any of the large amount of perturbative vacua (the dense “discretuum” or “landscape”) is in any sense preferred over the rest

So “perturbative vacuum” here is not perturbative vacuum?

Posted by: Urs Schreiber on October 26, 2009 7:15 PM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

but why shouldn’t it be true that a classical solution to a typeII/heterotic supergravity theory defines, to the given α′-order, a worldsheet CFT whose sum of correlators over genera computes something close to the perturbative S-matrix computed about these classical solutions?

With enough unbroken spacetime supersymmetry, the moduli space of vacua that you see, classically, is unaffected by quantum corrections, and is the moduli space of vacua of the quantum theory.

In that case, what you say is exactly true.

With slightly less unbroken supersymmetry, the moduli space of vacua that you see, classically, will not be lifted, quantum-mechanically. But its geometry, and even its topology, can change.

Still, there is at least a limit where what you say holds approximately (up to corrections that are small, at weak string coupling).

With $\mathcal{N}=1$ or $\mathcal{N}=0$ supersymmetry, the classical moduli space bears little, if any, resemblance to the moduli space of the quantum theory. And that’s true, even at weak string coupling.

In fact, for $\mathcal{N}=0$, there’s generically a 1-loop contribution to the vacuum energy (and hence a graviton/dilaton tadpole). So the classical background isn’t a solution at the 1-loop level.

Posted by: Jacques Distler on October 26, 2009 8:04 PM | Permalink | PGP Sig | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

With enough unbroken spacetime supersymmetry, the moduli space of vacua that you see, classically, is unaffected by quantum corrections, and is the moduli space of vacua of the quantum theory.

[…]

Okay, I get it now. Thanks.

Do you have some suggested good reference for me so that I can work on closing this gap in my education in more detail?

Actually, you should write a book on string theory one day. The stuff you taught on your blog and in other web discussion are by far the best kinds of accounts on the given issues. One day I should collect them on the $n$Lab. Actually, you should write a book together with Dan Freed! That would be guaranteed to be a lasting classic for a true structural understanding of the theory.

Posted by: Urs Schreiber on October 26, 2009 8:28 PM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

Is this book project somehow a spinoff of the book The Structural Foundations of Quantum Gravity? That book was put together by three philosophers of physics who advocate ‘structural realism’. I felt fine about putting a paper in there because I don’t have a big quarrel with structural realism, and my paper was in some loose sense advocating some sort of structuralism.

I think without an editor who has a strong vision of what ‘structural foundations’ are, it would be unfortunate to include this phrase in the title of a collection of papers. Quite possibly you have such a vision, Urs! But it would be a bit sad to adopt this title merely because ‘some publishing company’ thought that another book with this phrase in the title would sell. If you’re going to be in charge of this book, you should pick a title that exactly fits your vision. Don’t let publishers push you around: you’ll do most of the work, and they’ll make most of the money.

Posted by: John Baez on October 25, 2009 8:15 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

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?

Posted by: Urs Schreiber on October 25, 2009 10:35 AM | Permalink | Reply to this

### Re: Structural Foundations of Quantum Field and String Theory

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.

Posted by: John Baez on October 25, 2009 5:07 PM | Permalink | Reply to this

### Mathematical Foundations of Quantum Field and String Theory

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:

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.

Posted by: Urs Schreiber on October 25, 2009 8:55 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

Ok, busy reading the Preface. Hmm…,

The history of theoretical fundamental physics is the story of a search process for the suitably mathematical notions and structural concepts that naturally model the phyical structures in question. It may be worthwhile to recall some examples:

* the identification of differential cohomology as the underlying structure of gauge theory.

Isn’t that a bit “disingenuous”, to steal a much-used epithet nowadays? The way you write, it seems as if it is a done-and-dusted-commonly-accepted-fact-by-now that differential cohomology is the mathematical notion which models gauge theory. But my impression was that differential cohomology is still quite a new idea to many physicists and mathematicians, even though it may be old hat to you :-) Correct me if I’m wrong!

Posted by: BH Bartlett on October 25, 2009 9:35 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

Thanks!!

Isn’t that a bit “disingenuous” […]?

Not sure if it is. But certainly I want to avoid that any reader thinks it could be. So thanks for flagging this. I have now slightly modified that bit in the text. Could you have another look and see if it is better now?

it seems as if it is a done-and-dusted-commonly-accepted-fact-by-now that differential cohomology is the mathematical notion which models gauge theory.

We are still waiting for it to become dusted, that’s right. But otherwise it’s true. ;-)

Posted by: Urs Schreiber on October 25, 2009 9:48 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

By the way, please be critical but charitable on me with this preface. I wrote this in a bit of a haste, while I really need to work on these two referee reports waiting for my attention…

Posted by: Urs Schreiber on October 25, 2009 9:51 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

I think it’s a fascinating preface, Urs. I’m hoping you take this book as an excuse to write a very long article or series of articles expanding on your vision of quantum field theory.

And I hope you get someone to write a long introduction to the cobordism hypothesis — preferably someone whose last name begins with ‘L’. And I hope you get someone to write a long introduction to the classification of rational conformal field theories, and also the Fuchs–Runkel–Schweigert work (preferably not one of those guys, because they have already written many such introductions, and it would be nice to see one written by somebody with a bit more distance from the subject). And I hope you get Stolz and Teichner to write a long review of their work.

But most of all I hope you write a bunch of stuff, and I hope you work very hard to minimize the prerequisites, so lots of people can understand it.

Posted by: John Baez on October 26, 2009 8:23 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

I think it’s a fascinating preface, Urs.

Thanks for the feedback. I was wondering how it came across.

As you indicate, the main aim of this exercise is to manage to sketch a project that looks worthwhile enough for those supposed to contribute to it to actually do. As you also indicate, while there is non-vanishing flexibility in who that might be, clearly the intended authors here won’t be random guys from the street.

And I hope you get someone to write a long introduction to the cobordism hypothesis — preferably someone whose last name begins with ‘L’

You mean you are hoping it won’t begin with ‘B’?

Hm, maybe the effect of this discussion here will just be as a timely warning to all likely candidates to start thinking up good excuses not to have time to contribute…

Posted by: Urs Schreiber on October 26, 2009 10:07 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

Urs wrote:

You mean you are hoping it won’t begin with ‘B’?

No — I just think a contribution from someone whose name begins with ‘L’ could be a wonderful thing. Of course he’s a busy guy, but maybe you could get some smart person to take the existing videos of his lectures and use them to improve the existing lecture notes, and then have Professor L polish them a bit and give them his blessing. I don’t know if this makes sense — it’s just a wild idea.

As for people whose name begins with ‘B’, Julie Bergner is giving a seminar on the cobordism hypothesis here at UCR, people are already taking notes, and she will eventually explain at least one definition of $(\infty,n)$-category. And it seems that a good explanation of that topic is becoming a much-sought-after thing.

There’s also someone else whose name begins with ‘B’ at UCR, who is very tired of writing expository papers, but could vaguely imagine enjoying writing a paper with you explaining the current state of higher gauge theory. Again, just a wild idea.

One thing I don’t quite see is whether you’re mainly seeking original research papers, or expository papers, or both.

I’m sure you have your own ideas about all these things… I just thought some random external perturbations could help trigger some new thoughts.

Posted by: John Baez on October 26, 2009 3:35 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

I just think a contribution from someone whose name begins with ‘L’ could be a wonderful thing.

Right, indeed. Secretly I have an extended topic list for this project where to each topic I have a list of concrete authors with concrete requests what I would like them to contribute. I was thinking that clearly I should keep that wish-list for myself. But then, maybe it would be a sneaky move to make such a Wanted list public. Or maybe not.

Julie Bergner is giving a seminar on the cobordism hypothesis here at UCR

Right, good point. I put here name on my list. Hadn’t thought of her yet as the cobordism person, to be frank, but that’s a good idea.

One thing I don’t quite see is whether you’re mainly seeking original research papers, or expository papers, or both.

I think I am looking for whatever paints the most coherent total picture of the recent development. Most of the things necessary for that have been published, so I would be thinking of expository reviews for these. But in some cases I know that the good material that fits into the story is still being written up. In these cases I would enjoy it if the corresponding author took the opportunity to prepare the material for the book.

So I guess the answer is: both.

But, you know, I can dream here all day, I eventually need to check back with my co-editors-to-be and with the company that would make the book and get a reality check.

There’s also someone else whose name begins with ‘B’ at UCR, who is very tired of writing expository papers, but could vaguely imagine enjoying writing a paper with you explaining the current state of higher gauge theory. Again, just a wild idea.

Right, so I am in the middle (hopefully really in the middle) of compiling the material for the next thing that I am supposed to publish. Which is the stuff being built up at

differential cohomology - contents

and at

$\infty$-Lie theory - contents

I am still working on this, but one day in the not too far future I will open a LateX file and start pouring this material into that to finalize it.

I’d be happy to collaborate on all and any aspect of that, be it in terms of writing better expositions and outlines, be it in terms of cranking out further theorems or be it in terms of describing examples and applications better and in more detail.

Partly this is just a matter of me finding the time to do it, partly I need to think more.

Posted by: Urs Schreiber on October 26, 2009 4:35 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and String Theory

The real non local structure behind QFT is NOT a gauge theory … just ask any twistor theorist.

Posted by: Kea on October 25, 2009 11:18 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

I see that David Corfield kindly edited the entry by fixing some things here and there.

Thanks, David! Very much appreciated.

Posted by: Urs Schreiber on October 26, 2009 10:09 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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…

Posted by: David Corfield on October 26, 2009 10:24 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on October 26, 2009 10:38 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Interesting! I’d like the historical section to expand to tell us more about what can be achieved by finding the mathematical structure underpinning of a piece of physics, e.g., you could explain what was accomplished by “the identification of symplectic geometry as the underlying structure of classical Hamiltonian mechanics”.

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

Posted by: David Corfield on October 26, 2009 10:18 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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

Symplectic geometry and classical Hamiltonian mechanics

Posted by: Urs Schreiber on October 26, 2009 11:13 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Eugene Lerman on October 27, 2009 4:01 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

I have two examples.

I copy-and-pasted the first one into Hamiltonian mechanics

Here is a telegraphic version of the story:

and that into symplectic geometry.

Posted by: Urs Schreiber on October 27, 2009 4:31 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

There are many variations of symplectic reduction, though M and W seems perhaps the most popular
There is also Sniyaticki-Weinstein
whihc I find more compatible with the cohomological version of BFV. By the way,
the latter does not require an equivariant moment map nor even a strict Lie algebra in sight, jsut first class constraints.

Posted by: jim stasheff on October 27, 2009 8:09 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Hello Urs,

sounds very exciting and worthwhile to me, please try to make it selfcontained for
someone who mostly specializes in operator algebras and Haag/Kaster aka “local observables” QFT :-)
Some minor questions/ remarks on the draft of the preface:
- “some folklore called the path integral”: Calling the path integral folklore could result
in some critisim/hostility from some parts of the physics community. I think it would be wise to refomulate
that passage along the line that “the path integral is an example of a highly useful heuristic concept that defies attempts to formulate a mathematical explanation”.
- We don’t know “What is a 2d conformal field theory?” I’m satisfied with the explanation of e.g. Schottenloher: “A Mathematical Introduction to Conformal Field Theory” (2ed.).
It’s probably a bit off topic, but would you care to explain what’s amiss with that?
- about AQFT you write: “nobody should trust an axiom system that hasn’t proven its worth yet by providing some useful theorems and describing some nontrivial examples of interest.”
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
(both in the Wightman and in the Haag/Kastler approach, as far as I know) are interesting enough to count as useful theorems, don’t they?

Good luck with this project, I’m looking forward to it!

Posted by: Tim vB on October 26, 2009 10:41 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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).

Posted by: Urs Schreiber on October 26, 2009 11:56 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Hello Urs,
thanks for the link, I wasn’t aware of this.
Why do you expect that the book will be so provocative? I cannot imagine any topic that would be less controversial :-)

Posted by: Tim vB on October 26, 2009 2:04 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Urs, who is the intended reader of this book? You will be covering a very wide range of mathematical disciplines, so do you expect to appeal to many different readers with varying focused interests, or to a less specialized group of readers with broad interests in everything? More to the point, exactly what qualifications are required of the reader? How large do you think your audience is? It might even be helpful to have the names of a couple of people you know in your mind as you pull this book together. Just a thought.

Posted by: Charlie C on October 26, 2009 4:07 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on October 26, 2009 4:47 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Would this include whether Noether’s theorem applies to exceptional groups ?

Posted by: joel rice on October 26, 2009 5:15 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Noether’s theorem in the form

symmetries $\Rightarrow$ conservation laws

works for any Lie group.

Posted by: Eugene Lerman on October 26, 2009 7:23 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

The main problem with a “call for papers” is that for most of us, mere mortals in science, most career reviewing procedures put very little weight on proceedings, chapters in books and alike, and prefer to count only the publications in recognized journals. I had already few publications published in a wrong place from that point of view, and probably would not contribute to a project like this, unless it is possibly about making a very short paper on something what is really easy and quick for me to write. I think that many other people may have the same concern.

Posted by: Zoran Skoda on October 28, 2009 9:32 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: John Baez on October 29, 2009 9:54 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

With some other tasks out of the way I have now found the time to further expand on that annotated outline of the intended topics of the book.

Details are at the following anchor points

(I) Cobordism representations

(II) Factorization algebras

(III) Constructions from backgrounds

We are now entering the process of contacting potential authors for this project.

Posted by: Urs Schreiber on November 5, 2009 1:57 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

We are closing in on finishing the preparation of this book: I am being told that it will “go to print” on July 15.

You can now find a brief indication of the actual content of the book – contribution by contribution – on the book’s website: here.

(Scroll down a bit to “Content of the book”.)

The brief paragraphs there I wrote today in between preparing a topos-theory exam and writing a referee report for something else, so they might not be perfect yet.

Posted by: Urs Schreiber on June 28, 2011 7:08 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on July 8, 2011 10:35 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Blake Stacey on July 9, 2011 3:05 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on July 9, 2011 6:38 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on July 10, 2011 11:35 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Inspiring words! But do you have an editor to look over the English, especially before you get to the details of the physics. E.g. in the first pages,

…it was fully established that QFT in general and worldvolume theories in particular are precisely the representation theory of higher categories of cobordisms with structure OR, dually, OF the theory of copresheaves of local algebras of observables, vertex operator algebras, factorization algebras and their siblings.

I’m having trouble parsing this. First, ‘worldvolume theories are the representation theory…’ sounds odd to me. Maybe ‘form the representation theory’.

Second, does the ‘OR…OF’ I’ve capitalized mean that worldvolume theories are (also ‘dually’) the representation theory of the theory of presheaves of local algebras of…? Or, do you mean ‘higher categories of copresheaves…’? There are so many ‘of’s it’s hard to tell.

…the full impact has possibly not come yet to the wide public perception that it deserves,…

This is rather clumsy.

But it’s worth someone looking right through, e.g.,

with the supposed outcome of the (path integral) quantization process thus identified precisely by axioms for QFT, it becomes possible to think about what this quantization process itself is precisely.

Perhaps, ‘…it becomes possible to consider the nature of this quantization process itself’.

Posted by: David Corfield on July 11, 2011 10:50 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

Inspiring words! But do you have an editor to look over the English

One AMS editor did go through it once, yes, but quite possibly there was too much to be caught in one go.

So thanks a lot for your suggestions! Much appreciated. I have uploaded a new version that tries to incorporate what you said.

If you have further suggestions, I’d be most interested in hearing them.

Posted by: Urs Schreiber on July 13, 2011 9:27 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

I noticed another minor point of English in the following passage.

In the form of the Haag-Kastler axioms the description of QFT through its local algebras of observables had been given a clean mathematical formulation [HaMü06] long time ago . This approach had long produced fundamental structural results about QFT, such as the PCT theorem and the spin-statistics theorem (cf. [StWi00]).

The portion

long time ago .

should be

a long time ago.

Posted by: Blake Stacey on July 13, 2011 7:06 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

I noticed another minor point of English in the following passage.

Thanks!!

(Unfortunately, I cannot add acknowledgements for all the helpful comments I am receiving here: we had decided beforehand to keep the acknowledgements to the minimum at which they are.)

(And due to perpetual delays, the book “goes to print” not this Friday, but coming Monday.)

Posted by: Urs Schreiber on July 14, 2011 8:57 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

There’s still an extra space before the period. Thinking about it a bit more, I wonder if “a long time ago” should be made more specific. The reference is five years old, but it’s a review article; maybe the formulation and application of the Haag–Kastler axioms should be dated more precisely.

Among other things, the result shows that the ”landscape of string theory vacua” – roughly the moduli space of constent perturbative string backgrounds (cf. [Do])– is more subtle an object than often assumed in the literature.

The first quotation mark is reversed, and constent should be consistent.

The author uses this to demonstrate compactness results about the resulting moduli space of ”quantum Riemann spaces”.

Again, the first quotation mark should go the other way.

Posted by: Blake Stacey on July 15, 2011 1:11 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

There’s still […]

Thanks for all this!

I believe I have fixed all these issues in the latest version. (Hopefully without introducing new inconsistencies, I am operating from varying hotel rooms with varying wifi-connections and varying small time supplies.)

In particular I have added the references to Haag-Kastler’s original article from 1964. Also to the relevant $n$Lab entries.

Posted by: Urs Schreiber on July 15, 2011 7:20 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Blake Stacey on July 11, 2011 8:11 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

I have now added to each contribution outline a link to the actual contribution itself, here.

Posted by: Urs Schreiber on July 22, 2011 2:25 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: David Corfield on July 22, 2011 10:22 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

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.

Posted by: Urs Schreiber on July 22, 2011 11:49 AM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

…there is a “stringy” (higher) analog of the fundamental duality between space and algebra at work here.

That’s as described at pp. 5-6, right? So a stringy affine scheme is a category of D-branes for a closed conformal field theory. What kind of category is this?

Gluing these categories together to make string schemes won’t be necessary?

If we take a closed CFT loosely as an algebraic model for free loop space and an open CFT as an model for path space with a given boundary condition, we see immediately that this stringy geometry has the so-called holographic phenomenon. For example, a D-brane associated to a point is just the based loop space which contains the information of the entire space. It is very different from the usual sheaf-theoretical geometry in which a single chart does not contain any global information. This aspect is somewhat similar to that of noncommutative geometry [Con], where the noncommutativity encodes the information of gluing data.

That last sentence alludes, I take it, to Connes’s point that

As mentioned above the notion of scheme is obtained by patching together the geometric counterpart of arbitrary commutative rings. Thus one might wonder at first why such patching is unnecessary in noncommutative geometry whose basic data is simply that of a noncommutative algebra. The main point there is that the noncommutativity present already in matrices allows one to perform this patching without exiting from the category of algebras.

Posted by: David Corfield on July 22, 2011 2:12 PM | Permalink | Reply to this

### Re: Mathematical Foundations of Quantum Field and Perturbative String Theory

That last sentence alludes, I take it, to Connes’s point

Yes, I think so.

But personally, I would put a different emphasis here than Alain Connes does:

I am not sure that it is correct to say that it is the non-commutativity that absorbs the need to glue. I think saying this is mixing up two different aspects.

The need to glue in algebraic descriptions arises as soon as the algebras of functions considered have plenty of elements only locally. Sheaves were discovered in complex geometry because holomorphic functions may be plenty locally, but not globally.

But for Riemannian geometry one considers smooth real-valued functions. These happen to be plentiful even globally. And that’s why Connes’s spectral geometry gets away with the global function algebras.

This point is noted for instance in Dominic Joyce’s recent redux of smooth geometry, as the geometry over formal duals of $C^\infty$-rings: in that category every manifold is an “affine scheme” over $C^\infty$-rings, even if it would be far from affine as an ordinary scheme. I think this is really what is going on.

For some reason all the abstract technology that Grothendieck injected into algebraic geometry never made people look much into how the theory behaves under change of big site, away from algebraic sites. Some crucial things do change: as we move to a smooth site for instance affine-ness becomes a much more flexible notion and many spaces that were non-affine before now become affine. This is really a tautology, but still, apparently it requires amplification.

So I think: eventually we will be gluing also the noncommutative and stringy spectral geometries that, say, Connes and Kong are discussing here. The only thing is: we may feel the need to do so only much later.

But still, there is such need visible already now: the example to keep in mind is T-folds.

The idea is: one 2d CFT encodes a given target space geometry. But there are more equivalences of 2d CFTs than one would associate naively with these target spaces: two CFTs may be equivalent by T-duality, and have non-isometric (and not even homeomorphic) Riemannian target spaces. But this seems to mean: we should consider more generally objects that locally are Riemannian manifolds, which are glued together not just by diffeomorphisms but also by T-duality transformations. This idea goes by the name T-fold .

One can already see how this is going to pan out: 2d CFTs whose target spaces are T-folds should be “CFT with defects” where the domain wall on the wordlsheet runs where one piece of string that maps to one Riemannian manifold connects to one which maps in d “T-duality double overlap” to the T-dual space.

Posted by: Urs Schreiber on July 22, 2011 4:13 PM | Permalink | Reply to this

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