Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

May 12, 2011

∞-Dijkgraaf-Witten Theory

Posted by Urs Schreiber

In another thread Tom Leinster would like to learn what a sigma-model in quantum field theory is. Here I want to explain this in a way that will make perfect sense to Tom, and hopefully even intrigue him. To do so, I will look at the σ\sigma-model called Dijkgraaf-Witten theory, which stands out as a canonical toy example with which to exhibit precisely those aspects of σ\sigma-models that mean something to abstractly-minded people like Tom and suppresses precisely all the other technical details.

In order to keep things interesting despite the toy model nature of the example, I shall use this occasion to talk about the general class of models that deserve to be called ∞-Dijkgraaf-Witten models. The next step in this hierarchy of models is what is called the Crane-Yetter model.

I’ll post the content here iteratively in small digestible bits in the comment section below.

An Idea-section is

One sequence of posts starts below at

Another one starts at

The discussion in the title of this thread finally starts at

There is also

Posted at May 12, 2011 9:55 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2400

75 Comments & 2 Trackbacks

Re: ∞-Dijkgraaf-Witten Theory

Thanks! I’m honoured.

Maybe I should mention that today (Friday) I will have no time for digesting even the smallest bit, because we’re having our Scottish Category Theory Seminar. But in a couple of days my digestive capacity should be back to normal.

Posted by: Tom Leinster on May 13, 2011 1:34 AM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

The Dijkgraaf-Witten theory entry in the nLab is missing the “gentle exposition” paragraph, although the header is already there - maybe, as a side effect of this discussion, we could fill that in. And maybe I can help with some particularly elementary questions.

Dijkgraaf-Witten theory in dimension n is the topological sigma-model quantum field theory…

I think I’ll let that pass on for now, John explained “sigma-model” over here, and the other terms will become clearer during the discussion, I hope.

…whose target space is the classifying space of a discrete group…

Since I don’t handle these concepts too often, I always need to refresh my memory, which is possible using the nLab and Wikipedia:

  • A discrete group is a group with the trivial discrete topology, where all subsets are both open and closed,

  • for a discrete group the classifying space is a pathconnected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups are trivial.

…and whose background gauge field is a circle n-bundle with connection on BG…

Here GG is our discrete group and BGBG is its classifying space, but what is the “background gauge field”? And I’m also not sure that I know what a “circle n-bundle” is…

(From my viewpoint the nLab entry for “background gauge field” throws in more unfamiliar terms than it explains :-)

Posted by: Tim van Beek on May 13, 2011 7:34 AM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

The Dijkgraaf-Witten theory entry in the nLab is missing the “gentle exposition” paragraph

Now it does.

maybe, as a side effect of this discussion, we could fill that in.

That’s indeed the idea.

I had planned to start posting the first “digestible bits” as announced yesterday night already, but then I got kicked out of the matrix and it took me until this morning to log myself back in. Sorry if this delay caused confusion.

The exposition starts below in the comment titled Exposition . A hyperlinked version appears in parallel in the nnLab entry here.

Posted by: Urs Schreiber on May 13, 2011 9:19 AM | Permalink | Reply to this

Exposition

Exposition

We give a leisurely exposition of the idea of σ-models, aimed at readers with a background in category theory but trying to assume no other prerequisites.

What is called an nn-dimensional σ-model is first of all an instance of an nn-dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are σ-models is that

  1. these arise from a simpler kind of field theory – called a classical field theory – by a process called quantization;

  2. moreover, this simpler kind of field theory is encoded by geometric data in a nice way: it describes physical configurations spaces that are mapping spaces into a geometric space equipped with some differential geometric structure.

We give expositions of these items step-by-step

  1. Quantum field theory

  2. Classical field theory

  3. Quantization

  4. Classical sigma-models

  5. Quantum sigma-models

  6. Examples

See the comments with these titles further below.

Posted by: Urs Schreiber on May 13, 2011 8:40 AM | Permalink | Reply to this

1. Quantum field theory

Quantum field theory

For our purposes here, a quantum field theory of dimension nn is a symmetric monoidal functor

Z:Bord n S𝒞, Z : Bord_n^S \to \mathcal{C} \,,

where

  • Bord n SBord_n^S is a category of n-dimensional cobordisms that are equipped with some structure SS: its

    • objects are (n1)(n-1)-dimensional topological manifolds;

    • morphisms are isomorphism classes of nn-dimensional cobordisms between such manifolds

    • and all manifolds are equipped with SS-structure . For instance SS could be Riemannian structure . Then we would call ZZ a Euclidean quantum field theory (confusingly). If SS is “no structure” then we call ZZ a topological quantum field theory .

  • 𝒞\mathcal{C} is some symmetric monoidal category;

We think of data as follows:

  • Bord n SBord_n^S is a model for being and becoming in physics (following Bill Lawvere’s terminology): the objects of Bord n SBord_n^S are archetypes of physical spaces that are and the morphisms are physical spaces that evolve ;

  • the object Z(Σ)Z(\Sigma) that ZZ assigns to any (n1)(n-1)-manifold Σ\Sigma is to be thought of as the space of all possible states over the space Σ\Sigma of a the physical system to be modeled;

  • the morphism Z(Σ^):Z(Σ in)Z(Σ out)Z(\hat \Sigma) : Z(\Sigma_{in}) \to Z(\Sigma_{out}) that Σ\Sigma assigns to any cobordism Σ^\hat \Sigma with incoming boundary Σ in\Sigma_{in} and outgoing boundary Σ out\Sigma_{out} is the propagator along Σ^\hat \Sigma: it maps every state ψZ(Σ in)\psi \in Z(\Sigma_{in}) of the system over Σ in\Sigma_{in} to the state Z(Ψ)Σ outZ(\Psi) \in \Sigma_{out} that is the result of the evolution of ψ\psi along Σ^\hat \Sigma by the dynamics of the system. Or conversely: the action of ZZ encodes what this dynamics is supposed to be.

Posted by: Urs Schreiber on May 13, 2011 8:48 AM | Permalink | Reply to this

Re: 1. Quantum field theory

(I hope I don’t interrupt the conversation that Urs would like to have with Tom.)

Since the propagator maps every state ψZ(Σ in)\psi \in Z(\Sigma_{in}) to a state in Z(Σ out)Z(\Sigma_{out}), we actually have a time evolution in discrete time, right?

Looking at a cobordism one might think that there is a continuous evolution along the n-th dimension, but physical states are defined on the borders only.

BTW: The definition given here is a generalization of the one that Michael Atiyah introduced for topological QFT, see e.g.

  • Michael Atiyah: “The Geometry and Physics of Knots”,

and I’d like to use this occasion to mention again the book

  • Joachim Kock: “Frobenius Algebras and 2D Topological Quantum Field Theories”.

The advantage of 2D manifolds and their cobordisms is of course that they are easy to visualize, and, accordingly, the book has a lot of nice pictures.

Posted by: Tim van Beek on May 13, 2011 6:59 PM | Permalink | Reply to this

Re: 1. Quantum field theory

Hi Tim,

you write:

Since the propagator maps every state ψZ(Σ in)\psi \in Z(\Sigma_{in})to a state in Z(Σ out)Z(\Sigma_{out}), we actually have a time evolution in discrete time, right?

I am not quite sure where you are coming from here. There is an evolution operator for each cobordism, and unless we equip these with some exotic discrete structure SS, this will not describe continuous time evolution.

Here is the basic example to keep in mind (I’ll add that to the entry as soon as I am back on a stable internet connection):

Let S:=RiemS := Riem be “Riemannian structure” and consider Bord 1 RiemBord_1^{Riem}. The basic kind of morphism in this category is a topological interval equipped with a length .

Therefore a functor

Z:Bord 1 RiemVect Z : Bord_1^{Riem} \to Vect

is fixed by

  1. assigning a vector space \mathcal{H} to the point;

  2. assigning to the interval of length tt some linear endomorphism

    U(t): U(t) : \mathcal{H} \to \mathcal{H}

    such that

    U(t 1+t 2)=U(t 2)U(t 1). U(t_1 + t_2) = U(t_2) \circ U(t_1) \,.

So a quantum field theory on Bord 1 RiemBord_1^{Riem} is a system of quantum mechanics.

(I hope I don’t interrupt the conversation that Urs would like to have with Tom.)

Not at all. We are on a public blog, after all. I enjoy discussing this with you or anyone else. Tom might be less interested in this here anyway, he only got into this because I fiendishly made him ask an on-topic question just to make up for his sin of posting joke pictures of half-naked fat men to an innocent conference announcement. (I am joking.)

Posted by: Urs Schreiber on May 13, 2011 7:22 PM | Permalink | Reply to this

Re: 1. Quantum field theory

The example of quantum mechanics is now here.

Posted by: Urs Schreiber on May 13, 2011 9:37 PM | Permalink | Reply to this

Re: 1. Quantum field theory

This example clarifies a very basic misunderstanding on my part, in the example we have for every t>0t \gt 0 a cobordism (more precisely: an isomorphism class of cobordisms) of length tt. So when I fix a quantum state h 0Hh_0 \in H, the theory contains the evolved state h th_t for every time t>0t \gt 0. Ergo: time evolution in continuous time.

Posted by: Tim van Beek on May 16, 2011 8:35 AM | Permalink | Reply to this

Re: 1. Quantum field theory

To be even more precise, QFTs to VECT out of Bord 1 RiemBord_1^{Riem} define a finite-dimensional hilbert space Z(pt)Z(pt), and an operator satisfying some unitarity-type condition for each t>0t \gt 0, which compose well. But the corresponding map (0,)End(Z(pt))(0,\infty) \to End(Z(pt)) need not be continuous, as defined so far.

So the overall picture is correct, but there are more technical things to do. One of the main ones is that, rather than considering just the category Bord 1 RiemBord_1^{Riem}, you should consider some sort of fibered category in which Bordisms are allowed to vary in families, and then you should insist that everything be continuous with respect to these families (so replace VECT by some fibered category as well). Another is to replace VECT by some category of good topological vector spaces, so that Z(pt)Z(pt) may be infinite-dimensional.

A final thing, which isn’t an issue in 1-dimensional QFT but becomes an issue higher up, is that objects of Bord nBord_n should not really be (n1)(n-1)-dimensional manifolds, but rather germs of nn-dimensional manifolds near (n1)(n-1)-dimensional manifolds. The point is that whatever diffeogeometric structure the bordisms have, it is somehow specifically nn-dimensional, and so the objects should be equipped with precisely the same kind of structure. (A final, final thing is that at present, Bord RiemBord^{Riem} doesn’t have identity maps, but this can be fixed formally.)

Posted by: Theo on May 18, 2011 2:05 PM | Permalink | Reply to this

Re: 1. Quantum field theory

To be even more precise, QFTs to VECT out of Bord 1 Riem define a finite-dimensional hilbert space Z(pt), and an operator satisfying some unitarity-type condition for each t>0, which compose well. But the corresponding map (0,∞)→End(Z(pt)) need not be continuous, as defined so far.

Yes. In the more detailed nnLab entry of which this discussion here is a shadow, it says (here):

If we demand that ZZ respects the smooth structure on the space of morphisms in Bord 1 RiemBord_1^{Riem} then there will be a linear map iH:i H : \mathcal{H} \to \mathcal{H} such that U(t)=exp(iHt)U(t) = \exp(i H t). This HH is called the Hamilton operator of the system.

(We are glossing here over some technical fine print in the definition of Bord 1 RiemBord_1^{Riem}. Done right we have that \mathcal{H} may indeed be an infinite-dimensional vector space. See (1,1)-dimensional Euclidean field theories and K-theory).

Probably I should further expand on this.

For physical QFTs with non-finite dimensional state space, the key point it to remove the unit for the duality of objects. This actually has a natural geometrical interpretation, amplified once by Stolz and Teichner: in the geometric case one needs to to something at the gluing points (as you mention), for instance add collars . Their presence then makes the cap (or cup, depending on convention) have non-zero extension.

One of the main ones is that, rather than considering just the category Bord 1 RiemBord_1^{Riem}, you should consider some sort of fibered category

Yes, indeed, in the full story Bord n RiemBord_n^{Riem} is an (,n+1)(\infty,n+1)-sheaf on a site such as that of smooth manifolds.

By the way, as you may know. there is a nice article going in the direction of cobordisms generalized to toposes by David Ayala: Geometric cobordism categories (pdf) .

He redoes Galatius-Tilman-Madsen-Weiss for cobordisms with a map to a “space” XX that is, essentially, a smooth infinity-groupoid ((,1)(\infty,1)-sheaf on smooth manifolds).

I have been hinting in my latest posts here that I will eventually push the discussion in that direction: where in \infty-Dijkgraaf-Witten theory one has classical field configurations given by cobordisms equipped with a map into a topological space, for “\infty-Chern-Simons theories” the field configurations are cobordisms equipped with morphisms into such an (,1)(\infty,1)-sheaf (namely that of smooth \infty-connections).

But the series of posts has to be interrupted for a moment. I need to prepare a lecture, teach, then tomorrow I have a guest, then Friday we have our QVEST seminar and then… we’ll see what happens.

Posted by: Urs Schreiber on May 18, 2011 2:33 PM | Permalink | Reply to this

Re: 1. Quantum field theory

It should perhaps be pointed out, in case there is someone in the audience who hasn’t encountered this sort of thing before, that the symmetric monoidal structure of Bord n SBord_n^S is disjoint union, so that Z(M 1M 2)Z(M 1)Z(M 2)Z(M_1 \sqcup M_2) \cong Z(M_1) \otimes Z(M_2).

Another interesting fact worth pointing out is that symmetric monoidality of ZZ implies that Z()Z(\emptyset) is the unit object of 𝒞\mathcal{C}, and therefore ZZ assigns to every nn-manifold MM without boundary, an endomorphism of the unit object. If 𝒞\mathcal{C} is, say, the category of vector spaces, then such an endomorphism is just a number (an element of the ground field). Thus a quantum field theory of this sort induces, in particular, an invariant of nn-manifolds, which is one reason why non-physicists are interested in this stuff.

Posted by: Mike Shulman on May 14, 2011 4:24 AM | Permalink | Reply to this

Re: 1. Quantum field theory

It should perhaps be pointed out

Right, I have now added a remark along these lines to the entry here, mentioning also that this manifold invariant is what in the context of QFT is called the partition function .

Posted by: Urs Schreiber on May 14, 2011 9:50 AM | Permalink | Reply to this

Re: 1. Quantum field theory

Urs wrote:

For instance SS could be Riemannian structure. Then we would call ZZ a Euclidean quantum field theory (confusingly). If SS is “no structure” then we call ZZ a topological quantum field theory.

The “topological” nomenclature also seem to be confusing. For TQFTs, the manifolds and cobordisms have smooth structure. I guess some people count this as no structure at all. Nevertheless, the terminology would be better aligned if topological quantum field theories were about topological manifolds. In that case, what are actually called TQFTs would be called “smooth” or “differentiable” QFTs.

Anyway, thanks for making this clear. Clashes of terminology are common enough (especially where two disciplines overlap), but of course it’s much better to be aware of them than not.

Posted by: Tom Leinster on May 16, 2011 6:57 PM | Permalink | Reply to this

Re: 1. Quantum field theory

The “topological” nomenclature also seem to be confusing. For TQFTs, the manifolds and cobordisms have smooth structure. I guess some people count this as no structure at all.

That’s true, but on the other hand it is accurate to say that SS as used above, is “no structure” in the case of topological QFT: SS is structure on top of the cobordism. A cobordism itself is already defined to be a smooth manifold up to diffeomorphisms rel boundary (possibly in a higher categorical sens of “up to”) and so if we have that and only that, then indeed SS is “no structure”.

By the way, after playing with conformal cobordisms for a long time, Stolz and Teichner in their big quest for the identification of a 2-dimension QFT with structure SS whose partition function computes topological modular forms eventually ended up considering instead genuine “Euclidean” QFTs, in that the extra structure considered is indeed flat Riemannian metric structure (or rather the supergeometry analog of that). So when they speak about “Euclidean QFT” as in

they really mean it. On the other hand, people familiar with the traditional terminology may easily be misled.

Posted by: Urs Schreiber on May 16, 2011 7:23 PM | Permalink | Reply to this

Re: 1. Quantum field theory

A cobordism itself is already defined to be a smooth manifold

OK. I didn’t know cobordisms were by definition smooth; I’d imagined that you could consider topological, differentiable, smooth, Riemannian, … cobordisms, just as you can for manifolds. But I get the point.

Posted by: Tom Leinster on May 16, 2011 8:25 PM | Permalink | Reply to this

Re: 1. Quantum field theory

I’d imagined that you could consider topological, differentiable, smooth, Riemannian, … cobordisms,

Sure, but “the cobordism category” is with smooth cobordisms, historically and usually nowadays by default. This is the case that solves the plain vanilla cobordism hypothesis, notably.

What is considered in detail, too, is such cobordisms equipped with extra “topological structure” in the sense of higher lifts of the structure group of the tangent bundle: B-bordisms.

Also standard cobordisms with specified singularities have received attention.

Wikipedia asserts here that a little bit is known about the cobordims ring for (genuinely) topological and for piecewise linear manifolds.

Posted by: Urs Schreiber on May 16, 2011 9:56 PM | Permalink | Reply to this

Re: 1. Quantum field theory

If one feels like rebelling against the notion of smooth structure as “default”, as I have occasionally, it may be worth contemplating Remark 2.4.30 in On the Classification of Topological Field Theories:

Throughout this paper, we have considered only smooth manifolds. However, it is possible to define analogues of the (,n)(\infty,n)-category Bord nBord_n in the piecewise linear and topological settings. … We do not know an analogue of the cobordism hypothesis which describes the topological bordism categories Bord n TopBord_n^{Top} for n4n \ge 4. Roughly speaking, the usual cobordism hypothesis (for smooth manifolds) can be regarded as an articulation of the idea that smooth manifolds can be constructed by a sequence of handle attachments: that is, every smooth manifold admits a handle decomposition. … However, there are topological 4-manifolds which do not admit handle decompositions (such as Freedman’s E 8E_8-manifold).

In other words, the importance of smooth structure is not necessarily due to any intrinsic importance of infinitely differentiable functions (versus continuous, PL, analytic, etc. ones), but rather that they turn out to have just the right mix of constraints and flexibility so that smooth manifolds can be used as a “presentation” of free nn-categories. Kind of like how there may be no intrinsic importance to, say, compactly generated topological spaces (as opposed to other conditions one might impose on a topological space, or even other notions of “space”), but they turn out to be give a particularly nice presentation of the theory of \infty-groupoids.

Posted by: Mike Shulman on May 16, 2011 10:47 PM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

Back in the QVEST thread, John wrote:

A σ\sigma-model is a theory of physics where you have a manifold MM standing for ‘spacetime’, another manifold XX, and a function

F:MX F: M \to X

which is called a ‘field’. Then there will be some differential equations that FF needs to satisfy.

Here are some unordered thoughts.

  • Purely mathematically, this sounds almost like “a σ\sigma-model is a map of manifolds”.
  • But only “almost”, because of that bit at the end about satisfying some differential equations.
  • I realize that I don’t know what it means for a map between manifolds to satisfy a differential equation. I know that’s basic stuff and I’m not asking for an explanation; I’m sure I could look it up. But there we are.
  • Although John more or less just said “a σ\sigma-model is a map of manifolds”, he also said that the domain was meant to represent spacetime. This reminds me of Lawvere’s definition of generalized element. A generalized element of an object XX, of type MM, is a map MXM \to X. One’s first reaction might be “but why do we need another word for map/morphism/arrow?” But it is actually useful terminology, because of the change of emphasis. The MM here is typically supposed to be some archetypal shape (such as a point or a circle), and the elements in XX of type MM are then the “figures in XX of shape MM” (e.g. points or loops in XX). That way of thinking seems like it might fit with MM being spacetime.
Posted by: Tom Leinster on May 13, 2011 9:00 AM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

Tom wrote:

Purely mathematically, this sounds almost like “a σ-model is a map of manifolds”.

Right, but I actually said “it’s a theory where you have a map of manifolds satisfying a differential equation”. I didn’t say it’s one specific map. I said it was a theory where you have a map.

It’s like Lawvere and his algebraic theories, like “the theory of a group”. If you ask “which group?”, he’ll throw back his head, laugh, and say you’re mixing up the abstract general and the concrete particular.

But I certainly could have been clearer, and now I’ll try.

First, I’m talking about a classical σ\sigma-model; Urs is mainly interested in quantum σ\sigma-models, but they bring in an extra layer of complexity because to explain those, we need to explain a bit about quantum physics as well as σ\sigma-models. Everything in physics comes in classical and quantum versions, but the classical version is always quicker to explain. I wanted to focus on what’s ‘σ\sigma-modelly’ about σ\sigma-models, which is that the involve a map between manifolds.

Second, let me try something a bit more precise: a classical σ\sigma-model consists of manifolds MM, XX, and a differential equation obeyed by some of the smooth maps σ:MX\sigma : M \to X.

This is oversimplified in various ways, but it’s good for a start.

In particular, you should think about things like

M= 2M = \mathbb{R}^2

X=X = \mathbb{R}

and this differential equation that could be obeyed by a map f:MXf: M \to X:

2fx 2+ 2fy 2=0 \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2 } = 0

This is called Laplace’s equation. It’s overkill to think of this as a σ\sigma-model, mainly because XX is a vector space and it’s too darn easy to study differential equations for functions valued in a vector space: people have been doing that for centuries!

A more ‘σ\sigma-modelly’ example would be if you replaced XX by a 2-sphere. There’s still an analogue of Laplace’s equation. And it has this meaning: you’ve got a rubber sheet MM (or a portion of MM), and you’re wrapping it around your sphere XX, and pulling it tight: what’ll it do? It’ll try to minimize its energy, and it does that when it obeys this equation.

People love to consider σ\sigma-models that generalize Laplace’s equation in this way. Buzzwords like ‘minimal surface’ and ‘harmonic map’ should bubble to the surface here…

If you’re very observant, you’ll notice that I’ve switched to an example of MM that feels more like ‘space’ than ‘spacetime’: you’re probably imagining the plane as the good old Euclidean space. In other words, there’s no ‘time’ in your mental image of a rubber sheet wrapped around a 2-sphere.

But that’s a trick physicists often do: replace spacetime by space. So it’s okay. But a more spacetime-like example would be

M= 2M = \mathbb{R}^2

X=X = \mathbb{R}

and this differential equation that could be obeyed by a map f:MXf: M \to X:

2fx 2 2ft 2=0 \frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial t^2} = 0

This is called the wave equation, and I’ve switched to using coordinates xx and tt on the plane, instead of xx and yy, to hint at how you’re supposed to think. This is another famous differential equation, which describes \mathbb{R}-valued waves moving along the xx line as time tt passes. And again, this generalizes to lots of other choices of MM and XX.

Posted by: John Baez on May 13, 2011 9:33 AM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

John,

I have a basic question. What is the difference between concrete general, concrete particular and general general? This is in relation to your comment earlier about “the theory of a group”.

Posted by: professor kim on May 14, 2011 4:14 AM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

That’s not what I’d call a basic question, professor!

The famous category theorist Lawvere makes a distinction between concrete particular, concrete general and abstract general. He doesn’t have a “general general”.

Few people use these terms except him. In my summary of Lawvere’s ideas in week200, I wrote:

And this gives me an excuse to explain another bit of Lawvere’s jargon: while a theory is an “abstract general”, and particular model of it is a “concrete particular”, he calls the category of all its models in some category a “concrete general”.

For example, the algebraic theory of groups is an abstract general, and any particular group is a concrete particular, but the category of all groups is a concrete general. I mention this mainly because Lawvere flings around this trio of terms quite a bit, and some people find them off-putting. There are lots of reasons to find his work daunting, but this need not be one.

In short, we have this kind of setup:

ABSTRACT GENERAL        CONCRETE GENERAL
theory                  models
syntax                  semantics

None of this should make any sense unless you either read week200 or already know Lawvere’s work on algebraic theories.

The algebraic theory of groups is a category containing an object called “the group”. If you ask “which group?” then you’re missing the point. “The group” is not any particular group; it’s an abstract thing that only has the structures that all groups have. In Lawvere’s jargon it’s not a concrete particular, it’s an abstract general.

Similarly, when I say a classical σ\sigma-model is a theory of a map F:MXF: M \to X obeying some differential equation, you’re not supposed to think of FF as any one particular map here.

So, I was joking about this with Tom, figuring he’d have the esoteric knowledge to get this joke.

A theory of physics can be a lot like a theory in Lawvere’s sense. Unfortunately, also physicists use “model” to mean “theory”. So, a σ\sigma-model is not like a “model” of a theory in Lawvere’s sense. It’s more like a theory in Lawvere’s sense!

Posted by: John Baez on May 14, 2011 7:45 AM | Permalink | Reply to this

abstract generals are concrete particulars

On the other hand, a theory in Lawvere’s sense is also a model of another theory in his sense! That is, there is a theory whose models are theories. So a σ-model can also be itself a theory and at the same time a model of another theory. (-:

Posted by: Mike Shulman on May 16, 2011 6:35 AM | Permalink | Reply to this

2. Classical field theory

Classical field theory

A special class of examples of nn-dimensional quantum field theories, as discussed above, arise as deformations or averages of similar, but simpler structure: classical field theories . The process that constructs a quantum field theory out of a classical field theory is called quantization . This is discussed below. Here we describe what a classical field theory is. We shall inevitably oversimplify the situation such as to still count as a leisurely exposition. The kind of examples that the following discussion applies to strictly are field theories of Dijkgraaf-Witten type. But despite its simplicity, this case accurately reflects most of the general abstract properties of the general theory.

For our purposes here, a classical field theory of dimension nn is

  • a symmetric monoidal functor

    exp(iS()):Bord n SSpan(Grpd,𝒞), \exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C}) \,,

    where

    • Bord n SBord_n^S is the same category of cobordisms as before;

    • Span(Grpd,𝒞)Span(Grpd, \mathcal{C}) is the category of spans of groupoids over 𝒞\mathcal{C}:

    • objects are groupoids KK equipped with functors ϕ:K𝒞\phi : K \to \mathcal{C};

    • morphisms (K 1,ϕ 1)(K 2,ϕ 2)(K_1, \phi_1) \to (K_2, \phi_2) are diagrams

      K^ K 1 K 2 𝒞, \array{ && \hat K \\ & \swarrow && \searrow \\ K_1 &&\swArrow&& K_2 \\ & \searrow && \swarrow \\ && \mathcal{C} } \,,

      where in the middle we have a natural transformation;

    • composition of morphism is by forming 2-pullbacks:

      (K^ 2K^ 1)=K^ 1 K 2K^ 2. (\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2 \,.

Let Σ^:Σ 1Σ 2\hat \Sigma : \Sigma_1 \to \Sigma_2 be a [[cobordism]] and

exp(iS()) Σ=( Conf Σ^ ()| in ()| out Conf Σ 1 exp(iS() Σ^) Conf Σ 2 V Σ 1 V Σ 2 𝒞) \exp(i S(-))_{\Sigma} = \left( \array{ && Conf_{\hat \Sigma} \\ & {}^{\mathllap{(-)|_{in}}}\swarrow && \searrow^{\mathrlap{(-)|_{out}}} \\ Conf_{\Sigma_1} &&\swArrow_{\exp(i S(-)_{\hat \Sigma})}&& Conf_{\Sigma_2} \\ & {}_{V_{\Sigma_1}}\searrow && \swarrow_{\mathrlap{V_{\Sigma_2}}} \\ && \mathcal{C} } \right)

the value of a classical field theory on Σ^\hat \Sigma.

We interpret this data as follows:

  • Conf Σ 1Conf_{\Sigma_1} is the configuration space of a classical field theory over Σ 1\Sigma_1: objects are “field configurations” on Σ 1\Sigma_1 and morphisms are gauge transformations between these. Similarly for Conf Σ 2Conf_{\Sigma_2}.

    Here a “physical field” can be something like the electromagnetic field. But it can also be something very different. For the special case of σ\sigma-models that we are eventually getting at, a “field configuration” here will instead be a way of an particle of shape Σ 1\Sigma_1 sitting in some target space.

  • Conf Σ^Conf_{\hat \Sigma} is similarly the groupoid of field configurations on the whole cobordism, Σ^\hat \Sigma. If we think of an object in Conf Σ^Conf_{\hat \Sigma} of a way of a brane of shape Σ 1\Sigma_1 sitting in some target space, then an object in Conf Sigma^Conf_{\hat Sigma} is a trajectory of that brane in that target space, along which it evolves from shape Σ 1\Sigma_1 to shape Σ 2\Sigma_2.

  • V Σ i:Conf Σ i𝒞V_{\Sigma_i} : Conf_{\Sigma_i} \to \mathcal{C} is the classifying map of a kind of vector bundle over configuration space: a state ψZ(Σ 1)\psi \in Z(\Sigma_1) of the quantum field theory that will be associated to this classical field theory by quantization will be a section of this vector bundle. Such a section is to be thought of as a generalization of a probability distribution on the space of classical field configurations. The generalized elements of a fiber V cV_{c} of V Σ 1V_{\Sigma_1} over a configuration cConf Σ 1c \in Conf_{\Sigma_1} may be thought of as an internal state of the brane of shape Σ 1\Sigma_1 sitting in target space.

  • exp(iS()) Σ^\exp(i S(-))_{\hat \Sigma} is the action functional that defines the classical field theory: the component

    exp(iS(γ)) Σ^:V γ| inV γ| out \exp(i S(\gamma))_{\hat \Sigma} : V_{\gamma|_{in}} \to V_{\gamma|_{out}}

    of this natural transformation on a trajectory γConf Σ^\gamma \in Conf_{\hat \Sigma} going from a configuration γ| in\gamma|_{in} to a configuration γ| out\gamma|_{out} is a morphism in 𝒞\mathcal{C} that maps the internal states of the ingoing configuration γ| Σ 1\gamma|_{\Sigma_1} to the internal states of the outgoing configuration γ| Σ 2\gamma|_{\Sigma_2}. This evolution of internal states encodes the classical dynamics of the system.

Posted by: Urs Schreiber on May 13, 2011 9:02 AM | Permalink | Reply to this

Re: 2. Classical field theory

Comparing this to the previous bit, it looks as though a “classical field theory” is a special case of a “quantum field theory”, where we just make a particular choice of the symmetric monoidal target category. Is that really true?

Posted by: Mike Shulman on May 14, 2011 4:26 AM | Permalink | Reply to this

Re: 2. Classical field theory

Comparing this to the previous bit, it looks as though a “classical field theory” is a special case of a “quantum field theory”, where we just make a particular choice of the symmetric monoidal target category. Is that really true?

Yes, indeed! I have added a remark emphasizing this to the entry

Notice that this way a classical field theory is taken to be a special case of a quantum field theory, where the codomain of the symmetric monoidal functor is of the special form Span(Grpd,𝒞)Span(Grpd, \mathcal{C}). For more on this see [[classical field theory as quantum field theory]].

But that entry [[classical field theory as quantum field theory]] still needs to be written.

By the way, as I say in the entry but should maybe highlight here again: this notion is following section 3 of Freed-Hopkins-Lurie-Teleman.

This is the FQFT-point of view on classical field theory. There is also the AQFT-point of view of classical field theory as a special case of a quantum field theory. This is exposed well in Costello-Gwilliam’s factorization algebra wiki.

This may help give another angle on seeing why this is true:

One way to look at quantization in mechanics is to say that you start with a Poisson manifold and then assign to it the deformation quantization of its algebra of functions, and think of this as the quantum algebra of observables.

One way to look at quantum field theory (the AQFT way, dual to the point I have been focusing on above) is to say that to each open subset of base space you assign a quantum algebra of observables.

So if one thinks of this as coming via quantization from a classical system, one has that there should be a classical field theory given by an assignment of Poisson algebras to open subsets of base space.

Therefore in both cases the functorial structure is similar: classical field theory is a special case of quantum field theory.

Posted by: Urs Schreiber on May 14, 2011 10:06 AM | Permalink | Reply to this

Re: 2. Classical field theory

deformations or averages

Those are conceptually very different - at least to me

Posted by: jim stasheff on May 16, 2011 2:08 PM | Permalink | Reply to this

3. Quantization

Quantization

We assume now that 𝒞\mathcal{C} has colimits and in fact bilimits (colimits coinciding with limits).

Then for every functor ϕ:K𝒞\phi : K \to \mathcal{C} the colimit

Kϕ𝒞 \int^{K} \phi \in \mathcal{C}

exists, and this construction extends to a functor

:Span(Grpd,𝒞)𝒞. \int : Span(Grpd, \mathcal{C}) \to \mathcal{C} \,.

We call this the path integral functor.

For

exp(iS()):Bord n SSpan(Grpd,𝒞) \exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})

a classical field theory, we get this way a quantum field theory by forming the composite functor

Z:=exp(iS()):Bord n Sexp(iS())Span(Grpd,𝒞)𝒞. Z := \int \circ \exp(i S(-)) : Bord_n^S \stackrel{\exp(i S(-))}{\to} Span(Grpd, \mathcal{C}) \stackrel{\int}{\to} \mathcal{C} \,.

This ZZ we call the quantization of exp(iS())\exp(i S(-)).

It acts

  • on objects by sending

    Σ in (V Σ in:Conf Σ in𝒞) Σ in:= KV Σ in \begin{aligned} \Sigma_{in} & \mapsto (V_{\Sigma_{in}} : Conf_{\Sigma_{in}} \to \mathcal{C}) \\ & \mapsto \mathcal{H}_{\Sigma_{in}} := \int^K V_{\Sigma_{in}} \end{aligned}

    the vector bundle on the configuration space over some boundary Σ in\Sigma_{in} of worldvolume to its space Σ in\mathcal{H}_{\Sigma_{in}} of gauge invariant sections. In typical situations this Σ in\mathcal{H}_{\Sigma_{in}} is the famous Hilbert space of states in quantum mechanics, only that here it is allowed to be any object in 𝒞\mathcal{C};

  • on morphisms by sending a natural transformation

    Σ^ (exp(iS()) Σ^:γV γ| inV γ| out) ( Kexp(iS()) Σ^: Σ 1 Σ 2) \begin{aligned} \hat \Sigma & \mapsto (\exp(i S(-))_{\hat \Sigma} : \gamma \mapsto V_{\gamma|_{in}} \to V_{\gamma|_{out}}) \\ & \mapsto (\int^K \exp(i S(-))_{\hat \Sigma} : \mathcal{H}_{\Sigma_1} \to \mathcal{H}_{\Sigma_2} ) \end{aligned}

    to the integral transform that it defines, weighted by the groupoid cardinality of Conf Σ^Conf_{\hat \Sigma} : the path integral .

Posted by: Urs Schreiber on May 13, 2011 9:57 AM | Permalink | Reply to this

Re: 3. Quantization

Can you give an example of the sort of category 𝒞\mathcal{C} that you have in mind here? In particular I’m not quite sure what you mean by “colimits coinciding with limits.” I know what it means for coproducts to coincide with products, as for instance described at biproduct – in the finite case that makes our category enriched over abelian monoids, and in the infinite case it makes it enriched over suplattices. But what does it mean for, say, a coequalizer to coincide with an equalizer?

Also, where are you using that assumption? Does it have something to do with making \int into a functor on Span(Gpd,𝒞)Span(Gpd, \mathcal{C})? (The action of that functor on morphisms is not immediately obvious to me.)

Posted by: Mike Shulman on May 14, 2011 4:35 AM | Permalink | Reply to this

Re: 3. Quantization

Can you give an example of the sort of category 𝒞\mathcal{C} that you have in mind here? In particular I’m not quite sure what you mean by “colimits coinciding with limits.”

I said that wrong, I need to say “has colimits, limits and biproducts”. But with this assumption left and right Kan extensions along morphisms of groupoids with suitable finiteness assumtions agree. This is what one needs.

So for our purposes here finite dimensional vector spaces is the toy exmaple to keep in mind.

Also, where are you using that assumption? Does it have something to do with making \int into a functor on Span(Grpd,𝒞)Span(Grpd, \mathcal{C})? (The action of that functor on morphisms is not immediately obvious to me.)

Yes, exactly. The natural transformation and the fact that we have limits and colimit only give one a zig-zag from V Σ in\int V_{\Sigma_{in}} to V Σ out\int {V_{\Sigma_{out}}}. One needs bilimits to turn this into a direct map. Once Johan Alm was visiting me and working these things out in great detail, but then we got a bit frustrated when Freed-Hopkins-Lurie-Teleman appeared and just claimed all this and much more.

This is really what John calls “degroupoidification” but for the general case that there are those background fields on each groupoid.

I’ll try to do something to make the details of this appear on the nnLab.

Posted by: Urs Schreiber on May 14, 2011 10:30 AM | Permalink | Reply to this

Re: 3. Quantization

I’ve heard people say that “first quantization is a mystery, but second quantization is a functor.” The operation you are calling “quantization” here certainly looks like a functor, although I suppose it may be mysterious to some of us as well… so which is it?

Posted by: Mike Shulman on May 14, 2011 4:39 AM | Permalink | Reply to this

Re: 3. Quantization

I’ve heard people say that “first quantization is a mystery, but second quantization is a functor.” The operation you are calling “quantization” here certainly looks like a functor, although I suppose it may be mysterious to some of us as well…

This is one of the reasons for being interested in the special class of systems called “σ\sigma-models”: on this restricted class quantization should be a functor.

You see, the statement “quantization is a mystery” refers to the fact that there is no uniqueness for the quantization of a Poisson manifold, instead there are non-naural choices involved. But the quantum systems that we care about in fundamental physics are not given by random Poisson algebras. They have a much richer geometric structure: they are σ\sigma-models.

So when you ask

so which is it?

the answer is “It is first quantization, but we are not looking at first quantization of entirely random systems, but just of nice systems.”

Posted by: Urs Schreiber on May 14, 2011 10:40 AM | Permalink | Reply to this

Re: 3. Quantization

That’s nice! Is the “second-quantization functor” related to this restricted first-quantization functor?

Posted by: Mike Shulman on May 16, 2011 6:05 AM | Permalink | Reply to this

Re: 3. Quantization

Is the “second-quantization functor” related to this restricted first-quantization functor?

Yes, there is a deep connection. I cannot yet provide a general abstract account of this, but here is the idea:

  1. one starts with an nn-dimensional σ\sigma-model quantum field theory;

  2. one obtains the corresponding cobordism representation – physicists say: the correlators - Z:Bord n𝒞Z : Bord_n \to \mathcal{C}

  3. one then builds the “perturbative SS-matrix” defined by ZZ: this is the “linear operator on the Fock space” of the space of states assigned to the boundaries by the above cobordism representation, which is supposed to have components Σ in Σ out\mathcal{H}_{\Sigma_{in}} \to \mathcal{H}_{\Sigma_{out}} given by

    S(Σ in,Σ out)= Σ inΣ^Σ outZ(Σ^), S(\Sigma_{in}, \Sigma_{out}) = \sum_{\Sigma^{in} \to \hat \Sigma \leftarrow \Sigma_{out}} Z(\hat \Sigma) \,,

    where the sum is over all cobordisms Σ^\hat \Sigma with the given in- and out- boundaries, and some integral measure is left implicit.

    We think of this as the sum over all probability amplitudes of an (n1)(n-1)-brane of shape Σ in\Sigma_{in} to propagate through whichever space it propagates, interacting with itself as given by the shape of Σ^\hat \Sigma and coming out in the shape Σ out\Sigma_{out}.

    So this is a generalized Feynman perturbation series (an ordinary one when n=1n = 1).

  4. One can then ask for a field theory on target space XX whose correlators, in turn, are perturbatively approximated by the above “S-matrix”. This one also calls the “effective target space theory” defined by the original σ\sigma-model. The quantum version of this background theory is the “second quantization” of the original σ\sigma-model.

The experimentally verified quantum field theory of the standard model of particle physics wasn’t historically obtained this way, but it can be conceived in this way, for n=1n = 1. This approach is then called the Worldline approach to QFT . A review reference is here.

This is the very starting point of perturbative string theory: you look at the above, notice that it is confirmed by nature for n=1n = 1 and then wonder what happens when instead you set n=2n = 2. Or maybe even n=3n = 3. Whether or not the outcome is still related to experiment, it is very rich. I have tried to give a description of perturbative string theory along these lines in the nnLab entry string theory.

While I don’t know for sure how to give a general abstract formulation of this second quantization, I have an idea (which I once talked about here on the blog):

Notice the analogy between the following two setups:

  1. A σ\sigma-model itself is specified by an morphism (background field)

    Xsomewhere, X \to somewhere \,,

    where XX is the space (some cohesive \infty-groupoid) inside which “trajectories” (physical configuration) lie, and the morphism specifies a probability amplitude for each such configuration (its action).

    Path integral quantization of this is supposed to spit out a morphism

    Bord nsomewhere Bord_n \to somewhere

  2. Second quantization starts with the morphism

    Bord nsomewhere Bord_n \to somewhere

    and now does another quantization, by summing over morphisms in Bord nBord_n, where now the morphism assigns a cumulative probability amplitude to morphism in Bord nBord_n.

Suppose now you could regard this output of the quantization process again as input, hence suppose you could regard Bord nBord_n again as a space inside which configurations lie and this morphism as an assignment of propability amplitudes to each such configuration. Hence suppose you could regard the output of quantization of the σ\sigma-model itself as another σ\sigma-model. Then there ought to be another path integral quantization of this second-order σ\sigma-model, for which a configuration is a morphism in Bord nBord_n, and for which a correlator is a sum over morphisms in Bord nBord_n, weighted by this functor. But that is just what the above “S-matrix” is supposed to be.

So I am thinking: if one manages to set up the theory of σ\sigma-models and their quantization in sufficiently abstract generality, then it should be true that quantization is a process that reads in a σ\sigma-model and spits out another σ\sigma-model, and then second quantization is literally nothing but applying this operation twice.

But so far that’s just an idea.

Posted by: Urs Schreiber on May 16, 2011 11:08 AM | Permalink | Reply to this

Re: 3. Quantization

@Urs

A sigma-model itself is specified by an morphism (background field)

X –>somewhere,

where X is the space (some cohesive infty-groupoid) inside which trajectories (physical configuration) lie, and the morphism specifies a probability amplitude for each such configuration (its action).

Shouldn’t that be Config_traj X –> amplitudes?

Posted by: jim stasheff on May 16, 2011 2:05 PM | Permalink | Reply to this

Re: 3. Quantization

Shouldn’t that be Config_traj X –> amplitudes?

Right, I should have gone into more details on that analogy with second quantization that I was drawing.

First of all, the morphism XB nU(1) diffX \to \mathbf{B}^n U(1)_{diff} sends probes of XX to the data that assigns amplitudes to trajectories in these probes. But, yes, this is equivalent to a morphism P nXB nU(1)\mathbf{P}_n X \to \mathbf{B}^n U(1) from the path nn-groupoid that sends trajectories in XX to amplitudes.

Let’s even restrict to the case that the background field is flat, for emphasis. Then the above morphism is equivalently Π(X)B nU(1)\mathbf{\Pi}(X) \to \mathbf{B}^n U(1), where now Π\mathbf{\Pi} is the fundamental \infty-groupoid functor. Let’s moreover forget all the smooth structure for the same of the argument, which I indicate by passing from boldface to non-boldface, so that we have Π(X)B nU(1)\Pi(X) \to B^n U(1). Then pass to the associated “nn-vector bundle” by compositing with an infinity-representation ρ:B nU(1)nVect\rho : B^n U(1) \to n Vect, so that we arrive at a background field given by

tra:Π(X)nVect. tra : \Pi(X) \to n Vect \,.

This encodes the parallel transport of a flat nn-vector bundle on XX (some people would call is a “representation up to homotopy” of Π(X)\Pi(X) on nVectn Vect).

Suppose this happens to land in the fully dualizable nn-vector spaces, meaning: the “finite dimensional” ones. Then this version of the cobordims hypothesis-theorem tells us that this morphism is adjoint to a symmetric monoidal nn-functor

hol:Bord n(X)nVect. hol : Bord_n(X) \to n Vect \,.

This is the “holonomy functor” of the original “parallel transport” functor: where Π(X)nVect\Pi(X) \to n Vect only assigns parallel transport to disk-shaped trajectories, Bord n(X)nVectBord_n (X) \to n Vect extends that to closed nn-dimensional trajectories in XX, by “tracing”.

This makes very vivid how this classical (“oth quantized”) data is analogous to the result of quantizing it, which is supposed to be a morphism

Z:Bord nnVect, Z : Bord_n \to n Vect \,,

where now on the left we have no longer cobordisms equipped with a map to XX, but just abstract cobordisms.

So it seems that quantization is in some way about finding extensions

Bord n(X) hol nVect Z Bord n, \array{ Bord_n(X) &\stackrel{hol}{\to}& n Vect \\ \downarrow & \nearrow_{Z} \\ Bord_n } \,,

where the vertical morphism is the forgetful one.

Only that this is a bit too simple minded. Not the least because we had to assume flatness of the classical datat and had to forget smooth structure to arrive at this suggestive picture.

But even if its suggestiveness is to be taken with a grain of salt, it is still saltily suggestive. I think.

Posted by: Urs Schreiber on May 16, 2011 3:59 PM | Permalink | Reply to this

4. Classical sigma-models

Classical sigma-models

A classical σ\sigma-model is a classical field theory such that

  • the configuration spaces Conf ΣConf_{\Sigma} are mapping spaces H(Σ,X)\mathbf{H}(\Sigma,X) in some suitable category – some cohesive higher topos in fact – , for XX some fixed object of that category called target space ;

  • the bundles V Σ:Conf Σ𝒞V_{\Sigma} : Conf_{\Sigma} \to \mathcal{C} “of internal states” over these mapping spaces are

    • the transgression to these mapping spaces…

    • …of an associated higher bundle…

    • …asscociated to a circle nn-bundle with connection on target space…

    • …encoded by a classifying morphism

    α:XB nU(1) \alpha : X \to \mathbf{B}^{n} U(1)

    into the circle nn-group in H\mathbf{H}; equipped with an nn-connection \nabla

    • …where the association is via a representation ρ:B nU(1)(n+1)Vect \rho : \mathbf{B}^{n} U(1) \to (n+1) Vect on nn-vector spaces, which is usually taken to be the canonical 1-dimensional one.

One calls (α,)(\alpha,\nabla) the background gauge field of the σ\sigma-model.

  • The action functionals exp(iS()) Σ^\exp(i S(-))_{\hat \Sigma} are given by the higher parallel transport of \nabla over Σ^\hat \Sigma.

So an nn-dimensional σ\sigma-model is a classical field theory that is represented, in a sense, by a circle nn-bundle with connection on some target space.

More specifically and more simply, in cases where XX is just a discrete ∞-groupoid] – the case of sigma-models of Dijkgraaf-Witten type, every principal ∞-bundle on XX is necessarily flat, hence the background gauge field is given just by the morphism

α:XB nU(1). \alpha : X \to \mathbf{B}^{n} U(1) \,.

Then for Σ^\hat \Sigma a closed nn-dimensional manifold, the action functional of the sigma-model on Σ\Sigma on a field configuration γ:Σ^X\gamma : \hat \Sigma \to X has the value

exp(iS(γ)) Σ^= Σ^[γ *α] \exp(i S(\gamma))_{\hat \Sigma} = \int_{\hat \Sigma} [\gamma^* \alpha]

being the evaluation of [γ *α][\gamma^* \alpha] regarded as a class in ordinary cohomology H n(Σ^,U(1))H^n(\hat \Sigma, U(1)) evaluated on the fundamental class of XX.

One says that [α][\alpha] is the Lagrangian of the theory.

Posted by: Urs Schreiber on May 13, 2011 1:05 PM | Permalink | Reply to this

Re: 4. Classical sigma-models

I was mostly following up to this point, but at “transgression to these mapping spaces” you slipped over into “XXXXXX XXXXX” territory for me. Is it possible to describe the “internal state” bundles more simply by, say, starting (rather than ending) with the “data” that determines them, and hopefully without reference to nn-vector spaces, cohesive higher toposes, or other things that I understand even less? Perhaps a low-dimensional example would help? (Is that maybe what you’re about to get to down below?)

Posted by: Mike Shulman on May 14, 2011 4:45 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

I was mostly following up to this point, but at “transgression to these mapping spaces” you slipped over into “XXXXXX XXXXX” territory for me. Is it possible to describe the “internal state” bundles more simply by, say, starting (rather than ending) with the “data” that determines them, and hopefully without reference to n-vector spaces, cohesive higher toposes, or other things that I understand even less? Perhaps a low-dimensional example would help? (Is that maybe what you’re about to get to down below?)

I discuss this on my personal web, but didn’t yet find the time to move it over to the expository discussion. Let me indicate how it works.

We have the following nice statement: transgression is just the truncated internal hom.

Here is how it goes:

if Σ\Sigma is a closed manifold of dimension dimΣdim \Sigma and B nU(1)\mathbf{B}^n U(1) is the circle n+1n+1-group, then

τ ndimσH(Π(Σ),B nU(1)) τ ndimΣGrpd(Π(Σ),B nU(1)) B ndimΣU(1), \begin{aligned} \tau_{n-dim \sigma} \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)) & \simeq \tau_{n-dim \Sigma} \infty Grpd(\Pi(\Sigma), B^n U(1)) \\ & \simeq B^{n-dim\Sigma} U(1) \end{aligned} \,,

where the first line is more or less a definition and the crucial statement is the equivalence in the second line.

So then define the transgression of the background gauge field (I write the necessarily flat version for discrete \infty-group GG)

c:Π(BG)BGB nU(1) \mathbf{c} : \Pi(\mathbf{B}G) \simeq B G \to B^n U(1)

to the mapping space Grpd(Π(Σ),BG)\infty Grpd(\Pi(\Sigma), B G) to be the composite of homming Π(Σ)\Pi(\Sigma) into this and then trucating:

τ ΣV:Conf Σ:=Grpd(Π(Σ),BG)Grpd(Π(Σ),c)Grpd(Π(Σ),B nU(1))τ ndimΣB ndimΣU(1). \tau_\Sigma V : Conf_\Sigma := \infty Grpd(\Pi(\Sigma), B G) \stackrel{\infty Grpd(\Pi(\Sigma), \mathbf{c})}{\to} \infty Grpd(\Pi(\Sigma), B^n U(1)) \stackrel{\tau_{n-dim \Sigma}}{\to} B^{n- dim \Sigma} U(1) \,.

This (necessarily flat in this case) circle (ndimΣ)(n-dim \Sigma)-bundle is the transgression of the background gauge field to the configuration space of fields over Σ\Sigma.

In particular if dimΣ=ndim \Sigma = n we get a morphism to U(1)U(1). This is the holonomy of the flat circle nn-bundle on BGB G over Σ\Sigma.

This is what I used to call integration without integration .

Domenico Fiorenza helped me prove this. Details are at ∞-Chern-Simons theory – The action funtionals.

Posted by: Urs Schreiber on May 14, 2011 11:08 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

Thanks, but I wasn’t specifically asking about the transgression; I was just using that phrase to indicate the point where I stopped understanding (i.e. I understood nothing after that). Now I’ve gone back and re-read it, and I think I have some idea of what you’re saying, but it’s very difficult for me to parse the way you spread sentences out over bullet points. It might also help to indicate clearly which terms are things we’re expected to understand, and which are ones we’re not expected to understand. (“Scare quotes” are good for marking out the latter.) Here’s an attempt at the sort of explanation I was looking for:


We’re going to build a particular kind of classical field theory, and call it a “classical σ-model”. We start by assuming we have some “space” XX, in a category of “spaces” which includes manifolds. We call XX the target space, and we define the configuration space Conf ΣConf_\Sigma of a manifold Σ\Sigma to be the mapping space Map(Σ,X)Map(\Sigma,X). That is, a “configuration” on a manifold Σ\Sigma is just an XX-valued function on Σ\Sigma.

We also assume that XX is equipped with a “circle nn-bundle”. [An example would be good here, to explain why this is a reasonable thing and how we should think of it.] Rather than define exactly what that is, we’ll observe that any such thing is characterized by a classifying map α:XB nU(1)\alpha \colon X\to \mathbf{B}^n U(1), so we can just think about classifying maps instead. Here U(1)U(1) is the circle group, and B n\mathbf{B}^n denotes its nnth delooping; thus such a map is also a sort of “continuous nnth cohomology of XX with coefficients in U(1)U(1)”.

Now, just like an ordinary principal U(1)U(1)-bundle has an associated vector bundle once we fix a representation of U(1)U(1) to be the fibers, any “circle nn-bundle” has an associated “nn-vector bundle” once we fix a “representation” ρ:B nU(1)nVect\rho\colon \mathbf{B}^n U(1) \to n Vect on “nn-vector spaces”. Just as for the ordinary U(1)U(1), here we usually pick the canonical 1-dimensional such “representation”. Finally, we define the bundles V Σ:Conf Σ𝒞V_\Sigma \colon Conf_\Sigma \to \mathcal{C} of “internal states” by trangression of these associated bundles.


At this point, I’m kind of getting lost in technicalities. I see your definition of the transgression, but firstly, I’m not clear what the target category 𝒞\mathcal{C} of the “internal state” morphism is supposed to be, and secondly, I have no motivation. What’s going on? Why are we doing these constructions? What do they have to do with familiar things? I see the spaces Σ\Sigma and XX in John’s and your description of particular examples, but where do the circle nn-bundle, the nn-representation, the transgression, and the differential equation come together?

Posted by: Mike Shulman on May 16, 2011 6:32 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

Mike wrote:

At this point, I’m kind of getting lost in technicalities.

Me a little bit earlier, I don’t understand what “transgression” is, even after all the explanations…

Mike asked:

…where do the circle n-bundle, the n-representation, the transgression, and the differential equation come together?

If Urs uses traditional nomenclature, then he has defined the Lagrangian already:

One says that [α] is the Lagrangian of the theory.

In oldspeak the Lagrangian is a functional living on the configuration space, and only those elements who minimize it, are solutions of the theory. Often it is possible to show that functions that minimize the Lagrangian functional are those functions that solve certain partial differential equations, the Euler-Lagrange equations. That’s where the differential equations enter the stage.

Posted by: Tim van Beek on May 16, 2011 8:47 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

Here’s an attempt at the sort of explanation I was looking for:

Ah, thanks, now I understand what you are after. Thanks for going through the trouble of indicating this. Since I have said all these things many times before, it is hard for me to know what can be assumed and what not. But I know exactly how you feel about this, I believe: I felt the same way, I believe, at the beginning when you started talking about type theory, for instance, or (in much smaller scope, though) when Mike Stay started talking about concurrency (and I seem to remember that I said so).

All right, so thanks to your detailed indication of what you would like me to say, I have now considerably expanded the nnLab section

I have started with literally copy-and-pasting the text that you suggested you would have liked to see, and then expanded and modified it as I saw the need. It now reads as follows:


Sigma model – Idea

We are going to build a particular kind of classical field theory, and call it a classical σ\sigma-model or, after quantization, a quantum σ\sigma-model. We start by assuming we have some “space” XX, in a category of “spaces” which includes smooth manifolds. We call XX the target space , and we define the “configuration space of fields” Conf ΣConf_\Sigma over a manifold Σ\Sigma to be the mapping space Map(Σ,X)Map(\Sigma, X). That is, a “configuration of fields” over a manifold Σ\Sigma is just like an an XX-valued function on Σ\Sigma.

We assign a dimension nn \in \mathbb{N} to our σ\sigma-model, take dimΣndim \Sigma \leq n and assume that target space XX is equipped with a “circle n-bundle with connection”.

For n=1n = 1 this is an ordinary circle bundle with connection and models a configuration of the electromagnetic field on XX. To distinguish this “field” on XX from the fields on Σ\Sigma we speak of a background gauge field . (This remains fixed background data unless and until we pass to second quantization .) A field configuration ΣX\Sigma \to X on Σ\Sigma models a trajectory of a charged particle subject to the forces exrted by this background field.

For n=2n = 2, a circle nn-bundle with connection is a circle 2-group principal 2-bundle, or equivalently a “bundle gerbe” with connection. This models a “higher electromagnetic field”, called a Kalb-Ramond field. Now Σ\Sigma is taken to be 2-dimensional and a map ΣX\Sigma \to X models the trajectory of a string on XX, subject to forces exerted on it by this higher order field.

This pattern continues. In the next dimension a membrane with 3-dimensional worldvolume is charged under a circle 3-bundle with connection, for instance something called the “supergravity CC-field”.

While one can speak of higher bundles in full generality and full analogy to ordinary principal bundles, it is useful to observe that any circle nn-bundle is characterized by a classifying map α:XB nU(1)\alpha : X \to \mathbf{B}^n U(1) in our category of spaces, so we can just think about classifying maps instead. Here U(1)U(1) is the circle group, and B n\mathbf{B}^n denotes its nnth delooping; thus such a map is also a sort of cocycle in “smooth nnth cohomology of XX with coefficients in U(1)U(1)”. The additional data of a connection refines this to a cocycle in ordinary differential cohomology of XX.

Such connection data \nabla on a circle nn-bundle defines – and is defined by – a notion of higher parallel transport over nn-dimensional trajectories: for closed nn-dimensional Σ\Sigma it defines a map hol:(γ:ΣX)exp(i Σγ *)U(1)hol : (\gamma : \Sigma \to X) \mapsto \exp(i \int_\Sigma \gamma^*\nabla) \in U(1) that sends trajectories to elements in U(1)U(1): the holonomy of \nabla over Σ\Sigma, given by integration of local data over Σ\Sigma. The local data being integrated is called the Lagrangian of the σ\sigma-model. Its integral is called the action functional .

In the quantum σ\sigma-model one considers in turn the integral of the action functional over all of configuration space: the “path integral”. In the classical σ\sigma-model instead one considers only the critical locus of the action functional (where the rough idea is that the path integral to some approximation localizes around the critical locus). Points in this critical locus are said to be configurations that satisfy the “Euler-Lagrange equations of motion”. These are supposed to be the physically realized trajectories among all of them, in the classical approximation.

Just like an ordinary circle group-principal bundle has an associated vector bundle, once we fix a representation of U(1)U(1) to be the fibers, any “circle n-bundle” has an associated “n-vector bundle” once we fix a “∞-representation” ρ:B nU(1)nVect\rho : \mathbf{B}^n U(1) \to n Vect on “n-vector spaces”. Just as for the ordinary U(1)U(1), here we usually pick the canonical 1-dimensional such “representation”. We define bundles V Σ:Conf Σ𝒞V_\Sigma : Conf_\Sigma \to \mathcal{C} of “internal states” by trangression of these associated bundles.

The passage from principal ∞-bundles to associated ∞-bundles is necessary for the description of the quantum σ\sigma-model: it assigns in positive codimension spaces of sections of these associated bundles. For a 1-categorical description of the resulting QFT ordinary vector bundles (assigned in codimension 1) would suffice, but the σ\sigma-model should determine much more: an “extended” quantum field theory. This requires sections of higher vector bundles. For instance for n=2n = 2 some boundary conditions of the σ\sigma-model are given by sections of the background 2-vector bundle: these are the twisted vector bundles known as the “Chan-Paton bundles” on the “boundary-D-branes” of the string. (…)

Posted by: Urs Schreiber on May 16, 2011 4:45 PM | Permalink | Reply to this

Re: 4. Classical sigma-models

I’m still digesting this, but I have one question now: what is special about the circle nn-group? Is that nn necessarily the same as the dimension of Σ\Sigma? What about nonabelian gauge fields; are they also σ\sigma-models?

Posted by: Mike Shulman on May 17, 2011 5:40 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

Oh, and thanks!! This is indeed more the sort of thing I was looking for. It’s interesting that recently around here, we seem to be seeing a lot of this sort of thing: people with one background having trouble explaining their implicit assumptions to people with another background. I guess that insofar as we keep at it until we make progress, that means we’re doing some good in terms of communicating between disciplines. And it sounds a bit cheesy, but that really is one reason I became a category theorist. (-:

Posted by: Mike Shulman on May 17, 2011 5:56 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

what is special about the circle nn-group? Is that n necessarily the same as the dimension of Σ\Sigma?

We can use more general background fields. The relevant constraint is this:

all of known classical and quantum mechanics assumes and demands that the action functional is locally a function with values in the real numbers. Going beyond that would be wildly speculative about the structure of physics, even if probably it’s not logically ruled out.

Given this, and given that the action functional is the holonomy of the transgression of the background gauge field to codimension 0, it follows that the background gauge field must have structure nn-group whose top categorical homotopy group (homotopy sheaf) is a discrete quotient of \mathbb{R} in degree (n1)(n-1).

for this, it need however not be plain B n1U(1)\mathbf{B}^{n-1} U(1), it could have also categorical homotopy groups in lower degree. For instance for n=2n = 2 an “orientifold” background gauge field is one with structure 2-group

AUT(U(1))=[U(1) 2] AUT(U(1)) = [U(1) \to \mathbb{Z}_2]

instead of

BU(1)=[U(1)1] \mathbf{B} U(1) = [U(1) \to 1]

(or maybe even two copies of that).

What about nonabelian gauge fields; are they also σ-models?

Here we have to distinguish between background fields on XX and fields on the worldvolume Σ\Sigma.

By the above, unless one considers maybe something very different from what i have been describing, the background gauge field has to have structure nn-group which is abelian in in degree (n1)(n-1), but could have nonabelian pieces in first degree.

What it cannot have is for instance for n=1n =1 and GG a nonabelian (Lie) group, a background GG-bundle with connection.

But we can have GG-gauge theory realized as a σ\sigma-model, where the GG-principal bundles are the fields on the worldvolume :

let GG be any (possibly nonabelian) smooth \infty-group and let

c:BGB nU(1) \mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)

be a cocycle in smooth cohomology on BG\mathbf{B}G (a smooth circle nn-bundle on the smooth BG\mathbf{B}G). By “\infty-Chern-Weil theory” this has a differential refinement

c^:BG connB nU(1) conn \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}

where the subscript indicates coefficients for \infty-connections. So a morphism

γ:ΣBG conn \gamma : \Sigma \to \mathbf{B}G_{conn}

to the target space object BG conn\mathbf{B}G_{conn} is equivalently a GG-principal \infty-bundle with connection on Σ\Sigma. For GG an ordinary Lie group this is an ordinary GG-principal bundle.

Notice the shift in perspective here: a morphism ΣX\Sigma \to X we interpret as a trajectory of shape Σ\Sigma in target space XX, but if XX is a non-concrete smooth \infty-groupoid such as BG conn\mathbf{B}G_{conn}, this is very exotic as far as “trajectories” go and is in fact equivalently a GG-connection on Σ\Sigma.

So a σ\sigma-model with target “space” X=BG connX = \mathbf{B}G_{conn} is really a (nonabelian, in general) GG-gauge theory on Σ\Sigma. Precisely, with the above c^\hat \mathbf{c} as its (abelian!) background gauge field, this is a gauge theory of higher Chern-Simons theory-type.

the “\infty-Dijkgraaf-Witten theory” that gives this thread its title, and to which I will come in more detail, is the special case of this when GG happens to be a discretely smooth \infty-group. This is really the crucial point that allows us to speak of σ\sigma-models and DW theory and CS theory here in the same context:

in a suitable cohesive \infty-topos, gauge theory is part of the theory of σ\sigma-models (at least for certain action functionals).

More discussion of all this is at infinity-Chern-Simons theory.

Posted by: Urs Schreiber on May 17, 2011 8:02 AM | Permalink | Reply to this

Re: 4. Classical sigma-models

Okay, that second approach coincides with the approach to quantum field theory starting from plain quantum mechanics that I have picked up from some introductory books. Namely, one thinks of the “position of a particle in 3-space”, rather than being something about 3-space, as a list of 3 “fields” (or a single 3\mathbb{R}^3-valued field) on a point—or on \mathbb{R}, if we introduce time-dependence. Then we can reintroduce “space” or “spacetime” by stretching out that point.

But it does seem curious that there is something special about the circle, and hence about electromagnetism. Why is it that for the case of a particle moving in a background field, we can deal with a background electromagnetic field but nothing more general?

Posted by: Mike Shulman on May 17, 2011 4:46 PM | Permalink | Reply to this

Re: 4. Classical sigma-models

one thinks of the “position of a particle in 3-space”, rather than being something about 3-space, as a list of 3 “fields” (or a single 3\mathbb{R}^3-valued field) on a point—or on \mathbb{R}, if we introduce time-dependence. Then we can reintroduce “space” or “spacetime” by stretching out that point.

Yes, exactly, that’s the point of view that all this here starts from. As a slogan: “Quantum mechanics is 1-dimensional quantum field theory – on the particle’s worldline.”

Posted by: Urs Schreiber on May 17, 2011 6:00 PM | Permalink | Reply to this

Re: 4. Classical sigma-models

But it does seem curious that there is something special about the circle, and hence about electromagnetism. Why is it that for the case of a particle moving in a background field, we can deal with a background electromagnetic field but nothing more general?

This is all in what we mean when we say “we can deal”. Let me try to indicate this more accurately.

Of course we can write down much more general models (action functionals) in traditional terms.

However, here I am trying to give a general abstract description of these models. That is supposed to involve saying: “the action functional (except for the kinetic term) is the transgression of the background gauge field connection to configuration space”. This makes sense in great generality. But it also restricts the possible action functionals that are examples (though a large class is).

BUT… there is more to the story: I have not so far said much at all σ\sigma-models with boundaries. A whole new universe of concepts appears as soon as we do that.

For instance the case that I am guessing you have in mind is: let \nabla be any GG-connection on target space XX for nonabelian GG and fix a finite-dimensional linear representation of GG. Then there is (in particular, restricting attention to this for simplicity right now) the holonomy functional

hol(,):[γ:S 1X]hol(,γ)=trPexp( S 1γ *). hol(\nabla,-) : [\gamma : S^1 \to X] \mapsto hol(\nabla,\gamma) = tr P \exp(\int_{S^1} \gamma^* \nabla) \,.

So we can write down the action functional for a nonabelian charged particle (a “quark”, say)

exp(iS()):[γ]exp(iS kin([γ]))hol(,γ). \exp(i S(-)) : [\gamma] \mapsto \exp(i S_{kin}([\gamma])) \; hol(\nabla, \gamma) \,.

This is not of the kind of action functionals that I have discussed so far. But it turns out that this is part of the 2-dimensional σ\sigma-model which I vaguely discussed, once we allow Σ\Sigma to have a boundary.

To roughly see this, recall from post C4 that a section of a line 2-bundle is a (nonabelian!) (and possibly twisted) vector bundle, and that these naturally appear on the boundary of the 2-dimensional Σ\Sigma. In the full theory with boundary the abelian line 2-bundle (with connection) induces nonabelian 1-bundles (with connection) and the action functionals then involve boundary terms that are more general than what I have described so far.

There is supposed to be a long story of such “boundary QFTs”. Some examples are listed, as you may recall, in that entry with the curious title holographic principle of higher category theory in the section Applications to QFT here.

Posted by: Urs Schreiber on May 17, 2011 6:30 PM | Permalink | Reply to this

Re: ∞-Dijkgraaf-Witten Theory

This is a case of Descartes before the horse

What is called an n-dimensional σ-model is first of all an instance of an n-dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are σ-models is that

even the toc lists quantum before classical!

Posted by: jim stasheff on May 13, 2011 1:44 PM | Permalink | Reply to this

Exposition of classical sigma-models

Above John was trying to explain the idea of typical classical σ\sigma-models; concretely, without general abstraction.

Of course this is the traditional theory that is descibed in the introductory physics textbooks. My original idea of this thread here had been different: the strategy was to exploit the fact that one can speak of the concept of σ\sigma-model in an entirely different way – a general abstract way – a way that does not require knowledge of the concept of differential equations but just of basic category theory; and thus get abstract category theorists into the boat. But for the sake of completeness I should also provide some of the traditional discussion. So I’ll do that now.

As before I’ll post this in bite-sized bits in the comments below.

Exposition of classical sigma-models

  1. The Newtonian particle

  2. The relativistic particle

  3. The relativistic (n-1)-brane

  4. The relativistic string

Posted by: Urs Schreiber on May 13, 2011 9:48 PM | Permalink | Reply to this

C1 - The Newtonian particle

The Newtonian particle

With hindsight, the earliest σ\sigma-model ever considered was also the very origin of the science of physics:

In order to describe the motion of matter particles in space, Isaac Newton wrote down a differential equation with the famous symbols

F=ma. \vec F = m \vec a \,.

More in detail, this is meant to describe the following situation:

  • write X:= 3X := \mathbb{R}^3 for the Cartesian space of dimension 3; think of this as a crude model for physical space;

  • write Σ:=\Sigma := \mathbb{R} for the Cartesian space of dimension 1; think of this as the abstract trajectory of a point particle;

  • write γ:ΣX\gamma : \Sigma \to X for any smooth function; think of this as an actual trajectory of a point particle in XX;

    write furthermore

    • v:=γ˙Hom(TΣ,TX)\vec v := \dot \gamma \in Hom(T \Sigma, T X) for the derivative of γ\gamma; think of this as the velocity of the particle; and

    • a:=γ¨\vec a := \ddot \gamma for the second derivative, the acceleration of the particle.

We call then the collection of all smooth functions

Conf:=C (Σ,X) Conf := C^\infty(\Sigma, X)

the configuration space of a physical model of a point particle propagating on XX.

In order to define the model – the model of some physical situation –

  • pick a vector field FΓ(TX)\vec F \in \Gamma(T X) on XX.

    Think of this as expressing at each point a force acting on the particle with trajectory.

    For instance F:=qE\vec F := q \vec E could be an electric field influencing the propagation of an electrically charged particle of charge qq.

In modern language we may say:

  • Σ\Sigma is the worldline of the particle;

  • XX is the target space (expressing the fact that it is the codomain of a function γ:ΣX\gamma : \Sigma \to X)

  • F\vec F is the background gauge field ;

and the collection (Σ,X,F)(\Sigma, X, \vec F) of all three is a σ\sigma-model .

Given this data, the space of solutions to the original differential equation

P:={γConf|F=ma} P := \{ \gamma \in Conf | \vec F = m \vec a \}

is called the covariant phase space of the model. The configurations in PConfP \subset Conf have the interpretation of being those potential configurations, that describe actual trajectories of particles observed in nature.

Posted by: Urs Schreiber on May 13, 2011 10:03 PM | Permalink | Reply to this

Re: C1 - The Newtonian particle

I like this. I think even category theorists need examples to understand general concepts. The examples don’t need to be fancy, and a lot of mathematicians know some Newtonian mechanics, so it makes a good example.

I guess your plan is to describe it in such a way that it’s natural and effortless to generalize from particle trajectories γ: 3\gamma : \mathbb{R} \to \mathbb{R}^3 to maps γ:ΣX\gamma : \Sigma \to X for other manifolds Σ,X\Sigma, X. If you do this right, people who know Newtonian mechanics and manifolds can quickly get to classical σ\sigma-models.

Posted by: John Baez on May 14, 2011 7:58 AM | Permalink | Reply to this

Re: C1 - The Newtonian particle

Given γC (Σ,X)\gamma\in C^\infty(\Sigma,X), the velocity, vHom(TΣ,TX)\vec{v}\in Hom(T\Sigma, TX), is canonically-defined. Defining a\vec{a} requires more data, which you need to supply, before the above makes sense.

Posted by: Jacques Distler on May 14, 2011 5:43 PM | Permalink | PGP Sig | Reply to this

Re: C1 - The Newtonian particle

Yup. But I’m sure Urs knows the structure people usually put on Σ and X to define a σ-model.

Posted by: John Baez on May 15, 2011 4:41 AM | Permalink | Reply to this

Re: C1 - The Newtonian particle

What people “usually do” is geared to defining the quantum theory. The classical theory, that Urs is developing, does not, for instance, require a Riemannian structure on XX; a connection on TXTX suffices.

(It does, of course, still require a (pseudo)-Riemannian structure on Σ\Sigma.)

Without that additional data, the “equation” F=ma\vec{F}= m\vec{a} doesn’t even make sense.

Posted by: Jacques Distler on May 15, 2011 6:05 AM | Permalink | PGP Sig | Reply to this

Re: C1 - The Newtonian particle

Defining a\vec a requires more data, which you need to supply, before the above makes sense.

For expositional purposes I want to start with the simple and simple-minded examples here and work up to the full case. To my mind, if we speak about Newtonian physics we have (as people did historically) implicitly canonically assumed trivial and trivialized tangent bundle and connection of 3\mathbb{R}^3. But not to be misleading (in case it was) I have now added the following parenthetical remark in the nnLab entry:

write a=γ¨\vec a = \ddot \gamma for the second derivative, the acceleration of the particle (strictly speaking this is the covariant derivative with respect to the trivial connection on the (canonically trivialized) tangent bundle over 3\mathbb{R}^3, see below for the fully fledged discussion).

A minute ago I have posted the next installment in this should-be or would-be exposition

C2 – The relativistic particle

This now does talk about the covariant derivative.

Posted by: Urs Schreiber on May 16, 2011 2:24 PM | Permalink | Reply to this

C2 - The relativistic particle

The relativistic particle

The Newtonian particle propagating on 3\mathbb{R}^3, discussed above, is a special and limiting case of a particle propagating on a 4-dimensional pseudo-Riemannian manifold: spacetime. For historical reasons (the same that led to the theory of gravity being called a theory of relativity ) this is called the relativistic particle .

The σ\sigma-model describing the relativistic particle is the following.

  • Target space is a pseudo-Riemannian manifold (X,g)(X,g), thought of as spacetime.

  • Parameter space Σ=\Sigma = \mathbb{R} is the real line, thought of as the abstract worldline of the particle.

  • The background gauge field is given by a circle bundle with connection, which for the moment we shall assume to be topologically trivial and hence be equivalently given by a differential 1-form AΩ 1(X)A \in \Omega^1(X). Its curvature exterior derivative F:=dAF := d A is the field strength of an electromagnetic field on XX.

  • The configuration space is the quotient

    Conf=C (,X)//Diff(), Conf = C^\infty(\mathbb{R}, X)//Diff(\mathbb{R}) \,,

    of smooth functions ΣX\Sigma \to X by diffeomorphisms ΣΣ\Sigma \stackrel{\simeq}{\to} \Sigma. Each point in cofiguration space is a trajectory of a particle in spacetime.

  • The covariant phase space – the subspace of the configuration space of those configurations that satisfy the equations of motion – is defined to be

    P={[γ]Conf|mg( γ˙γ˙/|γ˙|,)=qF(γ˙,)}Conf, P = \{ [\gamma] \in Conf |\;\; m g(\nabla_{\dot \gamma} \dot \gamma/ {\vert \dot \gamma}\vert,-) = q F(\dot \gamma, -) \;\; \} \subset Conf \,,

    where

    • \nabla denotes the covariant derivative of the Levi-Civita connection of the background metric gg on the tangent bundle TXT X of XX.

    • the equation is taken to hold in each cotangent space T γ(τ) *XT^*_{\gamma(\tau)} X for each τ\tau \in \mathbb{R}.

To see what this means physically, consider some special cases. First regard the case that the background field strenght vanishes, F=0F = 0. Then the equations of motion reduce to

γ˙γ˙=0. \nabla_{\dot \gamma} \dot \gamma = 0 \,.

This says that the trajectory γ\gamma exhibits parallel transport of its tangent vectors with respect to the Levi-Civita connection of the background metric. These curves are precisely the geodesics of the background geometry. This models motion under the force exerted by the field of gravity on our particle.

In the even more special case that XX is Minkowski spacetime, where we may find a global coordinate chart ( 4,η)(X,g)(\mathbb{R}^4, \eta) \simeq (X,g), this are exactly the straight lines in 4\mathbb{R}^4. Given any such, there is precisely one representative in the diffeomorphism class for which γ 4x 0\mathbb{R} \stackrel{\gamma}{\to} \mathbb{R}^4 \stackrel{x^0}{\to} \mathbb{R} is the identity, hence for which the worldline parameter coincides precisely with the chosen global time coordinate t:=x 0t := x^0 on 4\mathbb{R}^4. For those the equations of motions are again those of the free Newtonian particle a=0\vec a = 0.

Remaining in the case that XX is Minkowski space but allowing now a nontrivial background field, notice that we may write the 2-form FF always as

F=E idx idt+B iϵ ijkdx jdx j, F = \vec E_i \cdot d x^i \wedge d t + \vec B^i \epsilon_{i j k} d x^j \wedge d x^j \,,

where E 3\vec E \in \mathbb{R}^3 is the electric field strength vector and B 3\vec B \in \mathbb{R}^3 the magnetic field strength vector. The spatial part of the above equations of motion are in this case again as for a Newtonian particle

ma=qE+qv×B, m \vec a = q \vec E + q \vec v \times \vec B \,,

where in the second term we have the cross product of vectors in 3\mathbb{R}^3. The force on the right is the Lorentz force exerted by an electromagnetic field on a charged particle.

Notice that the equations of motion imply, generally, that the norm of γ˙\dot \gamma is constant along the trajectory

ddτg(γ˙,γ˙) =2g( γ˙γ˙,γ˙) =2g(g 1(ι γ˙F,),γ˙) F(γ˙,γ˙) =0. \begin{aligned} \frac{d}{d \tau} g(\dot \gamma, \dot \gamma) &= 2 g(\nabla_{\dot \gamma} \dot \gamma, \dot \gamma) \\ & = 2 g(g^{-1}(\iota_{\dot \gamma} F, -), \dot \gamma) \\ & \propto F(\dot \gamma, \dot \gamma) \\ & = 0 \end{aligned} \,.

Therefore a trajectory that solves the equations of motion and whose tangent vector is timelike or spacelike , respectively at any instant is so throughout. In particular, no choice of gravitational and electromagnetic background field strength can accelerate a physical particle from being timelike to being light-like.

Experiments around the second half of the 19th and the beginning of the 20th century established that this covariant phase space correctly describes the dynamics of gravitationally and electromagnetically charged relativistic particles. But also formally this phase space is not a randomly chosen space; instead, it is the critical locus of a (mathematically) natural action functional .

The points in the covariant phase space

P:={[γ]Conf|mg( γ˙γ˙/|γ˙|,)=qι γ˙F}Conf P := \{ [\gamma] \in Conf | m g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert,-) = q \iota_{\dot \gamma} F \} \subset Conf

happen to be the local critical points of the functional

S:Conf S : Conf \to \mathbb{R}

given by

S([γ]) :=S kin([γ])+S gauge([γ]) :=m Σdvol(γ *g)+q Σγ *A, \begin{aligned} S([\gamma]) & := S_{kin}([\gamma]) + S_{gauge}([\gamma]) \\ & := m \int_\Sigma dvol(\gamma^\ast g) + q \int_\Sigma \gamma^\ast A \end{aligned} \,,

where on the left we have the integral of the volume form of the pullback γ *gSym 2T *Σ\gamma^\ast g \in Sym^2 T^\ast \Sigma of the metric on target space to the worldline.

This is called the action functional of the relativistic particle σ\sigma-model. The first summand is called the kinetic action , the second is calle the gauge coupling action.

Typically one characterizes σ\sigma-models in terms of such action functionals, so that the covariant phase space is given as their critical locus. This usually yields a simpler and deeper description of the model.

Notably the above action functional has an evident generalization to the case where the background electromagnetic field is not given by a globally defined 1-form, but more generally by a cicle bundle with connection \nabla: if we pass to the exponentiated action functional

exp(iS()):ConfU(1) \exp(i S(-)) : Conf \to U(1)

exp(iS(γ))=exp(S kin(γ))exp(iq Σγ *A) \exp(i S(\gamma)) = \exp(S_{kin}(\gamma)) \;\; \exp(i q \int_\Sigma \gamma^* A)

the second factor is precisely the holonomy of \nabla over the worldline. Hence for general electromagnetic background gauge fields the action functional is (assuming for simplicity now closed curves with Σ=S 1\Sigma = S^1)

exp(iS(γ))=exp(S kin(γ))hol(,γ). \exp(i S(\gamma)) = \exp(S_{kin}(\gamma)) \;\; hol(\nabla, \gamma) \,.

This is the beginning of an important pattern: most σ\sigma-models are determined by a kind of higher gauge field \nabla on target space (a cocycle in the differential cohomology of target space) and their dynamics is determined by an action functional that is the higher holonomy functional of this gauge field.

At the same time the kinetic action functional factor is usually to be understood as part of the measure on configuration space ConfConf. For the particle this has been made precise: the path integral

γConfhol(,γ)exp(iS kin(γ))[dγ]:= γConfhol(,γ)dμ Wien \int_{\gamma \in Conf} hol(\nabla,\gamma) \;\;\exp(i S_{kin}(\gamma)) [d \gamma] := \int_{\gamma \in Conf} hol(\nabla,\gamma) d \mu_{Wien}

can be interpreted as the integral with respect to the Wiener measure on path space (after Wick rotation, at least). The kinetic part of the action functional is then absorbed into the Wiener measure dμ Wiend \mu_{Wien}

exp(iS kin(γ))[dγ]:=dμ Wien \exp(i S_{kin}(\gamma)) [d\gamma] := d \mu_{Wien}

and the path integral is just the “expectation value” (after Wick rotation) of the holonomy, taken over all trajectories.

Since there is a good general abstract theory of higher gauge fields and their higher holonomies, this suggests that there should be a general abstract theory of σ\sigma-models.

Posted by: Urs Schreiber on May 16, 2011 2:11 PM | Permalink | Reply to this

Re: C2 - The relativistic particle

Okay, this is helpful to me at least, although I need to take a little while to digest it and see why it reduces to what’s familiar. A few initial questions:

I can’t help feeling queasy about all those [γ][\gamma]s. Should it be obvious that everything you wrote down is independent of the choice of representative?

And is there an explanation for what is “covariant” about the “covariant phase space”?

Finally, at the beginning you said that with Σ=\Sigma=\mathbb{R} we think of maps ΣX\Sigma \to X as trajectories of particles. But then later you said that for simplicity, we can take Σ=S 1\Sigma=S^1. That seems like, among other things, it would require our spacetime to contain closed timelike curves?

Posted by: Mike Shulman on May 16, 2011 11:06 PM | Permalink | Reply to this

Re: C2 - The relativistic particle

Should it be obvious that everything you wrote down is independent of the choice of representative?

I had a normalization factor 1/|γ˙|1/\vert\dot \gamma\vert missing in some places, but fixed now. With that it should be obvious. I have typed up more details at relativistic particle. Will try to polish tomorrow, it’s too late here now.

And is there an explanation for what is “covariant” about the “covariant phase space”?

Yes. Often (and traditionally) what people call phase space is really a choice of parameterization of the invariantly defined phase space (simply the space of solutions of the equations of motion) by boundary data on a choice of Cauchy surface. The “covariant” is physspeak here for “without unnatural choices”. In a better world terminology would be different.

Finally, at the beginning you said that with Σ=\Sigma = \mathbb{R} we think of maps ΣX\Sigma \to X as trajectories of particles. But then later you said that for simplicity, we can take Σ=S 1\Sigma = S^1. That seems like, among other things, it would require our spacetime to contain closed timelike curves?

I said this at a point where I just consider the action functional, not its critical locus. In the quantum theory it is of considerable interest to consider the path integral over the configuration space of closed trajectories – this gives what is called the partition function – even if not a single one of these will be a solution to the equations of motion.

But the point where I say “for simplicity consider Σ=S 1\Sigma = S^1” all I am doing is being too lazy to discuss that with nontrivial background gauge fields and compact but non-closed Σ\Sigma, the action functional no longer takes values in U(1)U(1), strictly speaking, but in fiber homomorphisms of the circle bundle. But that’s not a big deal either way. I’ll improve the discussion of this point tomorrow.

Posted by: Urs Schreiber on May 17, 2011 12:12 AM | Permalink | Reply to this

Re: C2 - The relativistic particle

not to mention that once upon a time
physicists covariant was topologists contravariant and vice versa

Posted by: jim stasheff on May 17, 2011 1:29 PM | Permalink | Reply to this

Re: C2 - The relativistic particle

not to mention that once upon a time physicists covariant was topologists contravariant and vice versa

Right, but this problem here is yet another one: in physics the term “covariance” is used in two entirely unrelated ways. You are referreing to the usage as in “covariant tensor field”.

Here however it is in the sense of “covariant field theory”, “covariant quantization” etc. This means the absence of arbitrary choices such as choices of Cauchy surfaces, gauge choices, etc.

As far as I can tell the origin of this mess is the following: when people passed from Newtonian physics to special relativity, they would speak for instance of “the covariant form of Newton’s law” (as here) when they expressed it as an equation of vector fields on spacetime, instead of in a fixed Newtonian reference frame.

This is clearly still connected to the “covariance” of “covariant tensors”. But then as time passed, the word “covariance” in some circles became used more and more in the sense of “invariant”, taking from the above example the aspect that the “covariant formulation” is independent of a choice of frame.

But then, of course, all good things are by themselves invariant, and only become not so if one applies force to them. So in turn many concepts that should have originally been considered as the invariant intrinsic concepts that they are – such as phase space – now go with the prefix “covariant” – to indicate that one means the correct version, not any of the versions that require arbitrary choices.

It’s a bit of a terminological mess, but we will have to live with it.

Posted by: Urs Schreiber on May 17, 2011 1:58 PM | Permalink | Reply to this
Read the post ∞-Dijkgraaf-Witten Theory
Weblog: The n-Category Café
Excerpt: In another thread Tom Leinster would lilke to learn what a sigma-model in quantum field theory is. Here I want to explain this in a way that will make perfect sense to Tom, and hopefully even intrigue him. To...
Tracked: May 16, 2011 5:19 PM

C3 - The relativistic (n-1)-brane

The relativistic (n-1)-brane

It is hard not to consider the following generalization of the relativistic particle σ\sigma-model, that we discussed above:

notice that nothing in the structure of the relativistic particle’s action functional relies on the dimension of Σ\Sigma being 11. Instead, it is just the degree-1 case of the following family of types of classical σ\sigma-models, that make sense for all nn \in \mathbb{N}:

  • let Σ\Sigma be of dimension nn;

  • let target space be a pseudo-Riemannian manifold (X,g)(X,g) as before;

  • let the background gauge field be given by a smooth differential nn-form AΩ n(X)A \in \Omega^n(X) .

  • let configuration space be the (weak) quotient

    Conf Σ:=C (Σ,X)//Diff(Σ). Conf_\Sigma := C^\infty(\Sigma, X)//Diff(\Sigma) \,.

  • let then finally the action functional be given by

    S([γ])= Σdvol(γ g)+ Σγ A. S([\gamma]) = \int_\Sigma dvol(\gamma^* g) + \int_\Sigma \gamma^* A \,.

This is the same formula as for the relativistic particle as before, only that now the differential forms are taken to be of degree nn and integrals to be over nn-dimensional spaces.

Moreover, for each nn \in \mathbb{N} there is an analog of the generalization

{1forms}{circlebundleswithconnection} \{1-forms\} \hookrightarrow \{circle bundles with connection\}

to the generalization

{nforms}{circlenbundleswithconnection}. \{n-forms\} \hookrightarrow \{circle n-bundles with connection\} \,.

Given any circle nn-bundle with connection \nabla and closed Σ\Sigma of dimension nn, there is a higher holonomy functional

hol(,):(ΣγX)hol(,γ)U(1) hol(\nabla, -) : (\Sigma \stackrel{\gamma}{\to} X) \mapsto hol(\nabla, \gamma) \in U(1)

that extends the functional Aexp(i Σγ *A)A \mapsto \exp(i \int_\Sigma \gamma^* A).

Therefore more generally we say

  • the background gauge field on XX is a circle nn-bundle with connection \nabla;

  • the exponentiated action functional is

    [γ]exp(i Σdvol(γ *))hol(,γ). [\gamma] \mapsto \exp(i \int_\Sigma dvol(\gamma^* )) \;\;\; hol(\nabla,\gamma) \,.

For n=3n = 3 such a σ\sigma-model describes an analog of a relativistic particle which is not pointlike, but 2-dimensional (with 3-dimensional trajectory) hence which reminds one of a membrane. Inspired by this term, the general case has come to be known as the relativistic (n1)(n-1)-brane .

The case n=2n = 2 is called the relativistic string , which we consider in more detail below. This has received a lot of attention (in string theory ) not just because it is the next simplest in an infinite hierarchy of cases, but also because its quantum theory turns out to have various interesting features that seem to make it special. Moreover, many of the (n1)(n-1)-branes for other nn re-appear in one way or other in the study of the string (as its boundary D-branes in all dimensions 0n100 \leq n \leq 10, as its “strongly coupled” version: the M2-brane , or as its electric-magnetic dual: the NS5-brane ). If nothing else, the seemingly innocent step from n=1n = 1 to n=2n = 2 in the σ\sigma-model shows that there is a rich pattern of higher dimensional (σ\sigma-model) quantum field theories that are all interrelated in intricate ways.

Another important special case for the general discussion of σ\sigma-models is the case of the membrane, n=3n = 3, for which the background gauge field is a Chern-Simons circle 3-bundle for some GG-principal bundle on XX, for GG some suitable Lie group. In this case the gauge-coupling Lagrangian of the σ\sigma-model is, locally, the Chern-Simons form CS( 𝔤)CS(\nabla_\mathfrak{g}) of a GG-connection 𝔤\nabla_{\mathfrak{g}} on XX, hence the action functional is (locally) the Chern-Simons functional

(ΣγX) ΣCS(γ * 𝔤). (\Sigma \stackrel{\gamma}{\to} X) \mapsto \int_\Sigma CS(\gamma^* \nabla_\mathfrak{g}) \,.

Below we will see that when σ\sigma-models are considered internal to a suitable cohesive (∞,1)-topos, then there are universal σ\sigma-models of this Chern-Simons type, whose target space is no longer a smooth manifold, but a smooth ∞-groupoid incarnation of a classifying space BGB G.

Posted by: Urs Schreiber on May 17, 2011 11:17 AM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

@Urs:
circle 3-bundle for some G-principal bundle on X, for G some suitable Lie group. …whose target space is no longer a smooth manifold, but a smooth ∞-groupoid incarnation of a classifying space BG.

so why is the word `circle’ in there?

Posted by: jim stasheff on May 17, 2011 1:25 PM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

so why is the word `circle’ in there?

The group /=U(1)=S 1\mathbb{R}/\mathbb{Z} = U(1) = S^1 is the “circle group”. This is the structure group for “circle bundles”. The associated bundles under the canonical 1-dimensional (complex) representation are (complex) line bundle.

For n2n \geq 2, the “circle nn-group” is the (n1)(n-1)-fold delooping (of smooth \infty-groupoids) of the circle group, written B n1(/)=B n1U(1)\mathbf{B}^{n-1} (\mathbb{R}/\mathbb{Z}) = \mathbf{B}^{n-1} U(1) (where here and always, the boldface is to indicate the corresponding operation – here delooping – in the smooth refinement, here: in the \infty-topos of smooth \infty-groupoids).

Under the Dold-Kan correspondence Ξ:Ch 0sAbKanCplx\Xi : Ch_{\bullet \geq 0} \to sAb \to KanCplx this is presented by the chain complex of abelian Lie groups

B n1U(1)Ξ[U(1)000] \mathbf{B}^{n-1}U(1) \simeq \Xi[U(1) \to 0 \to 0 \to \cdots \to 0]

concentrated in degree (n1)(n-1) on the circle Lie group.

Under the canonical 1-dimensional nn-representation

ρ:B nU(1)nVect \rho : \mathbf{B}^n U(1) \to n Vect

we have associated to every circle nn-bundle a corresponding line nn-bundle.

Posted by: Urs Schreiber on May 17, 2011 1:36 PM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

What definition of “nn-vector space” are we using here?

Posted by: Mike Shulman on May 17, 2011 4:35 PM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

What definition of “nn-vector space” are we using here?

Recursively:

  • a (basis for an) nn-vector space is an algebra object AA in (n1)Vect(n-1)Vect, thought of as a presentation for the nn-vector space Mod AHom(BA,(n1)Vect)Mod_A \simeq Hom(B A , (n-1)Vect);

  • a morphism of nn-vector space is an AA-BB- bimodule object NN in (n1)Vect(n-1)Vect, thought of as a matrix presentation of the nn-linear map

    Mod A() ANMod B. Mod_A \stackrel{(-)\otimes_A N}{\to} Mod_B \,.

I have expanded the nnLab entry n-vector space a little more.

I had once described the idea in the appendix of my AQFT-article, then later it appeared in section 7 of Freed-Hopkins-Lurie-Teleman in their discussion of quantization of “\infty-Dijkgraaf-Witten”-theories.

Posted by: Urs Schreiber on May 17, 2011 5:38 PM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

I don’t know how you manage to assimilate all of these things in such a short amount of time. I’m only gradually starting to come to terms with Lurie’s first book and his work on the cobordism hypothesis; it’ll be quite a while before I make any significant headway into his second book or this FHLT thing.

Right now, what I’d like to do is ignore the nn-vector spaces and just think about ordinary vector spaces. Can I avoid getting into trouble doing that if I also restrict my attention to the quantum field theory of particles, rather than strings or branes?

Posted by: Mike Shulman on May 18, 2011 5:10 AM | Permalink | Reply to this

Re: C3 - The relativistic (n-1)-brane

Right now, what I’d like to do is ignore the nn-vector spaces and just think about ordinary vector spaces. Can I avoid getting into trouble doing that if I also restrict my attention to the quantum field theory of particles, rather than strings or branes?

Yes, for n=1n = 1 all you need are ordinary vector spaces.

But just for emphasis, you are expert on the n=2n=2-case already: 2Vect2 Vect is nothing but Prof(Vect)Prof(Vect). And nVectn Vect is morally like Prof((Prof ((n-1)Vect))Vect).

I don’t know how you manage to assimilate all of these things in such a short amount of time. I’m only gradually starting to come to terms with Lurie’s first book and his work on the cobordism hypothesis; it’ll be quite a while before I make any significant headway into his second book or this FHLT thing.

Well, the FHLT article picks up several threads that I have thought about for a long time, and which we have discussed here on this blog for a long time. It is embarrassing enough that I didn’t see some of its assertions myself, the least one should expect is that I recognize them when i see them.

As for Higher Algebra : I find the storyline real easy to follow, since it’s all very natural and enlightning. That doesn’t mean that I have made much of an attempt to follow each proof line-by-line, so far.

Posted by: Urs Schreiber on May 18, 2011 6:22 AM | Permalink | Reply to this

C4 - The relativistic string

The relativistic string

The important case n=2n = 2 of the general (n1)(n-1)-brane σ\sigma-model, discussed above, is called the string-σ\sigma-model . Even though this is just the first step after the relativistic particle, the theory of this σ\sigma-model is already considerably richer. Classically and all the more so after quantization. For the purposes of this exposition here I only briefly indicate the physical interpretation of the σ\sigma-model and then consider some qualitatively new higher gauge theory aspects, that appear in this dimension.

First notice that by the general reasoning of relativistic (n1)(n-1)-branes, the background gauge field is now given (if we assume for the moment a topological trivial class) by a 2-form, which is traditionally denoted BΩ 2(X)B \in \Omega^2(X) and called the B-field . Its 3-form curvature field strength is traditionally denoted H:=dBH := d B.

The action functional of the string’s σ\sigma-model for a pseudo-Riemannian target space (X,g)(X,g) with background gauge field BB is

[γ] Σdvol(γ g)+ Σγ B. [\gamma] \mapsto \int_\Sigma dvol(\gamma^* g) + \int_\Sigma \gamma^* B \,.

To gain insight into the physical meaning of this, consider the simple case that target space (X,g)(X,g) is Minkowski spacetime and that the worldsheet Σ=×S 1\Sigma = \mathbb{R} \times S^1 is the cylinder. With (τ,σ)(\tau,\sigma) the two canonical coordinates on Σ\Sigma, we still write

γ˙:= τγ \dot \gamma := \partial_\tau \gamma

for the derivative “along the trajectory” (along the \mathbb{R}-factor), but now we also have the derivative σγ\partial_\sigma \gamma which we may think of as being tangential to the string at any instant of its trajectory. A field configuration γ:ΣX\gamma : \Sigma \to X may be thought of as the trajectory of a circle propagating in XX.

The critical trajectories γ:ΣX\gamma : \Sigma \to X are found to be those that satisfy the 2-dimensional wave equation

g(γ¨,)g( σ 2γ,)=H(γ˙, σγ,) g(\ddot \gamma,-) - g(\partial_\sigma^2 \gamma,-) = H(\dot \gamma, \partial_\sigma \gamma,-)

on the worldsheet. Comparison with the equation of motion of the relativistic particle shows that H( σγ,,)H(\partial_\sigma \gamma, -,-) plays the role of an electromagnetic field strength 2-form. Hence the string behaves as if electric charge is spread out evenly along it.

For point particle limit configurations γ\gamma, where the string has vanishing extension in that σγ=0\partial_\sigma \gamma = 0, the above equation reduces again to free motion

γ¨=a=0 \ddot \gamma = \vec a = 0

and for general (X,g)(X,g) to the corresponding geodesic motion.

Therefore close to these point particle configurations the string looks like a little oscillating loop whose dynamics is that of its “center of mass” point, but slightly modified by the energy in the oscillations and the way these interact with the background fields. After quantization of the σ\sigma-model, these oscillations have a discrete ( quantized!) set of possible frequencies, and indeed each of the oscillation modes makes the string appear in the point particle limit as one species or other of a relativistic particle.

Next we have a look at aspects of higher gauge theory that appears in n=2n = 2.

The above 2-form BB is in general just the local connection form of a circle n-bundle with connection|circle 2-bundle with connection \nabla on XX, given (as its homotopy fiber) by a morphism of smooth ∞-groupoids α:XB 2U(1)\alpha : X \to \mathbf{B}^2 U(1). Equivalently this is a U(1)U(1)-bundle gerbe with connection.

There is a canonical 1-dimensional 2-representation of the circle 2-group BU(1)\mathbf{B} U(1) on 2-vector spaces:

ρ:BBU(1)2Vect. \rho : \mathbf{B} \mathbf{B}U(1) \to 2 Vect \,.

Hence the corresponding associated infinity-bundle|associated 2-bundle is classified by a morphism

ρ(g):XgB 2U(1)ρ2Vect. \rho(g) : X \stackrel{g}{\to} \mathbf{B}^2 U(1) \stackrel{\rho}{\to} 2 Vect \,.

One can consider the string σ\sigma-model for worldsheets with boundary. A careful analysis then shows that the consistent Dirichlet-type boundary conditions that can be added correspond, roughly, to certain subspaces of target space – called D-branes – that are equipped with a section V:1ρ(g)| DbraneV : \mathbf{1} \to \rho(g)|_{D-brane} of the background gauge field 2-vector bundle restricted to the DD-brane. Such a section is precisely a twisted vector bundle on the brane, where the twist is the class in third integral cohomology of the background gauge field. More generally, these twisted bundles are cocycles in twisted differential K-theory . Hence more differential cohomology appears on the target space for the string in the presence of string boundaries.

More generally, the structure 2-group of the background principal 2-bundle need not be the plain circle 2-group BU(1)\mathbf{B}U(1), which is given by the crossed module [U(1)1][U(1) \to 1]. Instead, it can be the automorphism 2-group AUT(U(1))AUT(U(1)) which is given by the crossed module (U(1)Aut(U(1)) 2)(U(1) \to Aut(U(1)) \simeq \mathbb{Z}_2). An AUT(U(1))AUT(U(1))-principal 2-bundle on XX is equivalently a double cover of XX, equipped with a circle 2-bundle that has a 2\mathbb{Z}_2-twisted equivariance under the 2\mathbb{Z}_2-action. Such a background gauge field structure is called an string orientifold background. This is a kind of higher structure that the relativistic particle alone cannot see.

More such higher structure appears as one passes to the supergeometry analogs of the σ\sigma-models that we have considered so far: the superstring. The presence of the additional fermion fields that this brings with it (both on target space as well as on the worldsheet) influences all the structures that we have considered so far. For instance a phenonemon called a fermionic quantum anomaly forces the above circle 2-bundle to become a twisted 2-bundle , where the twist is given by a “fivebrane charge 3-bundle”.

These are the first examples of a general phenomenon: as nn increases, a background gauge nn-bundle with connection may constitute considerably more structure then one might naively expect from a generalization of the ordinary notion of a connection. More examples of this phenomenon arise when we allow our target spaces to be general smooth ∞-groupoids, below. A good deal of the richness of higher σ\sigma-models is due to this.

Posted by: Urs Schreiber on May 17, 2011 3:00 PM | Permalink | Reply to this

Higher gauge theories as sigma-models

I will now bring the two threads above together, as there were

  1. Exposition of classical sigma-models

  2. Exposition of a general abstract formulation

As before, I’ll proceed by posting below bite-sized expositional discussion, now for the following topics:

  1. Higher geometric target spaces

  2. Chern-Simons theory as a sigma-model

  3. AKSZ theory as a higher Chern-Simons σ\sigma-model

  4. Dijkgraaf-Witten theory as a σ\sigma-model

  5. The Yetter model as a σ\sigma-model

  6. \infty-Dijkgraaf-Witten theory

  7. \infty-Chern-Simons theory

Posted by: Urs Schreiber on May 17, 2011 3:14 PM | Permalink | Reply to this

G1 - Higher geometric target spaces

Higher geometric target spaces

The classical σ\sigma-models discussed above all have target spaces that are smooth manifolds. However, we saw that from dimension d2d \geq 2 on, the background gauge fields on these target spaces are naturally no longer just principal bundles with connection: instead they are smoth principal 2-bundles, then smooth principal 3-bundles, etc. and eventually generally principal ∞-bundles with ∞-connections. But the total space of such higher smooth bundles is no longer in general a smooth manifold anymore: instead, the total space is a Lie groupoid for d=2d = 2 , then a Lie 2-groupoid for d=3d = 3 and eventually generally a smooth ∞-groupoid.

This means that – unless we would artifically treat the total space of a background gauge field bundle on different grounds than its base space – the general theory of σ\sigma-models should naturally include target spaces that are not just smooth manifolds.

At least from d=2d = 2 on, for instance, target spaces should be allowed to be Lie groupoids. This has a fairly long tradition: the proper étale Lie groupoids are precisely orbifolds : spaces that are locally isomorphic to sufficiently nice quotients of a Cartesian space by a group action. Orbifolds have received a lot of attention in the study of string sigma-models. The orientifold background gauge fields mentioned above involve in general a 2\mathbb{Z}_2-orbifold target space, for instance.

But once we pass to the higher geometry of Lie groupoids at all, there is no good reason to restrict back to those that are orbifolds. For instance for any Lie group GG there is its delooping Lie groupoid, the action groupoid of the trivial action of GG on the point, which we shall write

BG:=*//G; \mathbf{B}G := *//G \,;

and this should perfectly serve as a target space object for σ\sigma-models, too. Here the boldface notation is to indicate that this Lie groupoid is a smooth refinement of the classifying space BGTopB G \in Top of the Lie group: in fact, where BGB G gives isomorphism classes of smooth GG-principal bundles, BG\mathbf{B}G also remembers the isomorphisms themselves – and hence in particular the automorphisms – of these bundles: it is the moduli stack of smooth GG-principal bundles: for Σ\Sigma a smooth manifold we have that the groupoid of morphisms of smooth groupoids ΣBG\Sigma \to \mathbf{B}G (the correct morphisms, sometimes called Morita morphism to distinguish them from any incorrect notion) is that of smooth GG-principal bundles and smooth homomorphisms between these

SmoothGrpd(Σ,BG)GBund(Σ)Grpd, SmoothGrpd(\Sigma, \mathbf{B}G) \simeq G Bund(\Sigma) \in Grpd \,,

whereas the geometric realization BG|BG|B G \simeq \vert \mathbf{B}G\vert only sees the equivalence classes: [Σ,BG]π 0GBund(Σ)Set. [\Sigma, B G] \simeq \pi_0 G Bund(\Sigma) \in Set \,.

This indicates that (nonabelian) gauge theory on Σ\Sigma should have a formulation as a σ\sigma-model with target “space” XX the Lie groupoid X=BGX = \mathbf{B}G: a σ\sigma-model field ΣX=BG\Sigma \to X = \mathbf{B}G is a GG-bundle, and an isomorphism of field configurations is a gauge transformation of GG-bundles.

But a field configuration in GG-gauge theory on Σ\Sigma is not just a GG-principal bundle, but is a GG-bundle with connection . There is no Lie groupoid that that would similarly represent such connections as a target space object. But there is a smooth groupoid that does: BG conn\mathbf{B}G_{conn}, the groupoid of Lie algebra valued 1-forms.

Here by a smooth groupoid we mean a groupoid that comes with a rule for which of its collections of objects or morphisms are smoothly parameterized families . Technically this is a (2,1)-sheaf or stack on the site CartSp of Cartesian spaces and smooth functions between them. Among all smooth groupoids, Lie groupoids – and generally diffeological groupoids – are singled out as being the concrete objects. While it is useful to know if a given smooth groupoid is concrete or even Lie, quite independent of that all of higher differential geometry exists for general smooth \infty-groupoids just as well. Therefore if we can allow Lie groupoids as targets for σ\sigma-models, we can allow general smooth groupoids as well.

The non-concrete smooth groupoid BG conn\mathbf{B}G_{conn} that we just mentioned is defined by the following rule: for UU \in CartSp, a smoothly UU-paramezterized family of objects is by definition a 𝔤\mathfrak{g}-valued differential 1-form AΩ 1(U,𝔤)A \in \Omega^1(U, \mathfrak{g}) on UU, where 𝔤\mathfrak{g} is the Lie algebra of GG. A smoothly UU-parameterized family of morphisms g:A 1A 2g : A_1 \to A_2 is a smooth gauge transformation gC (U,G):A 2=gAg 1+gdg 1g \in C^\infty(U, G) : A_2 = g A g^{-1} + g d g^{-1} between two such form data. (This is “non-concrete” because the smooth UU-parameterized families UBG connU \to \mathbf{B}G_{conn} are not UU-families of points *BG conn* \to \mathbf{B}G_{conn}).

One then finds that the mapping space groupoid for this target X=BG connX = \mathbf{B}G_{conn} is the groupoid

SmoothGrpd(Σ,BG conn)GBund conn(Σ), SmoothGrpd(\Sigma , \mathbf{B}G_{conn}) \simeq G Bund_{conn}(\Sigma) \,,

whose objects are smooth GG-principal bundles with conection on Σ\Sigma, and whose morphisms are smooth morophisms of principal bundles with connection. This groupoid is the configuration space of GG-gauge theory on Σ\Sigma, for instance of GG-Yang-Mills theory or of GG-Chern-Simons theory :

SmoothGrpd(Σ,BG conn)Conf YangMills(Σ)Conf ChernSimons(Σ). SmoothGrpd(\Sigma, \mathbf{B}G_{conn}) \simeq Conf_{Yang-Mills}(\Sigma) \simeq Conf_{Chern-Simons}(\Sigma) \,.

Notice that this configuration space is now itself a groupoid: morphisms are gauge transformations. In fact, it is naturally itself a smooth groupoid (when we read the hom-object here as an internal hom in SmoothGrpdSmoothGrpd.) In the traditional physics literature these Lie groupoidal configuration spaces of fields are best known in terms of their infinitesimal approximation Lie(Conf(Σ))LieAldLie(Conf(\Sigma)) \in LieAld, which are Lie algebroids , and these in turn are best known in terms of their function algebras, called the Chevalley-Eilenberg algebras CE(Lie(Conf(Σ)))CE(Lie(Conf(\Sigma))): this dg-algebra is in physics called the BRST complex of the gauge theory. Its degree-1 generators, the cotangents to the morphisms of Conf(Σ)Conf(\Sigma), are called the ghost fields of gauge theory.

Of coure we already saw secretly groupoidal configuration spaces in the above list of examples of σ\sigma-models of relativistic branes. We said that their configuration spaces C (Σ,X)//Diff(Σ)C^\infty(\Sigma,X)//Diff(\Sigma) were quotients; but really they are to be taken as higher categorical quotients, known as homotopy quotients or weak quotients : they are the action groupoids of Diff(Σ)Diff(\Sigma) acting on C (Σ,X)C^\infty(\Sigma,X).

We will see in the examples below that there is, of course, no reason to stop after passing from target manifolds to smooth target groupoids. At least as the σ\sigma-model increases in dimension, it is natural to consider smooth target 2-groupoids, target 3-groupoids, … target n-groupoids and eventually smooth ∞-groupoids. The full context of smooth \infty-groupoids is the natural completion of traditional differential geometry to higher geometry .

Given that it does thus make sense to regard general smooth ∞-groupoids as target spaces for σ\sigma-models, the questions is if there are useful background gauge fields on such. This is indeed the case:

for instance we have a theorem that says that for GG a compact Lie group, there is, for every integral cohomology class cH n+1(BG,)c \in H^{n+1}(B G, \mathbb{Z}) of the classifying space of GG – a characteristic class for GG-principal bundles – up to equivalence a unique smooth lift c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) to a smooth circle n-bundle on the smooth BG\mathbf{B}G. Moreover, we have a theorem that for sufficiently highly connected Lie groups or smooth \infty-groups GG, this refines canonically to a circle n-bundle with connection on the differentially refined smooth moduli space BG conn\mathbf{B}G_{conn}, given by a morphism:

c^:BG connB nU(1) conn. \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} \,.

This assignment generalizes the classical Chern-Weil homomorphism: we may speak of the ∞-Chern-Weil homomorphism . The first example below shows that ordinary Chern-Simons theory is a σ\sigma-model that arises this way. Generally we may this speak of σ\sigma-models with target space a smooth ∞-groupoid and background gauge fields given by the ∞-Chern-Weil homomorphism this way as ∞-Chern-Simons theories.

The second example below shows that ordinatry Dijkgraaf-Witten theory is a σ\sigma-model that arises this way when GG is a discrete group. Generally we may thus speak of σ\sigma-models with target space a discrete ∞-groupoid and background gauge fields given by the ∞-Chern-Weil homomorphism this way as ∞-Dijkgraaf-Witten theories :

Posted by: Urs Schreiber on May 20, 2011 8:03 PM | Permalink | Reply to this

G2 - Chern-Simons theory as a sigma-model

Chern-Simons theory as a sigma-model

One of the earliest topological quantum field theories ever considered in detail is Chern-Simons theory . We introduce this from the point of view of σ\sigma-models with higher geometric target spaces as discussed above.

An ordinary (as opposed to higher) gauge theory is a quantum field theory whose field configurations on a manifold Σ\Sigma are connections on GG-principal bundles over Σ\Sigma, for GG some Lie group. The word gauge transformation is essentially the physics equivalent of the word isomorphism , referring to isomorphisms in a configuration space of a field theory and specifically to isomorphisms in configuration spaces of gauge theories: between such bundles with connection. The action functional of a gauge theory is to be gauge invariant meaning that it assigns the same value to configurations that are related by a gauge transformaiton. This means precisely that the exponentiated action is a functor

exp(iS()):GBund conn(Σ)U(1) \exp(i S(-)) : G Bund_{conn}(\Sigma) \to U(1)

from the groupoid of gauge field configurations and gauge transformaitons, to the circle group (regarded as a 0-truncated groupoid).

The first nonabelian gauge theory to receive attention was Yang-Mills theory : in that model Σ\Sigma is a 4-dimensional pseudo-Riemannian manifold modelling spacetime. The exponentiated action functional is given by the integral of differential 4-forms naturally associated with a connection and a Riemannian structure:

exp(iS YM()):(P,)exp(i Σ1e 2F F +iθF F ). \exp(i S_{YM}(-)) : (P, \nabla) \mapsto \exp(i \int_\Sigma \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla \rangle + i \theta \langle F_\nabla \wedge F_\nabla \rangle) \,.

Here

  • PP is any GG-principal bundle and \nabla a connection on it;

  • F Ω 2(P,𝔤)F_\nabla \in \Omega^2(P, \mathfrak{g}) is the Lie algebra-valued curvature 2-form of this connection;

  • ,:𝔤𝔤\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} is an invariant polynomial on the Lie algebra: a bilinear form that is gauge invariant when evaluated on curvature 2-forms – for 𝔤\mathfrak{g} a semisimple Lie algebra this would be the Killing form and for a matrix Lie algebra this is simply the trace operation on products of matrices;

  • \star is the Hodge star operator given by the pseudo-Riemannian metric structure on Σ\Sigma.

  • ee \in \mathbb{R} is some constant, called the coupling constant of the model;

  • θ\theta is another parameter called the theta-angle .

The first summand in the exponent, that depending on the pseudo-Riemannian structure, is the crucial term for the direct application of this as a model of phenomenologically observed physics: it controls the dynamics of three of the four force fields in the standard model of particle physics.

Instead of investigating this further, we shall here look at the case where 1e 2\frac{1}{e^2} is set to 0. While not directly of phenomenological relevance, this is of quite some interest for the general theoretical understanding of the space of all possible field theories. Since the resulting action functional

exp(iS tYM):(P,)exp(i ΣF F ) \exp(i S_{tYM}) : (P, \nabla) \mapsto \exp(i \int_\Sigma \langle F_\nabla \wedge F_\nabla \rangle)

no longer depends on any extra (pseudo-Riemannian) structure on Σ\Sigma this may be interpreted as defining a topological quantum field theory : one speaks of topological Yang-Mills theory .

This is not quite a σ\sigma-model in the sense that we have been discussing: while the configuration space of topological Yang-Mills theory does consist of maps into the target space X=BG connX = \mathbf{B}G_{conn} (the smooth moduli stack of GG-principal bundles with connection, as discussed above), there is no way that the above action functional is induced directly from the transgression of the higher holonomy of a circle n-bundle with connection on this target space. This is because, at least for semisimple Lie groups GG, these are nontrivial only for odd nn, whereas here we have n=dimΣ=4n = dim \Sigma = 4. (There is a way, though, to think of topological Yang-Mills as coming from a background gauge field on the moduli 2-stack for certain 2-bundles with connection. This we come to later, after having discussed \infty-Chern-Simons theory in general.)

But something closely related is true: exp(iS tYM)\exp(i S_{tYM}) is the integrated curvature functional of a circle 33-bundle with connection on BG conn\mathbf{B}G_{conn}: we shall call this the universal Chern-Simons circle 3-bundle .

This means the following: in generalization of how an ordinary circle bundle with connection \nabla has a curvature 2-form, a circle n-bundle with connection \nabla on a manifold XX has a curvature (n+1)(n+1)-form F Ω cl n+1(X)F_\nabla \in \Omega^{n+1}_{cl}(X). These curvature forms are closed, but not necessarily exact. Nevertheless, a generalization of the Stokes theorem holds true for them: for Σ\Sigma of dimension n+1n+1 and denoting by Σ\partial \Sigma the boundary of Σ\Sigma and by γ:ΣX\gamma : \Sigma \to X a Σ\Sigma-shaped trajectory in XX, we have that the integral of the curvature over Σ\Sigma equals the higher holonomy of \nabla over Σ\partial \Sigma:

exp(i Σϕ *F )=hol(,γ| Σ). \exp(i \int_\Sigma \phi^* F_\nabla) = hol(\nabla, \gamma|_{\partial \Sigma}) \,.

This property in fact characterizes equivalence classes of circle nn-bundles with connection. When conceiving of circle nn-bundles with connection as rules for assigning higher holonomy that satisfy this property, one speaks of Cheeger-Simons differential characters .

Therefore, if we can find a circle 3-bundle with connection on the moduli stack BG conn\mathbf{B}G_{conn} of GG-principal bundles with connection whose curvature 4-form at (P,)(P,\nabla) is F F \langle F_\nabla \wedge F_\nabla \rangle, then we can interpret topological Yang-Mills theory on a 4-dimensional Σ\Sigma with boundary as being given by a σ\sigma-model on Σ\partial \Sigma with background gauge field that circle 3-bundle.

For GG a connected and simply connected Lie group, such a circle 3-bundle indeed exists. Its characteristic morphism

12p^ 1:BG connB 3U(1) conn \frac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}

from the smooth moduli stack of GG-bundles with connection to the smooth moduli 3-groupoid of circle 3-bundles with connection is constructed and discussed in (Fiorenza-Schreiber-Stasheff). This is the differential refinement of the smooth first fractional Pontryagin class

12p 1:BGB 3U(1) \frac{1}{2}\mathbf{p}_1 : \mathbf{B}G \to \mathbf{B}^3 U(1)

which in turn is a smooth refinement of the fractional Pontryagin class

12p 1:BGB 3U(1)K(,4) \frac{1}{2} p_1 : B G \to B^3 U(1) \simeq K(\mathbb{Z}, 4)

of the classifying space BGB G.

To get a feeling for what this circle 3-bundle is like, we look at what its pull-back 12p^ 1(ϕ):ΣϕBG conn12p^ 1B 3U(1) conn\frac{1}{2}\hat \mathbf{p}_1(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G_{conn} \stackrel{\frac{1}{2} \hat \mathbf{p}_1}{\to} \mathbf{B}^3 U(1)_{conn} to Σ\Sigma along any field configuration ϕ:ΣX=BG conn\phi : \Sigma \to X = \mathbf{B}G_{conn} is like.

Notice that for simply conneced GG the classifying space BGB G has vanishing homotopy groups in degree k3k \leq 3. Therefore every GG-principal bundle PP on the 3-dimensional Σ\partial \Sigma is necessarily trivializable. In this case the configuration space of the σ\sigma-model is equivalent to the groupoid of Lie algebra valued forms

SmoothGrpd(Σ,BG conn)Ω 1(Σ,𝔤)//C (Σ,G) SmoothGrpd(\partial \Sigma, \mathbf{B}G_{conn}) \simeq \Omega^1(\partial \Sigma, \mathfrak{g})//C^{\infty}(\partial \Sigma,G)

on Σ\partial \Sigma. For AΩ 1(Σ,𝔤)A \in \Omega^1(\Sigma, \mathfrak{g}) a field configuration and F A=dA+12[AA]F_A = d A + \frac{1}{2}[A \wedge A] the corresponding curvature 2-form, the curvature 4-form of 12p^ 1(ϕ)\frac{1}{2}\hat \mathbf{p}_1(\phi) is F AF A\langle F_A \wedge F_A \rangle. Its connection 3-form CC satisfying dC=F AF Ad C = \langle F_A \wedge F_A \rangle is – up to a closed 3-form – the Chern-Simons 3-form

C=cs(A)=AF A+16A[AA]. C = cs(A) = \langle A \wedge F_A \rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle \,.

Therefore the action functional of the 3-dimensional σ\sigma-model given by the background gauge field 12p^ 1:BG connB 3U(1) conn\frac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} sends a 𝔤\mathfrak{g}-valued form to the integral of its Chern-Simons 3-form over Σ\Sigma

exp(iS()) Σ 3:Aexp(i Σ 3cs(A)). \exp(i S(-))_{\Sigma_3} : A \mapsto \exp(i \int_{\Sigma_3} cs(A)) \,.

In this form the action functional for Chern-Simons theory is usually written. As we have seen, more abstractly we may identify this with the higher holonomy functional of the universal Chern-Simons circle 3-bundle with connection on BG conn\mathbf{B}G_{conn}.

Posted by: Urs Schreiber on May 26, 2011 3:52 PM | Permalink | Reply to this

Exposition of quantum sigma-models

Exposition of quantum sigma-models

Above I have discussed some standard classical sigma-models and higher gauge theories as sigma-models, also mostly classically. In the following I talk about the quantization of these models (or some of them) to genuine quantum field theories : quantum σ\sigma-models .

  1. Quantum particle on the line

  2. Topological string on a smooth manifold and Chas-Sullivan string topology

  3. Topological A-model string and Gromov-Witten theory

  4. Quantum abelian Chern-Simons theory

Posted by: Urs Schreiber on May 26, 2011 9:13 PM | Permalink | Reply to this

Second quantization of sigma-models

Second quantization of sigma-models

Low dimensional σ\sigma-models that describe the dynamics of particles – or generally branes – propagating in a target space XX subject to the forces exerted by a background field, such as discussed above, are just the first ingredient in a description of the quantum physics on target space XX : one is interested in describing a quantum field theory (or possibly some higher analog, such as a string field theory ) on XX, such that the original branes described by the σ\sigma-model are quanta of the fields on XX. The σ\sigma-model quantum field theory on the worldvolumes Σ\Sigma is supposed to induce, in turn, another quantum field theory, or something similar, but now on XX. This process – or its idea – goes by the name second quantization . We shall try to indicate below in which sense this should indeed be a direct iteration of the (“first”) quantization of the original σ\sigma-model itself, as described above. But we will (have to) be more vague and schematic than before.

Assume now that we started with a classical σ\sigma-model and have quantized it to obtain the functorial QFT

Z:Bord n SVect, Z : Bord_n^S \to Vect \,,

which we take here to be 1-categorical (un-extended), not to overburden the discussion. . Assume also for simplicity that the corresponding target space is of the form X=Y×X = Y \times \mathbb{R}, where the second factor is the time-axis.

Using this, define a space of states of a the second quantization to be the Fock space

Sym ( [Σ n1]Z(Σ n1)) :=Sym (𝒱) 1𝒱(𝒱𝒱) sym(𝒱𝒱𝒱) sym, \begin{aligned} Sym^\bullet(\oplus_{[\Sigma_{n-1}]} Z(\Sigma_{n-1}) ) & := Sym^\bullet( \mathcal{V} ) \\ & \simeq \mathbf{1} \oplus \mathcal{V} \oplus (\mathcal{V} \otimes \mathcal{V})_{sym} \oplus (\mathcal{V} \otimes \mathcal{V} \otimes \mathcal{V})_{sym} \cdots \end{aligned} \,,

the free commutative tensor algebra over the vector space 𝒱\mathcal{V} of states of the given single particle, string or, generally, brane. This has a contribution for each equivalence class of connected shapes Σ n1\Sigma_{n-1} of this brane. For instance for n=1n = 1 there is just the point Σ 0*\Sigma_0 *, but for the string with n=2n = 2 there is one contribution from the closed string Σ 1=S 1\Sigma_1 = S^1 and one contribution from the open string Σ 1=[0,1]\Sigma_1 = [0,1]; and in turn several contributions of this type if the open string carries boundary D-brane labels. (In the presence of fermions the vector spaces appearing here are super vector spaces and the Fock space construction is the corresponding super algebra version.)

This space of states is that of a quantum theory whose classical field configurations are given by many states of the original quantum σ\sigma-model. We therefore say that the original brane of shape Σ n1\Sigma_{n-1} is a quantum or a single excitation of the field theory that it induces on target space XX: where the original σ\sigma-model describes the dynamics of a single brane on XX, its second quantization describes the dynamics of many and interacting copies of that brane.

Of course Sym ():VectVectSym^\bullet(-) : Vect \to Vect is a functor. This is traditionally stated as the notorious slogan “second quantization is a functor”, usually meant to state a contrast with first quantization. Notice however, firstly, that the (“first”) quantization of σ\sigma-models – at least for the version discussed above – also is a functor (what is not functorial is the quantization of a random Poisson manifold, but σ\sigma-models carry much more structure than that), and, secondly, that the Sym Sym^\bullet-construction is at best half of what second quantization is about: the other half is the description of the interaction of the many branes:

for X=Y×(t,t)X = Y \times (-t,t) a piece of target space, the would-be second quantized theory on XX should assign a morphism

S:Sym (𝒱)Sym (𝒱) S : Sym^\bullet(\mathcal{V}) \to Sym^\bullet(\mathcal{V})

between the space of states of the incoming space YY, to the outgoing space YY, here taken to be the same. Usually one assumes here t=t = \infty and calls SS the S-matrix of the theory, where “S” is for scattering : for (v 1 inv r in)Sym r𝒱(v^{in}_1 \otimes \dots v^{in}_r) \in Sym^r \mathcal{V} the state of rr incoming brane quanta and (v 1 outv s out)Sym s𝒱(v^{out}_1 \otimes \dots v^{out}_s) \in Sym^s \mathcal{V} a state of ss outgoing brane quanta, the matrix element

((v 1 outv s out),S(v 1 inv r in)) \left( \left(v^{out}_1 \otimes \dots v^{out}_s\right) , S \left(v^{in}_1 \otimes \dots v^{in}_r\right) \right)

(using an inner product space-structure on 𝒱\mathcal{V}) is the probability amplitude for rr-many branes in the given states propagating through XX, interacting among each other, thereby transmuting into other states – hence scattering off each other – and eventually emerging again in the state (v 1 outv s out)(v^{out}_1 \otimes \dots v^{out}_s).

The idea is that this scattering happens in all possible ways that it can happen, with each way weighted by the probability amplitude for it to happen as seen by the quantum σ\sigma-model ZZ: the S-matrix is taken to be given by an expression like

S:= Σ nBord n SZ(Σ)dμ(Σ), S := \int_{\Sigma_n \in Bord^S_n} Z(\Sigma) \,\, d\mu(\Sigma) \,,

and called the brane perturbation series . For instance the Feynman perturbation series for n=1n = 1, or the string perturbation series for n=2n = 2: we sum over all possible SS-structured cobordisms Σ\Sigma – each representing an interaction “channel” for branes of shape inΣ\partial_{in} \Sigma to scatter into shape outΣ\partial_{out} \Sigma – the linear maps S(Σ):𝒱 inΣ𝒱 outΣS(\Sigma) : \mathcal{V}_{\partial_{in} \Sigma} \to \mathcal{V}_{\partial_{out} \Sigma} weighted by some measure dμ(Σ)d\mu(\Sigma), and all regarded as giving one single endomorphism on Sym (𝒱)Sym^\bullet(\mathcal{V}) in the evident way.

Except for simple cases, plenty of technical subtleties may have to be taken care of here in order to get anything close to being well defined. For instance there is an integral over some suitably compactified moduli space of SS-structured cobordisms involved. For sufficiently simple but still nontrivial σ\sigma-models this has been made fully precise (for instance for n=1n = 1 in standard QFT perturbation theory with renormalization or for n=2n = 2 in the example of Gromov-Witten theory), and intuition and motivation is drawn from these cases, but in full generality it remains an open problem to fully realize this idea.

This S-matrix-construction provides at least the rudiments of a quantum field theory on target space XX obtained by second quantization of a σ\sigma-model describing brane dynamics on XX.

In application to phenomenological physics we think of XX here as our spacetime. Everything that is measured in particle accelerator experiments is explained with such a construction for n=1n = 1. Everything that perturbative string theory hypothesizes is a refinement of this theory relevant at energies not visible in current accelerator experiments is described with such a construction for n=2n = 2. A little bit of investigation has gone into exploring the n=3n = 3-case. In principle one could investigate this further for n4n \geq 4. but so far the jump in complexity given by the step from n=1n = 1 to n=2n = 2 has kept mankind busy enough. There is already an intricate interrelation of quantum field theories showing up at this level. For instance the second quantization of the 2-dimensional A-model string σ\sigma-model (see there) has be shown to be Chern-Simons theory, which we have seen, above, may itself be understood as a 3-dimensional σ\sigma-model. The second quantization of the non-topological string σ\sigma-model is more complicated but follows this pattern. It contains even Yang-Mills theory and gravity, and this is what has driven much of the interest in this structure.

Given the evident importance of the “brane perturbation series” or “second quantization” of σ\sigma-models it would be desireable to have a more general abstract and systematic description of it, in the spirit of the above discussion of general abstract first quantization. Here is an observation that might be suggestive:

we had amplified that the input data for a classical (“0-quantized”) σ\sigma-model is a background field on a target space XX. At least in nice cases this background field is entirely encoded in its higher parallel transport and holonomy-assignment, which is a map

tra :Bord n(X)nVect tra_\nabla : Bord_n(X) \to n Vect

from nn-dimensional bordisms in XX to n-vector spaces (we had mostly discussed an equivalent characteristic morphism :XB nU(1) conn\nabla : X \to \mathbf{B}^n U(1)_{conn} and should eventually discuss in more detail how both perspectives are related…).

The process of (“first”) quantization of this σ\sigma-model involves some kind of extension of this functor through the projection to abstract cobordisms

Bord n S(X) tra Vect Z Bord n S \array{ Bord_n^S(X) &\stackrel{tra_\nabla}{\to}& Vect \\ \downarrow & \nearrow_{Z_\nabla} \\ Bord_n^S }

given by the path integral

Z := γBord n Stra (γ)dμ(γ) Z_\nabla := \int_{\gamma \in Bord^S_n} tra_\nabla(\gamma) \;\; d \mu(\gamma)

over all trajecgtories γ:ΣX\gamma : \Sigma \to X of given shape Σ\Sigma. We saw that second quantization reads in this Z Z_\nabla, in turn, and from the perturbation series

S:= ΣBord n SZ(Σ)dμ(Σ) S := \int_{\Sigma \in Bord^S_n} Z(\Sigma) \; d \mu(\Sigma)

produces a target space quantum field theory. Hence apparently there is a pattern of iterated path integrals, the first over morphisms in Bord n S(X)Bord_n^S(X), the second over morphisms in Bord n SBord_n^S:

  • classical (0-quantized) σ\sigma-model (higher parallel transport of background gauge field):

    tra :Bord n S(X)nVect; tra_\nabla : Bord_n^S(X) \to n Vect \,;

  • quantum (1st quantized) σ\sigma-model (FQFT):

    Z = γBord n S(X)tra (γ)dμ(γ):Bord n SnVect; Z_\nabla = \int_{\gamma \in Bord^S_n(X)} tra_\nabla(\gamma) \, d\mu(\gamma) : Bord_n^S \to n Vect \,;

  • second quantized model (S-matrix):

    S= ΣBord n SZ (Σ)dμ(Σ). S = \int_{\Sigma \in Bord^S_n} Z_\nabla(\Sigma)\, d\mu(\Sigma) \,.

Remember that these formulas are to be taken with a grain of salt. Quite some additional effort is in general needed to make them well-defined. Already in the well-understood case n=1n = 1 the path integral in the expression for Z Z_\nabla needs attention, then the single terms in the expression for SS may still need renormalization and after all that the sum still may need resummation , at least for models richer than for instance the \infty-Dijkgraaf-Witten theories, for which all integrals reduce to finite sums.

Posted by: Urs Schreiber on May 28, 2011 10:35 AM | Permalink | Reply to this

Sharpe on target stacks for sigma-models

I should add that maybe among the earliest people who considered string sigma-models on general smooth stacks (instead of just (global) orbifolds) with application in string theory and supergravity is Eric Sharpe. I have added the following references to the sigma model entry. Need to add many more, eventually.

  • Tony Pantev, Eric Sharpe, Gauged linear sigma-models for gerbes (and other toric stacks) (arXiv)

  • S. Hellerman, Eric Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters (arXiv)

Posted by: Urs Schreiber on June 8, 2011 8:16 PM | Permalink | Reply to this
Read the post AKSZ Sigma-Models
Weblog: The n-Category Café
Excerpt: The original definition of AKSZ sigma-model quantum field theories.
Tracked: August 5, 2011 1:18 AM

Post a New Comment