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September 12, 2006

Freed on Higher Structures in QFT, I

Posted by Urs Schreiber

I would like to talk about things related to 3-dimensional topological field theories, along the lines of what John mentioned a while ago.

Before doing so, however, I want to better understand a general phenomenon, which has originally been identified by Dan Freed and is more recently being pursued by Simon Willerton and Bruce Bartlett. It also appears in recent work by Sergei Gukov.

The observation is, roughly, that what physicists call an action functional for a nn-dimensional quantum field theory is really just one component of something that looks a little like an nn-functor, which assigns to dd-dimensional volumes (nd)(n-d)-Hilbert spaces.

I’ll briefly summarize what Freed and others have to say about this. What I would like to discuss then are some details of this concept. For instance, how to turn the above “looks a little like” into an “is”.

Parallel transport in a vector bundle is a functor

(1)P 1(X)Vect P_1(X) \to \mathrm{Vect}

which sends paths in base space XX to morphisms of vector spaces.

Propagation in quantum mechanics is a functor

(2)1CobHilb 1\mathrm{Cob} \to \mathrm{Hilb}

which sends 1-dimensional Riemannian cobordisms to morphisms of Hilbert spaces #.

Notice that these two concepts are closely related. The standard example is the single non-relativistic charged particle propagating on XX in the presence of an electromagnetic field.

The electromagnetic field itself is the functor

(3)tra EM:P 1(X)Trans(E)Vect, \mathrm{tra}_{\mathrm{EM}} : P_1(X) \to \mathrm{Trans}(E) \subset \mathrm{Vect} \,,

namely a line bundle EE with connection.

The propagation functor of our particle

(4)QM EM:1CobHilb QM_\mathrm{EM} : 1\mathrm{Cob} \to \mathrm{Hilb}

is obtained by performing the path integral over tra EM\mathrm{tra}_\mathrm{EM}:

(5)QM EM(t)=(K:(x,y)D(xγy)E xtra EM(γ)E y), QM_\mathrm{EM}(\bullet \stackrel{t}{\to} \bullet) = (K : (x,y) \mapsto \int D(x \stackrel{\gamma}{\to}y) \;\;E_x \stackrel{\mathrm{tra}_\mathrm{EM}(\gamma)}{\to} E_y ) \,,

where the right hand side denotes the operator defined by the integral kernel K(x,y)K(x,y) which is obtained by the path integral over all paths from xx to yy, using the Wiener measure DγD\gamma.

This operator acts on the Hilbert space

(6)QM EM()=Γ(E) \mathrm{QM}_\mathrm{EM}(\bullet) = \Gamma(E)

(or simply L 2(X)L^2(X) if E is trivial), the space of square integrable sections of the line bundle EE.

Notice how we can think of the space of sections of the bundle EXE \to X as something like

(7)"Γ(E)= xXE x". \text{"}\Gamma(E) = \oplus_{x\in X} E_x\text{"} \,.

So the Hilbert space assigned to the point in the quantum theory is indeed the “sum” of the vector spaces assigned to the many points of target space by the action! (See below for the relevance of this statement).

Notice how the parallel transport functor plays the role of the action functional (except for the kinetic contribution, which should really be thought of as being part of the measure of the path integral), whose integration yields the propagation functor.

Probably the right way to formalize this passage from parallel transport functors to propagation functors is to use a theory of spans as indicated in

E. Lupercio & B. Uribe
Topological Quantum Field Theory, String and Orbifolds
hep-th/0605255.

This is one of the things I want to better understand here. Especially the categorified version of the above (which plays a central role in Freed’s observation).

Before moving on, we should take the opportunity and identify some terminology which will be needed later on.

Notice how in our little theory of the electrically charged non-relativistic particle we were really describing a 1-dimensional quantum field theory.

The set of fields on a given cobordism t\bullet \stackrel{t}{\to} \bullet is the set of (suitably well behaved) maps from the interval [0,t][0,t] to XX, namely the set of parameterized paths in XX with parameter range [0,t][0,t]. Following Freed, we denote this space by

(8)C (t)=[[0,t],X]. \mathbf{C}_{\left(\bullet \stackrel{t}{\to}\bullet\right)} = \left[ \left[ 0,t \right], \; X \right] \,.

Accordingly, the space of fields over the point \bullet may be identified with XX itself

(9)C =X. C_\bullet = X \,.

(Notice that the electromagnetic field encoded in tra EM\mathrm{tra}_\mathrm{EM} is a background structure in the present case, and not part of the field content of our quantum theory. It would become so only after “second quantization” - whatever that really means.)

All of the above is supposed to make you want to say the following:

Parallel transport in an nn-vector bundle is an nn-functor

(10)P n(X)nVect. P_n(X) \to n\mathrm{Vect} \,.

Propagation in nn-dimensional quantum field theory is an nn-functor

(11)nCobnHilb. n\mathrm{Cob} \to n\mathrm{Hilb} \,.

Here nVectn\mathrm{Vect} and nHilbn\mathrm{Hilb} are to be thought of as (n+1)(n+1)-categories of some notion of nn-vector spaces and nn-Hilbert spaces, respectively. P n(X)P_n(X) is some notion of nn-paths in XX, and nCobn\mathrm{Cob} is a placeholder for nn-cobordisms with possibly extra structure on them, regarded as an nn-category (instead of as a 1-category).

For instance, you can think of a line bundle gerbe with connection as a “rank-1” 2-vector bundle with parallel transport, where to each point xXx \in X we associate a Vect\mathrm{Vect}-vector space A xMod{}_{A_x}\mathrm{Mod}, to each path xγyx \stackrel{\gamma}{\to} y a Vect\mathrm{Vect}-linear map A xModN γ A yMod{}_{A_x}\mathrm{Mod} \stackrel{N_\gamma}{\to} {}_{A_y}\mathrm{Mod}, and so on.

Notice that, for closed paths, this nn-functor associates

  • 2-vector spaces to 0-paths
  • 1-vector spaces to 1-paths
  • 0-vector spaces to 2-paths

Here we would like to understand in which sense an analogous reasoning lets us cook up from the data of an ordinary nn-dimensional QFT something like a propagation nn-functor

(12)nCobnHilb. n\mathrm{Cob} \to n\mathrm{Hilb} \,.

Crucial observations in this direction have been made long ago in

Daniel S. Freed
Quantum Groups from Path Integrals
q-alg/9501025

and

Daniel S. Freed
Higher Algebraic Structures and Quantization
hep-th/9212115 .

A nice overview of the key concepts is provided in section 5.4 of

Bruce Bartlett
Categorical Aspects of Topological Quantum Field Theories
math.QA/0512103.

Dan Freed takes the 2-dimensional Wess-Zumino(-Witten) model and in particular 3-dimensional Chern-Simons theory apart, to show that from the action functionals defining them, usually thought of as producing numbers from field configurations, one actually obtains a structure of the following sort:

An action denoted exp(iS())\exp(iS(\cdot)) which is a list of morphisms

(13)exp(iS()) : C X d (nd)Hilb ϕ exp(iS(ϕ)) \array{ \exp(iS(\cdot)) &:& \mathbf{C}_{X^d} &\to& (n-d)\mathrm{Hilb} \\ && \phi &\mapsto& \exp(iS(\phi)) }

which maps a field ϕ\phi defined over a closed dd-dimensional space to an (nd)(n-d)-Hilbert space exp(iS(ϕ))\exp(iS(\phi)).

In particular, if we agree to address elements of the ground field KK itself (usually K=K=\mathbb{C} here) as a 0-Hilbert space, then the action takes closed nn-dimensional spaces to numbers. This is the familiar notion of action, as used by ordinary physicists.

But now, we want the action also to assign data to lower-dimensional spaces. Very roughly the general idea for that is always the following:

Suppose we have an “action” defined on “field configurations” of nn-dimensional spaces. If our nn-dimensional space XX has an (n1)(n-1)-dimensional boundary X\partial X, then we may restrict our action to those fields on XX which have a specified restriction on X\partial X.

But this way the action becomes an element in the (nn-Hilbert) space of maps on fields on (n1)(n-1)-dimensional spaces. This space we identitfy as the value of our original action to (n1)(n-1)-dimensional spaces.

This is roughly the idea. Dan Freed spells it out in great detail for 2-dimensional WZW theory as well as for Dijkgraaf-Witten theory. I have, however, not seen a truly general statement of this idea. There should be some.

From the action, we want to produce a quantum field theory propagator. (Recall the toy example of the charged particle at the beginning.) This is now obtained by some sort of “path integral” involving the above generalized notion of action.

Accordingly, the quantum field theoretic structure we can distill here involves assignments of

  • integrals of numbers to nn-dimensional spaces (known, then, as the partition function)
  • “integrals of Hilbert spaces” to (n1)(n-1)-dimensional spaces
  • in general, “integrals of rr-Hilbert spaces” to (nr)(n-r)-dimensional spaces

Personally, I feel that a really good general understanding of the mechanism at work here has not yet been written down. Among other things, we expect all of the above structure to assemble into some nn-functor. Maybe we can discuss this in the comment section.

What has however nicely been worked out are a couple of concrete examples. As I said, most notably the Dijkgraaf-Witten model. This I shall talk about in a followup entry.

Posted at September 12, 2006 12:49 PM UTC

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24 Comments & 13 Trackbacks

Re: Freed on Higher Structures in QFT, I

Vague echoes of what John and I start discussing about a quarter of the way down here, concerning homotopy theory as the integral of an action in truth values and it’s categorifications. In a homotopy nn-type, as the dimension of the source and target of the homotopy goes up, the dimension of the solution comes down.

Posted by: David Corfield on September 12, 2006 4:57 PM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

Vague echoes of what John and I start discussing about a quarter of the way down here, […]

Indeed. After I had seen you talk about this, I was thinking a lot about related things. I wrote this entry as a reaction to this thinking.

At that time I also tried to understand path integrals in quantum mechanics from that point of view. But I failed.

Now I should mabye try again.

I just made a little correction to my discussion of the quantization of the electrically charged particle in the above entry. Because I finally realized that we can vaguely think of the space of (square integrable) sections of a bundle EXE \to X as a way to say

(1) xXE x, \oplus_{x\in X} E_x \,,

i.e. as a way to “sum over all” the fibers of the bundle.

This leads to a pretty cool picture here.

There should be a way to start with any nn-vector bundle with connection over xx, encoded in a transport nn-functor

(2)tra:P n(X)nVect. \mathrm{tra} : P_n(X) \to n\mathrm{Vect} \,.

This we may address as the (topological part of the) action of a charged nn-particle.

Now, as we quantize the nn-particle, we are doing something like “integrating” the entire nn-functor.

The single abstract point then gets assigned the “sum” (in the above sense) of all the nn-vector spaces that are the fibers on the nn-vector bundle over XX.

The 1-dimensional cobordisms cc between two points gets assigned the “sum” of all transport morphisms assigned by tra\mathrm{tra} to all paths in XX poarameterized by cc.

“And so on.”

Now I see that this is essentially what Freed is doing in his specific examples. But now I also think there must be a general abstract prescription that describes what it means to “path integrate”/”quantize” an nn-functor

(3)tra:P n(X)nVect. \mathrm{tra} : P_n(X) \to n\mathrm{Vect} \,.

In particular, I expect that where Freed discusses the Chern-Simons (or Dijkgraaf-Witten) model, we should identify the transport 3-functor that describes the 2-vector transport in the line bundle 2-gerbe with connection over BGBG.

I’ll think about this.

Posted by: urs on September 12, 2006 6:14 PM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

Where does the Lagrangian feature in your nn-transport set-up? What John and I were discussing is that your P n(X)P_{n}(X) is already the result of an ‘integral’ over some Lagrangian of truth values. Presumably Lagrangians can be related by maps between their targets, maps between rigs or nn-rigs.

Posted by: David Corfield on September 13, 2006 8:08 AM | Permalink | Reply to this

path integrals from Lagrangians

Where does the Lagrangian feature in your nn-transport set-up?

Thanks for the question. I tried to say this, but I should emphasize it more.

The cool point is: the transport nn-functor is the Lagrangian.

If you look at it from whis point of view:

the action for the ordinary electrically charged particles is, locally,

(1)exp(i|γ| 2+γ *A). \sim \exp(i \int |\gamma'|^2 + \int \gamma^* A) \,.

But the first contribution is just our Wiener measure (Wick rotated) on paths. Hence what remains of the Lagrangian is just the electromagnetic coupling.

That’s why the rather nifty formula at the end of my recent comment works. This formula really is the standard path integral for the charged particle (at least if you fill in the technical gaps that I glossed over).

So we really want a general concept of what it means to nn-path integrate an nn-transport functor, I think.

Now combine this with what we said about the nature of transport in terms of pseudofunctors (!) and with how we may understand ordinary integration in terms of limits over nerves under pseudofunctors (!).

I am beginning to suspect that we may find a general-abstract-nonsense formulation of the path integral over a the action defined by a (1-, for starters)transport functor by taking a certain colimit over the functor applied to the nerve of the category of paths (the pair groupoid, for instance).

For instance, using your and John’s idea of “changing the rig”, one can see that this way it is possible to express the classical limit of the path integral as a colimit over the nerve.

Posted by: urs on September 13, 2006 11:35 AM | Permalink | Reply to this

Re: path integrals from Lagrangians

So can you make sense of what was bugging John here (I don’t know if it still is):

Part of what’s been bugging me a lot about higher gauge theory is that I don’t understand how Lagrangians fit into it. Usually people write down a Lagrangian as a function of some fields, which lets you compute an action, and minimizing the action give you equations of motion. You can do this in higher gauge theory too. BUT, usually the action for an ordinary gauge theory is required to be INVARIANT UNDER THE GROUP OF GAUGE TRANSFORMATIONS, since then gauge transformations will map solutions of the equations of motionto solutions. In higher gauge theory we have a 2-GROUP of gauge transformations. What does it mean for an action to be “invariant” under a 2-group? 2-groups really want to act not on a mere set, but on a CATEGORY - and the proper notion of “invariance” is “weak invariance”, i.e. invariance up to a specified isomorphism satisfying some (understood) coherence laws.This suggests that actions in higher gauge theory should really take values in a category. And so, presumably, should Lagrangians. But, what category or categories??? Some categorification of the real numbers, maybe.

The problem is, I don’t see the physics pointing me towards any particular choice. Probably I’m just being dumb. It’s especially galling because I already think I know what one*result* of path-integral quantizing a higher gauge theory mightbe: a 2-Hilbert space of states! I wrote a paper on 2-Hilbert spaces once….

Hmm, this suggests that the appropriate “categorified transition amplitudes” lie not in C but in Hilb!!!

Posted by: David Corfield on September 13, 2006 12:47 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

[…] what was bugging John here […]

This is another sort of Lagrangian. Sorry, maybe I wasn’t clear about this.

I was talking about the Lagrangian for the dynamics of an nn-particle coupled to an nn-gauge field.

What John is talking about is the Lagrangian for the dynamics of the nn-gauge field itself.

This is, indeed, still a little mysterious.

Posted by: urs on September 13, 2006 2:03 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

What John is talking about is the Lagrangian for the dynamics of the n-gauge field itself.

This is, indeed, still a little mysterious.

Why?

The 1-gauge action is the trace of the holonomy around an infinitesimal loop, summed over all loops. That is precisely how the action is defined in lattice gauge theory. You have a notion of surface holonomy, right? So why not simply take the trace of the surface holonomy around an infinitesimal sphere, and sum over all such spheres?

Posted by: Thomas Larsson on September 21, 2006 4:26 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

So why not simply take the trace of the surface holonomy around an infinitesimal sphere, and sum over all such spheres?

That’s right, exactly, I agree. That’s what one should do.

After one has computed the surface holonomy, it yields an element of some 2-group (or 3-group, even #, in the non-fake flat case) GG, essentially given by the exponential of the 3-curvature H=d ABH = d_A B, as one expects.

You next want to hit that with a representation

(1)ρ:Σ(G)2Vect.\rho : \Sigma(G) \to 2\mathrm{Vect} \,.

I think the representations we want for the main physics applications are also available #.

For some weird reason nobody has yet taken the time and apply a notion of trace of a 2-vector space to that in order to get a gauge-invariant expression.

This shouldn’t be hard either, the right idea of 2-trace is probably that promoted by Kapranov and Ganter #.

So, it should just be a matter of putting one and one together. Maybe if I find the time…

Posted by: urs on September 21, 2006 4:44 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

This is of course exactly what I did on the lattice, and others, like NEW frequenter Peter Orland, did before me. I know that you don’t like my notion of surface holonomy, for reasons which never became clear to me, but at least it leads to a straightforward lattice action.

Posted by: Thomas Larsson on September 22, 2006 7:24 AM | Permalink | Reply to this

Re: path integrals from Lagrangians

Thomas Larsson wrote

[…] my notion of surface holonomy […]

You are considering 2-transport# of the form

(1)tra:P 2Σ(Vect). \mathrm{tra}: P_2 \to \Sigma(\mathrm{Vect}) \,.

Here Σ(Vect)\Sigma(\mathrm{Vect}) is the suspension of Vect\mathrm{Vect}.

So your 2-transport associates

- nothing to points

- vector spaces to paths, with composition of paths corresponding to the tensor product of vector spaces

- linear maps between vector spaces to surfaces cobounding paths.

In the special case where all vector spaces here are taken to be 1-dimensional we may take the domain P 2P_2 to be the 2-groupoid of thin-homotopy classes of 2-paths in some topological space XX. In this case, as shown here, such a 2-transport is the same (if smooth and locally trivializable) as an abelian gerbe with connection.

If, however, one allows the vector spaces involved to have dimension larger than 1, there is no obvious way to have the domain P 2P_2 consist of “continuous” paths in some sense (though it might be possible if we admit tra\mathrm{tra} to respect composition only weakly…).

For that reason, you decide to concentrate on a fixed lattice of the form n\mathbb{Z}^n and let P 2P_2 be the category generated from the 2-graph of elementary edges and plaquettes in this lattice.

One can do that, and it does yield a 2-transport

(2)P 2Σ(Vect). P_2 \to \Sigma(\mathrm{Vect}) \,.

In this case there is indeed a notion of 2-trace which makes the surface transport over the surface of a little cube invariant under pseudonatural isomorphisms of tra\mathrm{tra}: it suffices to take the ordinary trace on 2-morphisms.

Posted by: urs on September 23, 2006 1:59 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

You can take the opposite perspective, and demand that your continuum 2-gauge theory reduce to some nice generalization of lattice gauge theory upon lattization. There are not so many natural lattice actions, and mine is undoubtedly the simplest one. If your lattice version becomes ugly and contrived (IIRC, so was Pfeiffer’s), then maybe there is room for improvements in you continuum formulation as well.

Posted by: Thomas Larsson on September 25, 2006 12:20 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

If your lattice version becomes ugly and contrived (IIRC, so was Pfeiffer’s)

Sorry, I don’t think that I’d agree with this.

All 2-conections I have seen have a nice lattice formulation. In fact, it’s usually in the lattice formulation that they look most natural #, since on the lattice the arrow-theoretic setup becomes very hands-on. That’s one reason why it is desireable to understand the continuum theory, too, in lattice-like language (as in Anders Kock’s formalism on whose possible categorification I commented here).

Since you are trying to find weak spots: how about the demand that the lattice action must scale with a change of lattice spacing? If you have a continuum limit, that’s automatic. But for general 2-functors with values in Σ(Vect)\Sigma(\mathrm{Vect}) it would seem that it is hard to change the chosen lattice in a compatible way.

My experience suggests that nn-transport with values in nn-monoids instead of nn-groups is more suited to describe propagation in nn-dimensional quantum field theory than “parallel” transport in nn-bundles.

An easily accesible hint in this direction is that local trivialization of 2-transport with values in 2-monoids (like Σ(Vect)\Sigma(\mathrm{Vect}), for instance) leads to structures known in 2-dimensional topological field theory #.

Posted by: urs on September 25, 2006 12:48 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

Since you are trying to find weak spots: how about the demand that the lattice action must scale with a change of lattice spacing? If you have a continuum limit, that’s automatic. But for general 2-functors with values in S(Vect) it would seem that it is hard to change the chosen lattice in a compatible way.

Thats what the vector s^\mu, which describes the string direction, does for you. dA ~ a^3, [A, A] ~ a^4, s ~ 1/a, and thus F = dA + [A, s.A] ~ a^3. Without it, it won’t work. Note that the introduction of a privileged direction between the two marked points on a Wilson surface does not violate reparametrization invariance.

Posted by: Thomas Larsson on September 25, 2006 4:17 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

Thats what the vector s μs^\mu, which describes the string direction, does for you.

A vector space V eV_e is assigned to a given edge xeyx \stackrel{e}{\to} y, right?

Now say we want to refine the lattice, such that ee now looks like

(1)(xey)=(xe 1r 1e 2r 2e 3y). (x \stackrel{e}{\to} y) = (x \stackrel{e'_1}{\to} r_1 \stackrel{e'_2}{\to} r_2 \stackrel{e'_3}{\to} y) \,.

Which vector space should be assigned to e 1e'_1?

Posted by: urs on September 25, 2006 4:30 PM | Permalink | Reply to this

Re: path integrals from Lagrangians

If one link takes values in V, three links takes values in the threefold tensor product V^3 - this is a different vector space, as you know. This has no direct bearing on the action, which always takes values in C = V^0.

Surprisingly, and what is not in my paper, the N-fold tensor product does have a nice continuum limit, in the following sense. If the finite-dimensional group G acts on V, then G^N acts on V^N. The continuum limit of G^N is the loop group LG, which acts on modules LV. So the natural continuum limit of V^N is LV.

The thing that disturbs me about your setup, except that I’m usually lost in your highbrow math, is that you seem to dismiss what I regard as my main physical insight: that a Wilson surface depends not only on the surface itself, but also on two marked points on it. The vector between these points is reparametrization invariant. It is this idea that I have implemented on the lattice and, less successfully, in the continuum.

Long ago, Teitelboim proved a no-go theorem which states that in the continuum limit, p-gauge theory is equivalent to the abelian theory. By introducing the preferred vector between the marked points, I circumvent the axioms of his theorem. It seems to me that you dismiss this vector as unphysical, perhaps because the Nambu-Goto action only depends on the area and not on marked points, and hence you throw away the structure which allows you to avoid Teitelboim’s curse. From this perspective, it is not surprising that you have problems finding a pure gauge action.

Posted by: Thomas Larsson on September 27, 2006 9:04 AM | Permalink | Reply to this

Re: path integrals from Lagrangians

If one link takes values in VV, three links takes values in the threefold tensor product V 3V^3

Exactly, and for dim(V)>1\mathrm{dim}(V) \gt 1 the space VV is never isomorphic to V 3V^{\otimes 3}.

So if you insist on assigning the same vector space VV to every edge, then the vector space assigned to a given path depends on how finely this path is approximated by edges.

Looking at parameterized paths in VV instead (which, by the way, should be the continuum limit when we take direct sums instead of tensor products) might be a way to set up a formalism that describes what you keep describing in heuristic terms. As with all of the issues here, we have talked about that step before.

Posted by: urs on September 27, 2006 10:03 AM | Permalink | Reply to this

path integrals from colimits

David wrote:

Vague echoes of what John and I start discussing about a quarter of the way down here, […]

I replied:

At that time I also tried to understand path integrals in quantum mechanics from that point of view. But I failed.

Now I should mabye try again.

I am trying again. First observations relating what David had in mind with what Freed has observed and what I was discussing can be found here.

Comments are welcome.

Posted by: urs on September 13, 2006 5:49 PM | Permalink | Reply to this

a strange kind of sum

There is some discussion going on behind the scenes by private email. As a reaction to that, I want to make the following comment:

I’ll describe a construction which should give a suggestive and neat way to build a 1dQFT functor from a 1d transport functor.

But part of the construction looks slightly worrisome. I guess it’s on the right track, but not quite the full truth yet.

The worrisome part is this: I will need to talk about equality of objects in Vect\mathrm{Vect}. You will see below what this is needed for, so if you see a better way to achieve the same purpose, please let me know.

So here is the worrisome part:

I go ahead and define a modification \oplus' of the direct sum \oplus on Vect\mathrm{Vect}.

For VV different from WW set

(1)VW=VW. V \oplus' W = V \oplus W \,.

But for V=WV=W set

(2)VW=V. V \oplus' W = V \,.

Accordingly, on morphisms set

(3)(Vϕ 1V)(Wϕ 2W)=(Vϕ 1V)(Wϕ 2W) (V \stackrel{\phi_1}{\to} V') \oplus' (W \stackrel{\phi_2}{\to} W') = (V \stackrel{\phi_1}{\to} V') \oplus (W \stackrel{\phi_2}{\to} W')

if VWV \neq W and VWV'\neq W'.

If however V=WV=W and V=WV'=W' then set

(4)(Vϕ 1V)(Wϕ 2W)=(Vϕ 1+ϕ 2V). (V \stackrel{\phi_1}{\to} V') \oplus' (W \stackrel{\phi_2}{\to} W') = (V \stackrel{\phi_1 + \phi_2}{\to} V') \,.

Accordingly for the mixed cases where for instance V=WV=W but VWV'\neq W'.

I guess I can define this, but it doesn’t look “universal” at all, in any sense.

But if I allow myself to define this, then I can make the following concise construction.

Let tra:P 1(X)Vect\mathrm{tra} : P_1(X) \to \mathrm{Vect} denote a vector bundle with connection, as before. (The following makes detailed sense as stated only of we take the sets of objects and morphisms in P 1(X)P_1(X) to be finite.)

Also as before, for any Riemannian 1-cobordisms t\bullet \stackrel{t}{\to} \bullet, let C tC_t be the space of maps from [0,t][0,t] to morphisms in P 1(X)P_1(X).

Then the 1dQFT obtained from quantizing the transport functor above (to be thought of as the action functional) by means of the path integral is nothing but the 1-functor

(5)1Cob Vect (t) (xγyC t)(E xtra(γ)E y), \array{ 1\mathrm{Cob} &\to& \mathrm{Vect} \\ (\bullet \stackrel{t}{\to}\bullet) &\mapsto& \oplus'_{ (x\stackrel{\gamma}{\to} y \in C_t) } \;\; (E_x \stackrel{\mathrm{tra}(\gamma)}{\to} E_y ) } \,,

where I am suppressing the measure on paths, for simplicity.

In words: using the modified direct sum \oplus', the 1dQFT functor is just the \oplus'-sum of all images under tra\mathrm{tra} of all morphisms in P 1(X)P_1(X).

Hm…

Posted by: urs on September 12, 2006 9:19 PM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

If you’re mainly interested in describing topological quantum field theories as nn-functors

Z:nCobnHilbZ : n\mathrm{Cob} \to n\mathrm{Hilb}

you might want to look at this:

Maybe you have already. We wrote this in 1995 as an attempt to generalize Freed’s ideas to higher dimensions. In our “Cobordism Hypothesis”, we propose a purely algebraic description of nCobn\mathrm{Cob}, which should make it easier to construct nn-functors

Z:nCobnHilbZ : n\mathrm{Cob} \to n\mathrm{Hilb}

But, right now you seem to be focusing on something Freed studies: namely, building such nn-functors starting from a Lagrangian. This is very interesting too, especially in how it generalizes the usual ideas of geometric quantization from line bundles to gerbes, 2-gerbes, and so on.

All this stuff should generalize to non-topological quantum field theories, too - but it gets trickier.

(I’ll be sort of quiet until Sunday, since my wife and I will be visiting the nearby town of Hangzhou. This was the capital of China during the second half of the Sung Dynasty. It’s full of history, and it borders a wonderful lake called the West Lake. Next Wednesday I fly back to Riverside.)

Posted by: John Baez on September 13, 2006 8:06 AM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

If you’re mainly interested in […]

Thanks for the link. I believe I have read that once, but I should maybe look at it again now.

What I am interested in here is

1) understanding how to pass from an nn-bundle with connection to the quantum theory of the nn-particle coupled to it,

2) how to conceive Freed’s observation in a more systematic fashion, maybe something like a suitable colimit. I’ll prepare some notes on that.

By the way, I learned that Shanghai is sister city of Hamburg.

Posted by: urs on September 13, 2006 4:06 PM | Permalink | Reply to this
Read the post Quantum n-Transport
Weblog: The n-Category Café
Excerpt: An attempt to understand the path integral for an n-dimensional field theory as a coproduct operation over transport n-functors.
Tracked: September 14, 2006 2:10 PM

Re: Freed on Higher Structures in QFT, I

Urs wrote:

Now I see that this is essentially what Freed is doing in his specific examples. But now I also think there must be a general abstract prescription that describes what it means to “path integrate”/”quantize” an n-functor
tra:P n(X)→nVect.

Well, my understanding of the paradigm is passing from the classical to the quantum action is taking the space of sections : where `space of sections’ is applied relative to the dimension of the category you’re working in, i.e. a section is either a functor or a 2-functor and so on, satisfying the appropriate categorified notion of `section’.

In particular, I expect that where Freed discusses the Chern-Simons (or Dijkgraaf-Witten) model, we should identify the transport 3-functor that describes the 2-vector transport in the line bundle 2-gerbe with connection over BG.

Definitely!

Posted by: Bruce Bartlett on September 25, 2006 4:46 PM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

Hi Bruce,

thanks for your comment!

You wrote:

the paradigm is passing from the classical to the quantum action is taking the space of sections

I can see how that is meant, and how it produces nice results in given examples, because it is usually sort of “obvious” what the relevant space of sections would be.

But when I think about it, I feel that I still have to add some insight that goes beyond applying a mechanical algorithm.

Maybe I am just being dumb.

Here is a specific example which may illustrate what I am struggling with:

Say we have a space XX and a 2-functor P 2(X)Bim(Vect)P_2(X) \to \mathrm{Bim}(\mathrm{Vect}), which sends points of XX to algebras, paths to bimodules of these algebras and surfaces to bimodule homomorphisms.

What precisely is the space of sections at each level?

I can sort of guess what it is. I also think that after discretizing a little the prescription indicated here yields an answer.

But I am still not really satisfied with that. I want some general well-defined procedure that takes me from an assignment like P 2(X)somethingP_2(X) \to \text{something} to the relevant spaces of sections.

Can you help me with that?

Posted by: urs on September 25, 2006 5:00 PM | Permalink | Reply to this

Re: Freed on Higher Structures in QFT, I

Bruce Bartlett emphasized, that

the paradigm is passing from the classical to the quantum action is taking the space of sections

I replied that

I can see how that is meant, and how it produces nice results in given examples, because it is usually sort of “obvious” what the relevant space of sections would be.

But when I think about it, I feel that I still have to add some insight that goes beyond applying a mechanical algorithm.

I am still thinking about this. Today I noticed that the following might be a useful way to look at the quantum theory defined on the space of sections of a vector bundle with connection - in such a way that it nicely categorifies.

As I might have mentioned before (it’s getting boring, isn’t it…), I am thinking of this vector bundle EXE \to X with connection as a functor

(1)tra :P 1(X)Vect \mathrm{tra}_\nabla : P_1(X) \to \mathrm{Vect}

or, in fact, as the induced 2-functor with values in what I denoted Vect˜\tilde \mathrm{Vect} here.

(And I am thinking of hermitian vector spaces everywhere.)

Now let

(2)tra 0 \mathrm{tra}_0

be the trivial-as-can-be functor which assigns \mathbb{C} to every point and the identity morphism to every path in XX.

Then we can pull a nice trick:

A section ee of the vector bundle, together with its covariant derivative 1-form with respect to \nabla # is precisely a morphism

(3)e:tra tra 0. e : \mathrm{tra}_\nabla \to \mathrm{tra}_0 \,.

As I said, I want to think of the transport here as a 2-functor (assigning curvature to surfaces). So this morphism is a pseudonatural transformation, given on each path xγyx \stackrel{\gamma}{\to} y in XX by

(4)E x tra (γ) E y e x d e(γ) e y = . \array{ E_x & \stackrel{ \mathrm{tra}_\nabla(\gamma) }{\to}& E_y \\ e_x \downarrow \;\; &d_\nabla e(\gamma) \Downarrow& \;\; \downarrow e_y \\ \mathbb{C} &=& \mathbb{C} } \,.

Simple - but neat. This categorifies in an obviuous way without any further work. Plus, it automatically provides us with a nn-category structure on the space of nn-sections, simply because all the

(5)e:tratra 0 e : \mathrm{tra} \to \mathrm{tra}_0

live in the corresponding functor category.

(Ordinary 1-sections in this picture already form a 1-category. But you can see that it is just the pair groupoid on the set of these sections, refecting the fact that sections may be added together. So it fits in with what one expects.)

Better yet, since the ee as defined above automatically contains the covariant derivative nn-form, we may proceed as I indicated # with putting a Hilbert space structure on the space of sections and out drops the scalar product

(6)(e,e)= X(d e x,d e x) (e,e) = \int_X (d_\nabla e_x,d_\nabla e_x)

which is - depending on your taste - the first quantized energy of the charged nn-particle, or its action in the second quantized theory.

Posted by: urs on September 29, 2006 5:35 PM | Permalink | Reply to this

from n-transport to n-Hilbert spaces

I’ll continue talking here about my thoughts concerning the issue of

how to systematically and naturally pass from an nn-bundle with connection to the quantum theory of the corresponding charged nn-particle.

Above # I noticed that a neat way to talk about the (Hilbert) space of sections of a hermitian vector bundle with connection, represented by a 1-transport tra \mathrm{tra}_\nabla, is as the category of functors

(1)[tra ,tra 0], [\mathrm{tra}_\nabla, \mathrm{tra}_0] \,,

where tra 0\mathrm{tra}_0 represents the trivial bundle with trivial connection.

At the end of the previous comment, I still referred to equipping this category with a scalar product “by hand”.

Since upon categorification, this operation “by hand” will become less obvious, it would be good to have an s&n description (“systematic and natural” :-) for this, too.

It looks like I am growing quite fond of the following solution:

The point of working with hermitian vector bundles is that this is a category with duals, hence in particular with duals of morphisms, given by taking the adjoint with respect to the scalar product.

If the category TT has duals, every functor

(2)tra:P 1(X)T \mathrm{tra} : P_1(X) \to T

has a dual

(3)tra :P 1(X)T op \mathrm{tra}^\dagger : P_1(X) \to T^\mathrm{op}

obtained by composing tra\mathrm{tra} with dualization on morphisms.

Moreover, for

(4)e:tratra 0 e : \mathrm{tra} \to \mathrm{tra}_0

we have

(5)e :tra 0tra e^\dagger : \mathrm{tra}_0 \to \mathrm{tra}^\dagger

(where I use the fact that tra 0 =tra 0\mathrm{tra}_0^\dagger = \mathrm{tra}_0).

This means that to tra:P 1(X)T\mathrm{tra} : P_1(X) \to T we can naturally associate the functor

(6)tra ×tra:P 1(X)T op×T. \mathrm{tra}^\dagger \times \mathrm{tra} : P_1(X) \to T^\mathrm{op} \times T \,.

The right hand side here urges us to perform one further operation: apply the Hom\mathrm{Hom}-functor. In the present case TT is closed and we get an “inner product”

(7)(tra,tra):P 1(X)tra ×traT op×T opHomT. (\mathrm{tra},\mathrm{tra}) : P_1(X) \stackrel{\mathrm{tra}^\dagger \times \mathrm{tra}}{\to} T^\mathrm{op}\times T^\mathrm{op} \stackrel{\mathrm{Hom}}{\to} T \,.

But we were really interested in the inner product of “sections”:

(8)e 1,2:tratra 0. e_{1,2} : \mathrm{tra} \to \mathrm{tra}_0 \,.

Sure enough, we can form also the functor

(9)(tra 0,tra 0):P 1(X)tra ×traT op×T opHomT (\mathrm{tra}_0,\mathrm{tra}_0) : P_1(X) \stackrel{\mathrm{tra}^\dagger \times \mathrm{tra}}{\to} T^\mathrm{op}\times T^\mathrm{op} \stackrel{\mathrm{Hom}}{\to} T

and e 1,e 2e_1,e_2 provide us with a natural transformation

(10)(tra,tra)(e 1,e 2)(tra 0,tra 0). (\mathrm{tra},\mathrm{tra}) \stackrel{(e_1,e_2)}{\to} (\mathrm{tra}_0,\mathrm{tra}_0) \,.

The fun thing now is that, if one writes out the details of this, the natural transformation

(11)(e 1,e 2) (e_1,e_2)

does indeed encode precisely the desired (fiberwise) inner product on these sections together with (I think) indeed the desired inner product on the covariant derivatives (d trae 1,d trae 2)(d_\mathrm{tra} e_1,d_\mathrm{tra} e_2) of these sections.

That’s pretty cool, I think. Very s&n, and yet exactly what we are after.

If anyone has the impression I am nothing but beginning to reproduce something that is already well known, please be so kind and drop me a note.

Posted by: urs on October 1, 2006 2:38 PM | Permalink | Reply to this
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