"Area Metric" Manifolds | The String Coffee Table

November 29, 2005

“Area Metric” Manifolds

Posted by Urs Schreiber

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David Roberts made me have a look at the recent preprint

F. Schuller & M. Wohlfarth
Canonical differential structure of string backgrounds
hep-th/0511157.

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People are trying to find the right mathematical formalisms to capture ‘higher’ stringified structures.

The best example is Hitchin’s currently very popular generalized geometry, a mathematical framework motivated by and modeled after the fact that the metric and the Kalb-Ramond 2-form appear on equal footing in string theory. The authors of the above paper propose another way to unify a symmetric and an antisymmetric rank-2- tensor into some generalized geometric structure.

An ordinary metric $g$ can be regarded as a linear map from vectors to covectors

(1)

g : T M \to T^{*} M .

In this spirit, Schuller and Wohlfarth define an area metric $G$ to be a linear map from 2-vectors (no, not those 2-vectors :-) to 2-forms:

(2)

G : Λ^{2} T M \to Λ^{2} T^{*} M .

An area metric manifold is then a manifold $M$ equipped with such an area metric.

Every ordinary metric $g$ gives rise to an area metric $G_{g}$ in the obvious way. Similarly every nondegenerate 2-form $B$ on $M$ gives rise to an area metric $G_{B}$ .

Consider a ‘worldsheet’ $Σ : [0,1]^{2} \to M$ with tangent 2-vector

(3)

Ω = \partial_{1} Σ \land \partial_{2} Σ .

It is a well known and widely used fact that studying the Polyakov action in a background with gravity $g$ and Kalb-Ramond fields $B$ leads one to use the ‘generalized metric’ tensor $g \pm B$ .

Using the above language this fact appears in form of the statement that the Polyakov action in such a background is classically equivalent to an action given by the term

(4)

\propto \int d σ^{2} \sqrt{G_{g + B} (Ω, Ω)},

formally analogous to the Nambu-Goto point partcicle action.

This is not too surprising, it seems, being essentially a new coordinate-independent formulation of well-known formulas, I think. But it could point us to useful new concepts.

The authors of the above paper propose that such a useful new concept is that of area derivative and area parallel transport.

They discuss in detail various generalizations of the concept of a covariant derivative to the context of manifolds with area metric. For the applications to string actions the most important one is the canonical area derivative described in section IV. In the case where this is area metric compatible (in analogy to how the Levi-Civita connection is metric compatible) this a gadget that maps two infinitesimal surfaces $Σ, Ω \in Λ^{2} T M$ to the 1-form $D_{Σ} Ω$ given by

(5)

D_{Σ} Ω (v) = d G (Ω) (Σ \land v)

for $v \in T M$ .

Using this object, it turns out that the classical equations of motion of the Polyakov string in metric and KR-background simply read

(6)

D_{Ω} Ω = 0

and hence look completely similar to the geodesic equation of motion of a free point particle.

I would like to better understand to what extent this operator $D_{Ω}$ is the right thing to consider, generally. For instance, what does it mean that $D_{Σ} Ω$ is a 1-form, something I find somewhat counterintuitive. Naïvely I would have expect it to be an element in $Λ^{2} T M$ .

Hence what I would like to understand is to what extent area metric geometry and in partiular the ‘canonical area derivative’ follow from some general abstract reasoning, possibly along the lines of our recent discussion on higher Poisson geometry and higher geometric quantization.

That would help understand in which sense $D_{Ω} Ω = 0$ is just a rewriting of known things or a concept leading to something genuinely new.

Posted at November 29, 2005 2:43 PM UTC

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8 Comments & 2 Trackbacks

Re: “Area Metric” Manifolds

I haven’t absorbed this yet, but it seems fascinating to me, because in the “BF theory” approach to general relativity that spin foam people like, the basic field is not the metric but a Lorentz-Lie-algebra-valued 2-form, the B field, which plays the role of an “area frame field” - assigning internal bivectors to tiny parallelograms, instead of assigning internal vectors to tiny vectors, as a (co)frame field does.

Does string theory really involve an area metric?

Of course the other reason this “Lie-algebra-valued 2-form” is so fascinating, is that such things naturally show up in 2-connections.

Which of course is part of my sneaky plan to unify string theory, spin foam models and n-categories. :-)

By the way, Urs: why don’t you put up an entry about our new paper? And while you’re at it, you can advertise this even shorter version:

Higher Gauge Theory: 2-Connections

… my talk at the Union College Mathematics Conference this weekend.

Posted by: john baez on December 1, 2005 9:29 PM | Permalink | Reply to this

Re: “Area Metric” Manifolds

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a Lorentz-Lie-algebra-valued 2-form, the $B$ field, which plays the role of an ‘area frame field’ - assigning internal bivectors to tiny parallelograms

So I guess here you are identifying ‘internal bivectors’ with the Lorentz generators associated to the Lorentz-rotation that these bivectors generate?

Does string theory really involve an area metric?

It surely involves special cases of area metrics. I don’t know if all area metric can be written in the form $G_{g + B}$ coming from a metric tensor and a 2-form.

Posted by: Urs on December 2, 2005 12:34 PM | Permalink | Reply to this

Re: “Area Metric” Manifolds

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John Baez asked:

Does string theory really involve an area metric?

I replied:

It surely involves special cases of area metrics. I don’t know if all area metric can be written in the form $G_{g + B}$ coming from a metric tensor and a 2-form.

I have now talked with Mattias about this. Apparently it is not known if in arbitrary dimensions every area metric can be written as one coming from an ordinary metric and a 2-form. But it seems unlikely. At least one would expect the degree of freedom corresponding to the dilaton to play a role, too.

Special cases are different. Every area metric in three dimensions can be obtained from a single ordinary metric because in 3D areas are dual to their normal vectors.

What is known (as recalled in the introduction of the paper) is that every area metric decomposes into a 4-form and an ‘algebraic curvature tensor’, i.e. a rank-4 tensor with the symmetry properties of a Riemann curvature tensor.

Certainly the 4-form part drops out in Nambu-Goto-like terms $G (Ω, Ω)$ when $Ω = \partial_{1} Σ \land \partial_{2} Σ$ .

Concerning the global formulation of this (if it contains the KR field the area metric cannot be meaningful globally in general) one should probably do the following.

Instead of looking at 2-functors from thin-homotopy classes of surfaces, one should look at 2-functors from surfaces just modulo orientation-preserving parameterization. They should come fom area metrics. Maybe even vice versa. Given any target 2-category for such a 2-functor which locally looks like $(ℝ \to 1)$ or something we could then turn the usual crank and obtain the ‘gauge transformation’ laws and transition laws for the area metric.

Looks like a rather straightforward research project…

As a warmup, one could look at suitably nice 1-functors from paths up to orientation-preserving reps to $ℝ$ or something. These should come from a metric and a 1-form on target space. For instance they would associate the action of a particle in a gravitational and EM field to each path.

Here the question would be: Can all such functors be obtained from a metric and a 1-form on target space?

Posted by: Urs on December 8, 2005 5:50 PM | Permalink | Reply to this

Spin Foam and ‘Wilson Surface’

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unify string theory, spin foam models and n-categories

I know next to nothing about spin foams. I have seen on your transparencies that you are claiming that topological strings are a special case of spin foam models. This means a spin foam must be something much more general than I thought it is.

On these transparancies you say a spin foam model is something which assigns group irreps to faces and intertwiners to edges of a 2D complex, like a spin network assigns irreps to edges and intertwiners to vertices.

To my mind a ‘spin network’ is nothing but the idea of a Wilson line, generalized to allow ‘Wilson networks’. (Where the word ‘spin’ here really is a remnant of thinking only of the gauge group $SU (2)$ .)

Therefore I would expect that a ‘spin foam’ should really be the network generalization of a Wilson surface! Namely an observable in a 2-gauge theory with a notion of parallel surface transport.

Such a Wilson 2D network I would expect to assign 2-group irreps to bigons, probably. It should also assign something which I am not sure of what it should be to edges and also something to the vertices.

In any case, it should be something that I can hit with a 2-connection to produce a number, like I can hit a spin network with a 1-connection to produce a number.

But that’s not what you consider to be a spin foam, is it? Maybe in special cases?

(Naively, I would define a 2-representation of a 2-group as a 2-functor from the 2-group regarded as a 2-category to the 2-category of 2-vector spaces. Though that might be a little too restrictive. For instance there does not seem to be an interesting rep of $(U (1) \to 1)$ this way.)

Is there a crisp functorial definition of ordinary spin networks? That might help me to fomulate my idea/question above more precisely.

Posted by: Urs on December 2, 2005 4:01 PM | Permalink | Reply to this

Re: Spin Foam and ‘Wilson Surface’

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Urs writes:

I know next to nothing about spin foams. I have seen on your transparencies that you are claiming that topological strings are a special case of spin foam models. This means a spin foam must be something much more general than I thought it is.

Well, let’s start at the beginning…

On these transparancies you say a spin foam model is something which assigns group irreps to faces and intertwiners to edges of a 2D complex, like a spin network assigns irreps to edges and intertwiners to vertices.

Not quite. A “spin foam” is a 2d complex with faces labelled by irreps and edges labelled by intertwiners. A “spin foam model” is a rule that assigns amplitudes to spin foams - typically by computing an amplitude for each vertex, edge and face, and then multiplying them all.

But actually we can (and should) be a bit more general in our definition of “spin foam”. Instead of using objects and morphisms in a category of group representations to label the faces and edges of our 2d complex, we can use objects and morphisms in some more general kind of category. A popular example is the category of representations of a quantum group.

The details of what sort of category we can use depend on the features of our spin foam. Normally our category should be “monoidal” - it should have “tensor products”, as evident from page 2 of my talk. As you can see from the picture here, we need these tensor products to handle spin foams where several faces meet along an edge.

But topological string theories are a specially degenerate case where the only 2-complexes that get nonzero amplitudes are 2-manifolds with corners. In these 2-complexes, at most two faces meet along an edge. When two faces meet along an edge, we can label this edge with a morphism f: x -> y, where x and y are the objects labelling the two faces. So, our category doesn’t need tensor products!

So, topological string theories are a bit funny compared to the really interesting spin foam models… but they still fit in the general framework. The details should appear in Lauda and Pfeiffer’s next paper.

I could and should say a bunch more, but I’m getting tired - and maybe whoever is reading this is getting tired, too. :-)

Posted by: john baez on December 7, 2005 6:33 AM | Permalink | Reply to this

Re: Spin Foam and ‘Wilson Surface’

Thanks for your reply. If you allow, I would get back to my question concerning a functorial understanding of spin networks.

There is something like a 1D cobordism category where we allow the cobordisms to join and split. Alternatively this can be thought of as an open 2D cobordism category.

This is a monoidal category under disjoint union. Isn’t a spin network a monoidal functor from this to some monoidal category of group reps?

I note that planar spin networks (under the name ‘Wilson network’) appear in the algebraic CFT description of Schweigert/Fuchs/Runkel et al.. There the decorations live in a possibly more general modular tensor category.

In the planar case, i.e. when the spin network can be drawn in a plane, there seems to be a neat 2-functorial reformulation of it. Switch to the dual of the graph, replace objects by 1-morphisms, etc. Then the above 1-functor form cobordisms to a monoidal category becomes a 2-functor from some 2-category of surface elements to the same monoidal category, but now regarded as a (weak) 2-category with a single object.

I should draw pictures to transport this idea. But maybe you see what I am trying to get at.

Posted by: Urs on December 8, 2005 3:05 PM | Permalink | Reply to this

Re: Spin Foam and ‘Wilson Surface’

Sorry to take so long to reply, Urs!

You wrote:

If you allow, I would get back to my question concerning a functorial understanding of spin networks.

There is something like a 1D cobordism category where we allow the cobordisms to join and split. Alternatively this can be thought of as an open 2D cobordism category.

This is a monoidal category under disjoint union. Isn’t a spin network a monoidal functor from this to some monoidal category of group reps?

That sounds about right for spin networks in the plane. For spin networks in 3d you want to say “braided monoidal” in your last paragraph, and for spin networks in 4d or higher - or abstract spin networks - I think you want to say “symmetric monoidal”.

There’s also the subtlety that people usually prefer spin networks whose edges are labelled by irreducible group representations… but this makes it harder to get a really slick description of spin networks, so let’s ignore it.

Personally I think of things in a different but equivalent way to the way you just invented - a way that sounds more “algebraic” and less “topological”. It relies on the fact that there’s a (symmetric) monoidal functor

eval: Net(G) → Rep(G)

from the category whose morphisms are spin networks to the category whose morphisms are intertwining operators between representations of G. I hope you see what this functor is. It’s the functor people are implicitly using whenever they evaluate a Feynman diagram and get an operator!

Anyway, both your idea and the above functor doesn’t use anything about Rep(G) except that it’s a (symmetric) monoidal category! Hand me a monoidal category M and we get a monoidal category M where the morphisms are “Feynman diagrams” with edges labelled by objects in M, and vertices labelled by suitable morphisms in M.

And, there’s a god-given (symmetric) monoidal functor

eval: M → M

To understand what’s going on here, it’s good to look at a simpler example of the same idea: the “canonical presentation” of a group G. For any group G there’s a free group G whose generators are all the elements of G. An element of G is a “formal” product of elements of G. And, there’s a homomorphism

eval: G → G

sending these formal products to actual products in G! The kernel of this homomorphism gives all the relations in the canonical presentation of G.

Am I making sense? Do you see the similarity? Do you see that M is in some sense the “canonical presentation” of M?

I should probably check to see if we’re communicating before I go on… I’ve already explained the basic idea, but then one can go ahead and formalize it.

Posted by: john baez on December 17, 2005 3:48 AM | Permalink | Reply to this

Re: Spin Foam and ‘Wilson Surface’

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For spin networks in 3d you want to say ‘braided monoidal’ in your last paragraph, and for spin networks in 4d or higher - or abstract spin networks - I think you want to say ‘symmetric monoidal’.

Yes, thanks.

labelled by irreducible group representations

In the algebraic CFT approach the simple objects in a modular tensor category play a special role. Unless I mixed up what I just learned, these correspond to irreps if the category is a representation category. Anyway, in that context some of the edges are required to be labelled by simple objects (namely those edges which represent field insertions). But, as you say, there does not yet seem to be any slick way at all to encode all the instructions for labelling graphs which are used in this context (as far as I am aware).

It’s sort of funny. Spin networks play an absolutely fundamental role for CFT and hence for string theory. But they are not called this way in the algebraic CFT context. They are called ‘Wilson networks’ or ‘Wilson graphs’ instead.

category whose morphisms are spin networks to the category whose morphisms are intertwining operators between representations of $G$ . I hope you see what this functor is

Yes, I believe I understand. It seems one can introduce a third concept here, which is what I was thinking of, namely that of a spin network without labels, i.e. a bare Feynman graph. Let me call the obvious category of such bare graphs (which is similar to but richer than that of 1D cobordisms) maybe just $Net$ .

What I was thinking of is a functor

(1)

F : Net \to Rep (G) .

which sends ‘points’ ( $≃$ edges without branchings) to objects in $M$ and sends nets between points to morphisms in $M$ .

What you describe is, if I understand correctly, something like an intermediate step in this functor. There is an obvious way in which $F$ above restricts to a functor

(2)

\tilde{F} : Net \to Net (G) .

Then ‘my’ functor is the composition of $\tilde{F}$ with your functor

(3)

Net \overset{\tilde{F}}{\to} Net (G) \overset{eval}{\to} Rep (G) = Net \overset{F}{\to} Rep (G)

Right?

Do you see the similarity?

Yes, I believe so. We can think of the free group $G$ as a degenerate variant of a spin network. It is like a network with no branching and no parallel strands, with a single label for edges and with each vertex (which all have precisely one ingoing and one outgoing edge) labelled by an element in $G$ .

but then one can go ahead and formalize it.

I would eventually like to regard all the networks appearing here as ‘string diagrams’ of 2-categories or 3-categories, if possible.

I know how this works for simple monoidal target categories like $Vect$ . I am not sure yet for instance how a non-trivially braided monoidal category would naturally be interpreted dually.

Posted by: Urs on December 17, 2005 1:50 PM | Permalink | Reply to this

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November 29, 2005