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August 17, 2006

SuGra 3-Connection Reloaded

Posted by Urs Schreiber

As John already mentioned, part of the purpose of this blog is to have a place for the REALLY-DRY-DISCUSSIONS™ that we enjoy so much.

I guess it’s like with instant coffee. Dry by itself, but with some hot water added one can get pretty excited by it.

So before adding any genuinely new content here, we should allow for a place to coherently continue some discussions we are already having spread out over the blogosphere. Apart from the Klein 2-geometry meta-exercise, which David will take up us soon as he returns from his vacation next month, this involves in particular an exchange of observations concerning the relation between nn-connections and (super)gravity theories that John and myself are involved in.

All is based on the age-old observation that certain free graded-commutative differential algebras - FDAs for short - constitute a surprisingly efficient tool for reasoning about supergravity.

Dry topic, isn’t it?

Let’s add some hot water.

It turns out that a couple of well-known sophisticated concepts in algebra that keep appearing all over the place in mathematical physics can actually be understood in a unified way as different incarnations of categorifications of elementary algebraic concepts.

For instance L L_\infty algebras. They are nothing but (semistrict) categorified Lie algebras.

Or free graded-commutative differential algebras. They are just the Koszul dual of the L L_\infty guys, something that follows from general abstract nonsense on operads.

You might argue that you don’t care how much abstract nonsense is equivalent to known structures. If it’s equivalent, why not stick to the familiar concepts?

The point is: only the nn-category theoretic bird’s eye point of view reveals the message that god has written in large letters all over structures like supergravity. I claim.

For instance, in the Sugra-FDA community people have tried to heuristically understand the apparently unreasonable effectiveness of FDAs in terms of what they call “soft group manifolds”. That’s because the crucial algebraic structures in this business look almost - but not quite - like Maurer-Cartan equations of left-invariant differential forms on Lie groups.

So from this point of view one tries to regard the field content of some supergravity theory as a collection of something like differential forms on something like group manifolds. Except that everything is in a funny way “softer” than for honest groups.

I claim that this is not a useful point of view. The main reason is this:

These collections of fields that physicists usually write down, like the graviton field, the gravitino maybe, some 3-form field, etc., are usually really just local representatives of the true - globally defined - fields they represent. It’s like writing down a 1-form for representing the electromagnetic field. In general this only tells you what is going on in contractible patches of spacetime.

That’s fine. But one needs more. There needs to be a way to glue all these local fields together to well-defined global thing.

In phenomenological physics, one usually gets away with completely ignoring this aspect. There is as yet no robust observable evidence of our surrounding spacetime having nontrivial topology. So who cares?

One should care for two reasons:

1) For practical matters, it might well be - who knows? - that there really are small extra dimensions. If so, one naturally expects these to be compact. Hence most likely they will have nontrivial topology. And the effects that small extra dimensions have on observations at practical energy scales are all entirely due to the global topological effects. The effects of the local physics of small extra dimensions would be obserble only at ever higher energies.

In a word - if you are at all interested in a theory of supergravity that lives in, well, eleven dimensions, you should better not ignore the implications of nontrivial topology of spacetime.

2) The other reason is much better. If it doesn’t work globally even in theory - even if you will never be able to check it experimentally - even if the theory has nothing to do with the real world - it’s bound to be nonsense.

So the question is this: does thinking of the local field content of supergravity as a collection of “differential forms on a soft group manifold” tell you how to lift your theory from local patches to the full thing?

If it does, I don’t see it. I’d say it does not.

Instead, I claim that what is called a “soft group manifold” in supergravity is precisely - in disguise - the data of a local nn-connection with values in some Lie nn-algebra.

So, in particular, I should maybe add that the problem I am referring to here goes beyond understanding spinors as sections of spin bundles. We need to understand not sections of ordinary 1-bundles - but of 3-bundles (or 2-gerbes, if you like). And indications are that we need 2-gerbes coming from twisted nonabelian 1-gerbes. So we better get this formalism under control.

As an example, a Lie(G)\mathrm{Lie}(G)-valued 1-form on a contractible patch of spacetime would be a local 1-connection with values in a 1-algebra. If you want to see instanton effects in your Yang-Mills theory, you will have to be able to glue two of such guys consistently on overlapping domains of definition.

And this generalizes. From 1-forms to 2-forms to … nn-forms, taking values in 2-algebras, 3-algebras… nn-algebras.

And this immediately tells us what’s really going on. The theory for how to turn this into something globally defined has been worked out.


You may or may not believe in what I am saying here. If not, you are lacking the hot water to turn our dry discussion into something thrilling.

But in any case, this is part of the reason why John and I think it is interesting to identify, for instance, the gauge 3-group of supergravity as such, and to understand what it all means. And that’s what we are doing in this discussion.


Below the fold I simply compile some of the already existing parts of the disucssion, taken from the String Coffee Table (I, II) and John’s This Week’s Finds (I, II).

The idea is to reserve the comment section of this entry for further discussion along these lines. Sort of as a sub-forum. So it may happen that we post comments to this thread not today, not tomorrow, but maybe in five weeks, or in half a year. Depending on how things develop. If you are intersted in following the discussion, you might want to subscribe to the nn-Category Café comment RSS feed. This will alert you automatically when new comments come in.

(I am just saying this because with all these non-dry blogs around I got the impression that people tend to abandon a comment section of an entry just because it is no longer on top of the index page. )

So here is what I originally wrote on the Coffee Table, slighly abridged, followed by some comments that were exchanged between me and John, and, at the end, John’s comments on how Koszul duality plays a major role in this business.

Urs said:


In

Leonardo Castellani
Lie derivatives along antisymmetric tensors, and the M-theory superalgebra
hep-th/0508213

the author implicitly shows that

1)

the central extension by membrane charges {Q α,Q β}=i(CΓ a) αβP a+(CΓ ab) αβZ ab \lbrace Q_\alpha, \, Q_\beta \rbrace = i(C\Gamma^a)_{\alpha\beta} P_a + {(C\Gamma_{ab})_{\alpha\beta}Z^{ab}} of the super-Poincaré algebra in eleven dimensions defines a semistrict Lie 3-algebra;

2)

the local field content of 11D supergravity defines the local data for a connection on a 3-bundle with this gauge 3-group.

Recall (\to) that we expect on general grounds (\to) M-branes to couple to a 3-bundle (2-gerbe) with some gauge 3-group (\to).
1) survey of FDAs, Lie nn-algebras and nn-connections and nn-bundles

Free differential graded algebra (we should really say “free differential graded commutative algebras”, as Ezra Getzler kindly emphasized to me today), FDA for short, are essentially the same (\to) as

\bullet semistrict Lie nn-algebras and Lie nn-algebroids (\to, \to)

\bullet nn-term L L_\infty algebras .

More precisely, from L L_\infty-algebras and Lie nn-algebras, which come with complexes of vector spaces with lots of graded brackets on them, we obtain free differential graded algebras simply by passing to the dual vector spaces and equipping them with a differential whose nilpotency is equivalent to the (intricate) system of higher Jacobi almost-identities defining the original structure.

This is nice, for two reasons:

1) FDAs are much easier to compute with than Lie nn-algebras.

2) Lie nn-algebras have a much clearer conceptual interpretation in higher gauge theory (\to) than their mere FDA structure suggests.

So we can pass between the two pictures as convenient. In particular, if we encounter considerations in just one picture, we know how to translate it to the other.

The conceptual understanding of Lie nn-algebras allows us to easily understand their role in higher gauge theory.

An (integrable) connection on an nn-bundle (\to) is, by definition, something that allows us, consistently, to perform parallel transport (\to) over nn-dimensional volumes.

In other words, it is a morphism tra:P n(X)Trans n(T) \mathrm{tra} : P_n(X) \to \mathrm{Trans}_n(T) from the nn-groupoid of nn-paths in the base space XX to the transport nn-groupoid of the nn-bundle TXT\to X with connection (\to).

But this setup is easily differentiated. Passing to infinitesimals, nn-groupoids become nn-algebroids. Hence, infinitesimally, an (integrable) nn-connection on an nn bundle is a morphism

dtra:p n(X)trans n(T) d\mathrm{tra} : p_n(X) \to \mathrm{trans}_n(T)

of the corresponding algebroids (\to).

Knowing this, we may pass to the dual FDA description of this situation, and study connections on nn-bundles in terms of morphisms of FDA algebras (differential graded algebras).

Motivated by the Poisson σ\sigma-model, Thomas Strobl and collaborators have looked at such morphisms (\to) from the point of view of gauge theory.

One finds a couple of nice, unifying structures in this context.

i) First of all, one should note that a morphism dtra:ABd\mathrm{tra} : A \to B of nn-algebroids corresponds to a chain map (d B,B )(d A,A )(d_B, B^\bullet) \to (d_A,A^\bullet) of the corresponding dg-algebras.

ii) Naturally, then, 1-morphisms of nn-algebroid morphisms correspond to chain homotopies, 2-morphisms to homotopies of homotopies, and so on.

iii) If we look at the double complex (Q:=d A±d B,A B )(Q := d_A \pm d_B , A^\bullet \oplus B^\bullet), these conditions read as follows:

- a map Φ\Phi of dg-algebras (=FDAs) has to be QQ-closed [Q,Φ]=0[Q,\Phi] = 0.

- a map of complexes ϵ:B A 1\epsilon : B^\bullet \to A^{\bullet-1} is a 1-morphisms of maps of dg-algebras, with ϕ=ϕ+[Q,ϵ]\phi' = \phi + [Q,\epsilon] (where the bracket is graded, hence now an anticommutator).

- similarly, a map ϵ p:B A p\epsilon_p : B^\bullet \to A^{\bullet-p} is a pp-morphisms of (the underlying) Lie nn-algebras, relating two (p1)(p-1)-morphisms that differ by [Q,ϵ p][Q,\epsilon_\p] (\to).

iv) the failure of a morphism Φ\Phi to be a chain map in degree pp is, when this map is interpreted as a connection on an nn-bundle, precisely the pp-form curvature (for p<np \lt n also known as “fake curvatures”)

v) Bianchi identites in the gauge theory sense are nothing but Q 2Φ=0Q^2\Phi = 0.

vi) Infinitesimal gauge transformations in the gauge theory sense are nothing but exact morphisms [Q,ϵ][Q,\epsilon].

vii) more generally, infinitesimal transformations are generated by generalized Lie derivatives {Q,i t}\{Q,i_t\}. These are symmetries of Φ\Phi iff [L t,Φ]:=[{Q,i t},Φ]=0[L_t,\Phi] := [\{Q,i_t\},\Phi] = 0.

2) translating Castellani’s paper into Lie nn-algebra language

We can now, step by step, go through Castellani’s paper hep-th/0508213 and interpret the FDA constructions there in the context of nn-connections on nn-bundles.

\bullet equations (2.1) and (2.2) are the dual formulation of a certain semistrict Lie nn-algebra, which plays the role of the nn-algebra of the gauge nn-group. The number nn is detrmined, in this paper, by the highest pp-form degree appearing, as n=pn = p.

For p=1p=1 we get only 1-forms and the formalism described charged points (section 13.5).

For p=2p=2 we get 1- and 2-forms. If the 1-forms are trivial and the 2-form is abelian this described the Kalb-Ramond gerbe connection that the fundamental string couples to (section 13.6)

For p=3p=3 we get 1-, 2, and 3-forms. The 3-form of 11D supergravity should be a realization of this (compare section 13.8).

Indeed, that’s the case the Castellani studies in section 3 of his paper.

\bullet We may interpret all the constants appearing there intrinsically. In particular, the coefficients C i A 1A 2A 2C^i{}_{A_1 A_2A_2}, which relate the pp-forms to the connection 1-form encode nonvanishing Jacobiators (measuring the failure of the Jacobi identity to hold).

\bullet The concept referred to as soft group manifolds in the last paragraph of page 2 is secretly precisely the concept of a map from the dg-algebra characterizing a group to that of the ordinary deRham complex, i.e. a morphisms from nn-paths to an nn-group characterizing an nn-connection.

\bullet Equations (2.6) and (2.7) give the curvatures, which encode the failure of this map to be a chain map.

\bullet Equation (2.16) is a realization of the statement that Lie derivates split into a pure gauge part and a contraction of the curvature [[Q,i v],Φ]=±[[Q,ϕ],i v]±[Q,[i v,Φ]]. \left[ \left[ Q,i_v \right] , \Phi \right] = \pm \left[ \left[ Q,\phi \right] , i_v \right] \pm \left[ Q, \left[ i_v,\Phi \right] \right] \,.

\bullet The closure of the algebra of generalized Lie derivatives, given in the particular example in equations (2.28)-(2.30), is guaranteed by the general structure [[Q,i v],[Q,i w]][Q,[i v,[Q,i w]]]. \left[ \left[ Q,i_v \right] , \left[ Q,i_w \right] \right] \propto \left[ Q, \left[ i_v, \left[ Q,i_w \right] \right] \right] \,.

3) Castellani’s result

The crucial new result of the paper is given in section 3. With hindsight, given the above considerations, I think can rephrase this main result as follows.

The author notes that there is semistrict Lie 33-algebra (\simeq 3-term L L_\infty-algebra \simeq a certain dg-algebra) whose Lie algebra generated by the [Q,i v][Q,i_v] is precisely the super-Poincaré Lie algebra in eleven dimensions, centrally extended by the central charge Z abZ^{ab} corresponding to membrane (“M-branes”).

Moreover, a 3-connection with values in that Lie 3-algebra encodes, locally, precisely the field content of 11D supergravity.

There is one more nice fact, which builds on an older, well known, result, as discussed for instance in

L. Castellani, R. D’Auria & P. Fré
Supergravity and superstrings: a geometric perspective
World Scientific, Singapore (1991),

namely that imposing the condition that the curvature [Q,Φ][Q,\Phi] of this 3-connection is horizontal, meaning that it takes values only in the algebra of objects of the gauge 33-algebra, is equilvalent to the equations of motion of the graviton, the gravitino and the vielbein.


John said:

The stuff about free differential graded algebras and L L_\infty algebras (= chain complexes that are Lie algebras up to coherent homotopy) is an example of a wonderful general pattern called “Koszul duality”.

One can also use Koszul duality to give efficient descriptions of A A_\infty algebras (= chain complexes that are associative algebras up to coherent homotopy) and C C_\infty algebras (= chain complexes that are commutative algebras up to coherent homotopy). There’s a nice treatment of it in Markl, Schnider and Stasheff’s book on operads and physics.

But, Koszul duality has many other aspects as well! I understand some of these but not others. I feel I still need to dig down into its essence. It’s one of those grand patterns that manifests in many different contexts.

Anyway, I’m digressing. What I want to understand now is this Lie 3-superalgebra associated to 11d supergravity. You write:

The author notes that there is semistrict Lie 3-algebra … whose Lie algebra generated by the [Q,i v][Q,i_v] is precisely the super-Poincaré Lie algebra in eleven dimensions, centrally extended by the central charge Z abZ_{ab} corresponding to membranes (“M-branes”).

Could you please help me understand this Lie 3-superalgebra? I’d eventually like to know the objects, morphisms, and 2-morphism… or if you prefer, the 0-chains, 1-chains and 2-chains… and the bracket, Jacobiator, and Jacobiatorator. But, any step in this direction would be great!

What I really want to understand is why this structure can only be built in 11 dimensions. A purely algebraic explanation of “what’s so great about 11 dimensions” - that’s been a dream of mine for some time.

But first, I need to understand what this structure is!

It sounds like the 0-chains include the super-Poincare algebra in 11 dimensions.

What else? I know you said somewhere that the loop Lie algbra of E 8E_8 makes its appearance….

Could you please help out? At some point, I’m hoping that my understanding of exceptional groups will kick in and I’ll be able to see what’s special to 11 dimensions here. I understand how E 8E_8 is built using the rotation groups in 8 and 16 dimensions. I understand how E6 is related to 10-dimensional spacetime. But, I don’t know relations between these groups and 11-dimensional spacetime! It’s possible that Castellani’s construction will explain that.


Urs said:

Could you please help me understand this Lie 3-superalgebra?

Sure. It’s defined in equation (3.1) of hep-th/0508213.

In order to see how to read off the data in the form you are looking for, open for instance hep-th/0509163, page 342, example 13.1. and compare term-by-term.

Actually, that example applies to 2-groups, but the generalization of the pattern should be obvious.

So, the 3 Lie algebra in question

- has as objects the Lie algebra of the super Poincaré group

- has only trivial 1-morphisms

- has a 1-dimensional vector space of 2-morphisms on every 1-morphism.

Apart from the ordinary 2-ary bracket on objects, the only nontrivial bracket is the 4-ary one.

(I don’t know the category-theoretic name for that. Maybe “2-associator”? )

This is implicitly defined in the last line of equation (3.1) in Castellani’s paper. The bracket is nonvanishing precisely if two of its entries are generators of translations, and the other two are spinors. In components it is simply given by the commutator of “Gamma-matrices” C α¯β ab:=[Γ a,Γ b] α¯β. C^{ab}_{\bar\alpha \beta} := [\Gamma^a,\Gamma^b]_{\bar \alpha \beta} \,.

You ask:

why this structure can only be built in 11 dimensions

Because (as Castellani recalls right below equation (3.1)) the Fierz identity ψ¯Γ abψψ¯Γ aψ=0, \bar \psi \Gamma^{ab}\psi \, \bar \psi \Gamma^a \psi = 0 \,, which holds in D=11D=11, is what ensures that the above “2-associator” (I’ll be glad to use a better term if you suggest one) does satisfy its coherence law - in other words, that the differential of the dg-algebra defined in (3.1) does indeed square to zero.

The existence of this extension of the super-Poincaré dg-algebra is implied by an old result by Chevalley on extensions of dg-algebras, reviewed somewhere in the FDA review papers cited in Castellani’s papers.

The new result of Castellani’s papers is that one can re-extract the centrally extended super-Poincaré algebra from this 3-algebra, as he describes in section 3.1 and 3.2.

E 8E_8 makes its appearance

Yes, according to the Jurčo-Aschieri argument (\to) we expect the supergravity 3-form to contain a component of the lifting 2-gerbe of a twisted Chern-Simons 1-gerbe for the lift of crossed modules (ΩE 8PE 8)(Ω^E 8PE 8), (\Omega E_8 \to P E_8) \to (\hat \Omega E_8 \to P E_8) \,, i.e. from E 8E_8 to String E 8\mathrm{String}_{E_8}.

As Danny explains in his notes somewhere (possibly here), this 2-gerbe has as connection 3-form the Chern-Simons form for an E 8E_8-bundle (he explains it for Spin(n)\mathrm{Spin}(n) instead of E 8E_8).

So, it looks like we should take something like the direct sum of the super Poincaré 3-algebra and the String E 8\mathrm{String}_{E_8}-3-algebra for the description of 11D SUGRA.

As a consistency check, we note that this predicts that the SUGRA 3-form has a component which is an E 8E_8 CS form and a component coming from a Lorentz connection (aka spin connection). This is in accordance with what Diaconescu-Freed-Witten anomaly cancellation demands.

I was beginning to work out more details of this. But no chance - too much other things to do.

Here is a vague observation, though:

I think we can construct a 3-Lie algebra which in lowest two degrees is that of a trivial differential crossed module (LieGLieG), (\mathrm{Lie} G \to \mathrm{Lie} G) \,, thus giving rise to 2-connections with LieG\mathrm{Lie} G-valued 1- and 2-forms.

In addition, let there be the Lie algebra of U(1)U(1) in the next degree, with a 2-associator that leads to the 4-curvature dC 3+tr(B 2B 2). d C_3 + \mathrm{tr}(B_2 \wedge B_2) \,. Now demand all curvatures to vanish (fake curvature and everything).

Vanishing of fake curvature says that B 2=F A B_2 = F_A is the curvature of an ordinary GG-connection AA. Vanishing of the top-level curvature then says that dC 3dC_3 is the Pontryagin 4-form of the corresponding GG-bundle dC 3tr(F AF A). dC_3 \propto \mathrm{tr}(F_A \wedge F_A) \,. This is indeed the case for the Chern-Simons 2-gerbes that we are after. It implies that the 3-form is the CS 3-form of that GG-bundle.

Now use G=E 8G = E_8.


John said:

In your email to me you observed that detailed computations in this subject tend to bog down in a mess of Fierz identities. I’m glad you said this, because I’d been sort of embarrassed to admit that for me to understand this subject, the first thing I need to understand is where the d=11d = 11 Fierz identity comes from, and why it manages to make the Jacobiatorator satisfy the Jacobiatorator identity.

A note on terminology is probably warranted here:

Jacobi identity = identity satisfied by the bracket in a Lie algebra.

Jacobiator = 1-chain which replaces the Jacobi identity when we go from Lie algebras to Lie 2-algebras.

Jacobiator identity = identity satisfied by the Jacobiator in a Lie 2-algebra.

Jacobiatorator = 2-chain which replaces the Jacobiator identity when we go from Lie 2-algebras to Lie 3-algebras.

Jacobiatorator identity = identity satisfied by the Jacobiatorator in a Lie 3-algebra.

etcetera.

If this terminology seems too silly, which it probably is after “Jacobiator identity”, feel free to say l2 for bracket, l3 for Jacobiator, l4 for Jacobiatorator, etc.

Anyway, I don’t understand the d=11d = 11 Fierz identity and why it just luckily happens to be the Jacobiatorator identity in disguise.

But, I’ve confronted this sort of issue before in my work on the octonions. Usually people say the Lagrangian in super-Yang-Mills theory gets its supersymmetry in d=3d = 3, 44, 66 and 1010 because of certain special Fierz identities that hold in these dimensions. However, a more illuminating explanation involves the reals, complexes, quaternions and octonions - which “just happen” to have dimensions 2 less than the above listed numbers.

The reals, complexes, quaternions and octonions are all alternative algebras - not in the counterculture sense of “alternative”, but in the technical sense: the associator

[a,b,c]=(ab)ca(bc) [a,b,c] = (ab)c - a(bc)

is completely antisymmetric. And, this fact is secretly the same as the relevant Fierz identities!

This is explained somewhat in Robert Helling’s Addendum to my week104.

Similar facts underlie the existence of the exceptional Lie algebras F4, E6, E7 and E8, which are closely related to the Lie algebras of rotations in 9, 10, 12 and 16 dimensions - which “just happen” to be 8 more than the dimensions of the reals, complexes, quaternions and octonions. I understand this pretty well.

So, I want to think of d=11d = 11 spinors in terms of the octonions, and see what the Fierz identity you mention is “really saying”.


Urs said:

see what the Fierz identity you mention is really saying

I see what you are after. While I don’t know the full answer, I could point out that a useful representation-theoretic explanation and list of the D=11D=11 Fierz identities is given in section 3 of

R. D’Auria & P. Fré
Geometric supergravity in D=11D=11 and its hidden supergroup
NPB 201 (1982) 101-140
(pdf) .

On p. 115 they write down all the pp-form terms that one might naively expect and then use an irrep decomposition given on p. 112 to show that only the 2-form and the 11-form satisfy the required identity.

So they start by observing that for A (p)A^{(p)} some pp-form, the corresponding l p+1l_{p+1} bracket (the Jacobiatoratoratoratorator…) must be of the form

ψ¯Γ a 1a p1ψV a 1V a p1, \bar \psi \wedge \Gamma^{a_1\cdots a_{p-1}} \psi \wedge V_{a_1}\wedge \cdots V_{a_{p-1}} \,, where ψ\psi are spinor-valued 1-forms and VV vector-valued 1-forms.

This is, first of all, non-vanishing only for p=2p=2, 3, 6, 7, 10 and 11.

The identity to be satisfied by this guy is (ψ¯Γ a 1a p1ψ)(ψ¯Γ a iψ)V a 2V a p1=0, (\bar \psi \wedge \Gamma^{a_1\cdots a_{p-1}} \psi) \wedge ( \bar\psi \wedge \Gamma_{a_i} \psi) \wedge V_{a_2}\wedge \cdots V_{a_{p-1}} = 0 \,, where I have put some brackets just to highlight the structure of this expression.

In other words, this says that (ψ¯Γ a 1a p1ψ)(ψ¯Γ a iψ) (\bar \psi \wedge \Gamma^{a_1\cdots a_{p-1}} \psi) \wedge ( \bar\psi \wedge \Gamma_{a_i} \psi) must be a vanishing antisymmetric rank p2p-2-tensor.

So it boils down to checking if this term may contain any (p2)(p-2)-form contributions at all. We have four gravitinos, hence the representation ((12) 5) 4((\frac{1}{2})^5)^{\otimes 4}.

This can be decomposed in bosonic reps as indicated below equation 3.2 on p. 112.

More concretely, table 2 on p. 113 shows how to realize this decomposition by contracting the ψ\psi with gamma matrices.

Either way, the result is that (up to Hodge duality) precisely the 3-form, and 4-form reps do not appear. We cannot use 4-forms, since for them, as noted above, the Jacobiator vanishes in the first place. Hence we are left with the 3-form.

Maybe you can see a deeper truth by staring at that for a while and using some facts about triality and octonions.


John said:

Next I want to say a tiny bit about Koszul duality for Lie algebras, which plays a big role in the work of Castellani on the M-theory Lie 3-algebra, which I discussed in “week237”.

Let’s start with the Maurer-Cartan form. This is a gadget that shows up in the study of Lie groups. It works like this. Suppose you have a Lie group GG with Lie algebra Lie(G)\mathrm {Lie}(G). Suppose you have a tangent vector at any point of the group GG. Then you can translate it to the identity element of GG and get a tangent vector at the identity of GG. But, this is nothing but an element of Lie(G)\mathrm{Lie}(G)!

So, we have a god-given linear map from tangent vectors on GG to the Lie algebra Lie(G)\mathrm{Lie}(G). This is called a “Lie(G)\mathrm{Lie}(G)-valued 1-form” on GG, since an ordinary 1-form eats tangent vectors and spits out numbers, while this spits out elements of Lie(G)\mathrm{Lie}(G). This particular god-given Lie(G)\mathrm{Lie}(G)-valued 1-form on GG is called the “Maurer-Cartan form”, and denoted ω\omega.

Now, we can define exterior derivatives of Lie(G)\mathrm{Lie}(G)-valued differential forms just as we can for ordinary differential forms. So, it’s interesting to calculate dωd\omega and see what it’s like.

The answer is very simple. It’s called the Maurer-Cartan equation:

dω=ωω d\omega = - \omega \wedge \omega On the right here I’m using the wedge product of Lie(G)\mathrm{Lie}(G)-valued differential forms. This is defined just like the wedge product of ordinary differential forms, except instead of multiplication of numbers we use the bracket in our Lie algebra.

I won’t prove the Maurer-Cartan equation; the proof is so easy you can even find it on the Wikipedia:

14) Wikipedia, Maurer-Cartan form,

An interesting thing about this equation is that it shows everything about the Lie algebra Lie(G)\mathrm{Lie}(G) is packed into the Maurer-Cartan form. The reason is that everything about the bracket operation is packed into the definition of ωω\omega \wedge \omega.

If you have trouble seeing this, note that we can feed ωω\omega\wedge \omega a pair of tangent vectors at any point of GG, and it will spit out an element of Lie(G)\mathrm{Lie}(G). How will it do this? The two copies of ω\omega will eat the two tangent vectors and spit out elements of Lie(G)\mathrm{Lie}(G). Then we take the bracket of those, and that’s the final answer.

Since we can get the bracket of any two elements of Lie(G)\mathrm{Lie}(G) using this trick, ωω\omega \wedge \omega knows everything about the bracket in Lie(G)\mathrm{Lie}(G). You could even say it’s the bracket viewed as a geometrical entity - a kind of “field” on the group GG!

Now, since

dω=ωω d\omega = - \omega \wedge \omega

and the usual rules for exterior derivatives imply that

d 2ω=0 d^2\omega = 0

we must have

d(ωω)=0. d(\omega \wedge \omega) = 0 \,.

If we work this concretely what this says, we must get some identity involving the bracket in our Lie algebra, since ωω\omega \wedge \omega is just the bracket in disguise. What identity could this be?

THE JACOBI IDENTITY!

It has to be, since the Jacobi identity says there’s a way to take 3 Lie algebra elements, bracket them in a clever way, and get zero:

[u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0 [u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0

while d(ωω)d(\omega \wedge \omega) is a Lie(G)\mathrm{Lie}(G)-valued 3-form that happens to vanish, built using the bracket.

It also has to be since the equation d 2=0d^2 = 0 is just another way of saying the Jacobi identity. For example, if you write out the explicit grungy formula for d of a differential form applied to a list of vector fields, and then use this to compute d 2d^2 of that differential form, you’ll see that to get zero you need the Jacobi identity for the Lie bracket of vector fields. Here we’re just using a special case of that.

The relationship between the Jacobi identity and d 2=0d^2 = 0 is actually very beautiful and deep. The Jacobi identity says the bracket is a derivation of itself, which is an infinitesimal way of saying that the flow generated by a vector field, acting as an operation on vector fields, preserves the Lie bracket! And this, in turn, follows from the fact that the Lie bracket is preserved by diffeomorphisms - in other words, it’s a “canonically defined” operation on vector fields.

Similarly, d 2=0d^2 = 0 is related to the fact that d is a natural operation on differential forms - in other words, that it commutes with diffeomorphisms. I’ll leave this cryptic; I don’t feel like trying to work out the details now.

Instead, let me say how to translate this fact:

d 2ω=0d^2 \omega = 0 IS SECRETLY THE JACOBI IDENTITY

into pure algebra. We’ll get something called “Kozsul duality”. I always found Koszul duality mysterious, until I realized it’s just a generalzation of the above fact.

How can we state the above fact purely algebraically, only using the Lie algebra Lie(G)\mathrm{Lie}(G), not the group GG? To get ourselves in the mood, let’s call our Lie algebra simply LL.

By the way we constructed it, the Maurer-Cartan form is “left-invariant”, meaning it doesn’t change when you translate it using maps like this:

L g: GG xx \begin{aligned} L_g : & G \to G \\ & x \mapsto x \end{aligned}

that is, left multiplication by any element gg of GG. So, how can we describe the left-invariant differential forms on GG in a purely algebraic way? Let’s do this for ordinary differential forms; to get Lie(G)\mathrm{Lie}(G)-valued ones we can just tensor with L=Lie(G)L = \mathrm{Lie}(G).

Well, here’s how we do it. The left-invariant vector fields on GG are just

LL

so the left-invariant 1-forms are

L *L^*

So, the algebra of all left-invariant diferential forms on GG is just the exterior algebra on L *L^*. And, defining the exterior derivative of such a form is precisely the same as giving the bracket in the Lie algebra LL! And, the equation d 2=0d^2 = 0 is just the Jacobi identity in disguise.

To be a bit more formal about this, let’s think of LL as a graded vector space where everything is of degree zero. Then L *L^* is the same sort of thing, but we should add one to the degree to think of guys in here as 1-forms. Let’s use SS for the operation of “suspending” a graded vector space - that is, adding one to the degree. Then the exterior algebra on L *L^* is the “free graded-commutative algebra on SL *SL^*”.

So far, just new jargon. But this lets us state the observation of the penultimate paragraph in a very sophisticated-sounding way. Take a vector space LL and think of it as a graded vector space where everything is of degree zero. Then:

Making the free graded-commutative algebra on SL *SL^* into a differential graded-commutative algebra is the same as making LL into a Lie algebra.

This is a basic example of “Koszul duality”. Why do we call it “duality”? Because it’s still true if we switch the words “commutative” and “Lie” in the above sentence!

Making the free graded Lie algebra on SL *SL^* into a differential graded Lie algebra is the same as making LL into a commutative algebra.

That’s sort of mind-blowing. Now the equation d 2=0d^2 = 0 secretly encodes the commutative law.

So, we say the concepts “Lie algebra” and “commutative algebra” are Koszul dual. Interestingly, the concept “associative algebra” is its own dual:

Making the free graded associative algebra on SL *SL^* into a differential graded associative algebra is the same as making L into an associative algebra.

This is the beginning of a big story, and I’ll try to say more later. If you get impatient, try the book on operads mentioned in “week191”, or else these:

15) Victor Ginzburg and Mikhail Kapranov, Koszul duality for quadratic operads, Duke Math. J. 76 (1994), 203-272. Also Erratum, Duke Math. J. 80 (1995), 293.

16) Benoit Fresse, Koszul duality of operads and homology of partition posets, Homotopy theory and its applications (Evanston, 2002), Contemp. Math. 346 (2004), 115-215. Also available at http://math.univ-lille1.fr/~fresse/PartitionHomology.html

The point is that Lie, commutative and associative algebras are all defined by “quadratic operads”, and one can define for any such operad OO a “dual” operad O *O^* such that:

Making the free graded OO-algebra on SL *SL^* into a differential graded OO-algebra is the same as making LL into an O *O^*-algebra.

And, we have O **=OO^{**} = O, hence the term “duality”.

This has always seemed incredibly cool and mysterious to me. There are other meanings of the term “Koszul duality”, and if really understood them I might better understand what’s going on here. But, I’m feeling happy now because I see this special case:

Making the free graded-commutative algebra on SL *SL^* into a differential graded-commutative algebra is the same as making LL into a Lie algebra.

is really just saying that the exterior derivative of left-invariant differential forms on a Lie group encodes the bracket in the Lie algebra. That’s something I have a feeling for. And, it’s related to the Maurer-Cartan equation… though notice, I never completely spelled out how.

Addenda: Let me say some more about how d 2=0d^2 = 0 is related to the fact that dd is a canonically defined operation on differential forms. Being “canonically defined” means that dd commutes with the action of diffeomorphisms. Saying that dd commutes with “small” diffeomorphisms - those connected by a path to the identity - is the same as saying

dL v=L vd d L_v = L_v d

where vv is any vector field and L vL_v is the corresponding “Lie derivative” operation on differential forms. But, Weil’s formula says that

L v=i vd+di v, L_v = i_v d + d i_v \,,

where i vi_v is the “interior product with vv”, which sends pp-forms to (p1)(p-1)-forms. If we plug Weil’s formula into the equation we’re pondering, we get

d(i vd+di v)=(i vd+di v)d d (i_v d + d i_v) = (i_v d + d i_v) d

which simplifies to give

d2i v=i vd 2. d2 i_v = i_v d^2 \,.

So, as soon as we know d 2=0d^2 = 0, we know dd commutes with small diffeomorphisms. Alas, I don’t see how to reverse the argument.

Similarly, as soon as we know the Jacobi identity, we know the Lie bracket operation on vector fields is preserved by small diffeomorphisms, by the argument outlined in the body of this Week. This argument is reversable.

So, maybe it’s an exaggeration to say that d 2=0d^2 = 0 and the Jacobi identity say that dd and the Lie bracket are preserved by diffeomorphisms - but at least they imply these operations are preserved by small diffeomorphisms.


Urs said:

We know that the supergravity 3-form CC should really come from Chern-Simons 3-forms of a Lorentz and an E 8E_8 connection.

This information is not present in the classical FDA formulation of supergravity. Here is a general observation on connections on Chern-Simons 2-gerbes, which might be relevant.


Consider a Lie group GG. Its Lie algebra is encoded in the fda defined by da a+12C a bca ba c=0, \mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c = 0 \,, where {C a bc}\{C^a{}_{bc}\} are the structure constant in some chosen basis. Nilpotency of d\mathbf{d} is equivalent to the Jacobi identity.

Next, consider the crossed module GGG \to G. The FDA corresponding to its Lie 2-algebra is given by da a+12C a bca ba c+b a=0 db a+C a bca bb c=0. \begin{aligned} &\mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + b^a = 0 \\ & \mathbf{d} b^a + C^a{}_{bc}a^b b^c = 0 \end{aligned} \,. A 2-connection with values in this Lie 2-algebra is given by a 1-form A aA^a and a 2-form B aB^a and has curvatures F 1 =F A+B F 2 =d AB. \begin{aligned} F_1 &= F_A + B \\ F_2 &= \mathbf{d}_A B \end{aligned} \,. Without mentioning anything like 2-connections, but implicitly considering precisely this, such 2-connections have been studied for instance in hep-th/0204059. The 2-group aspect is discussed very nicely in hep-th/0206130. Suppose we demand this 2-curvature to vanish. This is equivalent to B=F A. B = -F_A \,. Hence a flat (GG)(G\to G)-2-connection is the same as an ordinary GG-connection. In fact, a trivial flat (GG)(G\to G)-2-bundle is the same as an ordinary GG-bundle. Next, let’s also add a generator in degree 3 to our fda, to get the Lie 3-algebra encoded by da a+12C a bca ba c+b a=0 db a+C a bca bb c=0 dc+k abb ab b=0, \begin{aligned} &\mathbf{d} a^a + \frac{1}{2}C^a{}_{bc}a^b a^c + b^a = 0 \\ & \mathbf{d} b^a + C^a{}_{bc}a^b b^c = 0 \\ & \mathbf{d} c + k_{ab} b^a b^b = 0 \end{aligned} \,, where k abk_{ab} is proportional to the Killing form on the Lie algebra of GG. This ensures that d\mathbf{d} is still nilpotent. A 3-connection with values in this Lie 3-algebra is a 1-form A aA^a, a 2-form B aB^a and a 3-form CC. Its curvature is F 1 =F A+B F 2 =d AB F 3 =dC+tr(BB). \begin{aligned} F_1 &= F_A + B \\ F_2 &= \mathbf{d}_A B \\ F_3 &= \mathbf{d} C + \mathrm{tr}(B \wedge B) \end{aligned} \,. Assume again that the curvature vanishes. In the first two degrees this is, as before, equivalent to B=F AB = - F_A. In third degree it now says that dCtr(F AF A). \mathbf{d} C \propto \mathrm{tr}(F_A \wedge F_A) \,. But this means that CC itself must be, up to a closed part, the Chern-Simons 3-form of AA CCS(A). C \propto \mathrm{CS}(A) \,. Therefore a 3-connection of the above sort is the local connection of a Chern-Simons gerbe corresponding to some GG-bundle with connection. Now take GG to be the product of E 8E_8 with the Lorentz group in D=11D=11 and there we go.

Posted at August 17, 2006 5:59 PM UTC

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32 Comments & 14 Trackbacks

Re: SuGra 3-Connection Reloaded

Thanks for gathering our discussion thus far in one place. I’ll have more to say about Koszul duality and Maurer-Cartan forms for Lie n-groups in while. But for right now:

Just as 11d M-theory mysteriously reduces to 11d supergravity in the classical limit, 10d string theories reduce to 10d supergravities… but a lot less mysteriously. Can the fields in these 10d supergravities be described as 2-connections the way the fields in 11d supergravity can be described as a 3-connection? Or perhaps a 2-connection plus some other fields (since the Kaluza-Klein mechanism takes gravity in n+1 dimensions and turns it into gravity coupled to other fields in n dimensions)?

I ask this because it would be nice to have some sort of “string theory 2-group” to go along with your “M-theory 3-group”, for at least two reasons. First, I understand how the super-Poincaré algebra in 10 dimensions can be built using the octonions. I don’t really understand it in 11 dimensions. Second, I hear the supersymmetry is visible “off shell” in 10d supergravity, unlike in 11d supergravity… and this suggests the supersymmetry should be easier to see in the basic algebraic structures one is working with, before the equations of motion get into the act. In fact, I’m hoping the first and second reasons here are related.

So far my only wild guess regarding a “string theory Lie 2-algebra” is one having the superPoincaré algebra in degree 0 and R in degree 1. This should give us the graviton, gravitino and B field. What other fields are there in the various 10d supergravities, like type IIA and type IIB?

Posted by: John Baez on August 18, 2006 2:16 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

It’s an old idea of Horava (subsequently developed by Adams and Evslin) that, just as E 8E_8 seems important for classifying the fields of M-theory, because IIA is really M-theory on a circle, the group LE 8LE_8 (the free loop group) should classify the fields of IIA. This turns out to be almost correct. If you look at the semidirect product of U(1)U(1) and LE 8LE_8, then you get a pretty good classification of the fields in IIA. Uday Varadarajan and myself worked this out in hep-th/0406128, but we’re not entirely sure what to do with it. It seems in contradiction with the generally believed idea that the fields of IIA should be classified by a twisted K-theory group.

Of course, given the recent work of Freed, Moore and Segal saying that the torsion parts of fluxes are unobservable, perhaps this isn’t such a stark disagreement.

Posted by: Aaron Bergman on August 18, 2006 2:30 AM | Permalink | Reply to this

LE8

I’ll make the following observation on LE^ 8\hat LE_8-bundles and the gerbes that we are after.

First, consider the toy example of PU(H)PU(H)-bundles and line bundle gerbes.

Given a PU(H)PU(H) bundle EE on XX, with connection \nabla, we obtain a line bundle gerbe with connection as follows.

First, associate to AA the algebra of compact operators on HH to obtain the algebra bundle AXA\to X, each of whose fibers looks like compact operators on HH.

(This is actually a bundle of special symmetric Frobenius algebras with Jandl structure.)

There is a 2-transport

(1)tra:P 2(X)Bim \mathrm{tra} : P_2(X) \to \mathrm{Bim}

associated to that as follows.

To each xXx \in X assign the fiber algebra A xA_x. To each path xyx \to y associate the A xA_x-A yA_y-bimodule which is A xA_x as an object with ordinary left action and right action twisted by the action of parallel transport in EE from yy to xx.

To each surface associate one of U(1)U(1)-worth of bimodule homomorphisms obtained by right composition with a lift of the PU(H)PU(H)-curvature around the surface to a U(H)U(H) curvature.

I claim that locally trivializing this 2-transport to locally obtain a 2-transport with values in 1DVect1D\mathrm{Vect} yields the corresponding line bundle gerbe as the transition data of this trivialization.

What is at work here is really a certain rep of the 2-group

(2)U(1)1. U(1)\to 1 \,.

Every strict 2-group (HG)(H \to G) has a canonical rep on bimodules induced by any ordinary rep of HH, as described here.

This works by forming the algebra generated by the operators of the ordinary rep of HH. For U(1)U(1) this algebra is \mathbb{C}. Hence every fiber of a 2-bundle associated by means of this rep should be equivalent to the algebra \mathbb{C}, where equivalence is equivalence in Bim\mathrm{Bim}. That’s Morita equivalence and hence tells us that the fibers have to look like algebras of compact operators on some Hilbert space.

We could imagine playing the same game with the 2-group

(3)LE^ 8E 8. \hat LE_8 \to E_8 \,.

To get a rep of this 2-group on bimodules of the kind that I am talking about, we need an ordinary rep of LE^ 8\hat LE_8 - like, say, the level 1 positive energy rep.

Using this, we associate to the LE^ 8\hat LE_8-bundle PYP \to Y on 10-dimensional target space YY - which Aaron and Uday obtain from KK-reduction of the E 8E_8 bundle up in d=11d = 11, a Hilbert bundle and a 2-transport on that in analogy to the above construction.

But probably this is not quite what we need.

We really want a twisted LE 8LE_8-gerbe in 10-dimensions whose lifting gerbe is the E 8E_8-Chern-Simons gerbe up in d=11d=11. That’s the old Jurčo-Aschieri argument.

So I think we really want to have 3-transport (on XX) with values in the strict 3-group

(4)U(1)LE^ 8PE 8. U(1) \to \hat LE_8 \to PE_8 \,.

And, actually, one day I should try to check if the strict Lie 3-algebra of this guy is equivalent to the semistrict Lie 4-algebra which I promoted above.

Given that (LE 8PE 8)(1E 8)(LE_8 \to PE_8) \simeq (1 \to E_8), at least at the level of 2-groups in Set\mathrm{Set}, this 3-group contains in it all the groups that appear when we go from d=11d=11 down to d=10d=10.

The U(1)U(1)-factor controls the abelian CS-2-gerbe whose connection is the SuGra 3-form. The E 8E_8 part resolves this in terms of E 8E_8 bundles.

So we really need to apply the above reasoning, but with the domain dimensions lifted by one unit.

Let’s see. Start with an E 8E_8 on X=Y×S 1X = Y \times S^1. Take loops everywhere to get an LE 8LE_8-bundle on LXLX. Maybe we want only loops that wrap the S 1S^1, sunno.

So we now construct that associated bundle of operators on this loop bundle, with algebras associated to loops, bimodules of these (obtained from twisting with the connection on the LE 8LE_8-bundle) to surfaces and a U(1)U(1)-worth of bimodule homomorphisms to cobounding volumes.

Sorry for rambling on, I am sure nobody wants to follow that. What I am talking about is - I think - the obvious lift of the 2-reps of 2-groups on bimodules that I mentioned before, to a rep of 3-groups on Alg \mathrm{Alg}_\mathbb{C}.

Posted by: urs on August 18, 2006 12:53 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Can the fields in these 10d supergravities be described as 2-connections the way the fields in 11d supergravity can be described as a 3-connection?

Depends on how strict you interpret the condition “in the same way”, I think.

The nice thing about 11D is that there should be a single 2-gerbe encoding everything.

What we need is a good concept of doing KK-reduction of a space XX which has an nn-gerbe living over it.

If we compactify on a circle, X=Y×S 1X = Y \times S^1, the result should be an (n1)(n-1)-gerbe on YY - plus lots of other stuff.

Namely, locally, we have the connection (n+1)(n+1) form on XX. Down on YY this becomes, by the usual gymnastics, a 2-form plus other stuff.

So - I think - one reason why RR forms are comparatively harder to understand than the Kalb-Ramond 2-form is that the latter is a connection on a 1-gerbe obtained from KK-reducing a 2-gerbe, while the RR-forms are “extra stuff” that only assembles itself coherently to a 2-gerbe connection up one dimension higher after undoing the KK-reduction.

Or so I’d think.

The E 8E_8-reasoning that Aaron mentions might be a way to say the same on the level of bundles representing gerbes.

Posted by: urs on August 18, 2006 11:50 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Hence a flat (GG)(G\to G)-2-connection is the same as an ordinary GG-connection. In fact, a trivial flat (GG)(G\to G)-2-bundle is the same as an ordinary GG-bundle.

This is akin to what I was trying to tell you in Vienna - I probably wasn’t quite on the money though.

Nice to see the n-categories really getting into gear.

Posted by: David Roberts on August 18, 2006 4:03 AM | Permalink | Reply to this

Curvators

[…] what I was trying to tell you in Vienna […]

Sorry, maybe there was some misunderstanding on my part.

In fact, in Vienna I was trying to generalize the principle behind this.

The principle is this:

Given any transport 1-functor tra :P 1(X)Σ(G)\mathrm{tra}_\nabla : P_1(X) \to \Sigma(G), you can construct from it a 2-functor

(1)curv :P 2(x)Σ(GG) \mathrm{curv}_\nabla : P_2(x) \to \Sigma(G \to G)

in the obvious way, namely by transport around little surfaces.

If you regard curv A\mathrm{curv}_A again as a parallel transport (2-)functor tra=curv \mathrm{tra}' = \mathrm{curv}_\nabla, you get a gerbe with connection, instead of a “bundle with curvature” - if you allow me to put it this way.

I believe this construction generalizes.

Given a surface transport

(2)tra :P 2(X)Σ(HtG) \mathrm{tra}_\nabla : P_2(X) \to \Sigma(H \stackrel{t}{\to} G)

we know its curvature lives in ker(t)\mathrm{ker}(t). I believe we can form the 3-group (ker(t)HtG)(\mathrm{ker}(t) \to H \stackrel{t}{\to} G) and obtain a 3-functor

(3)curv :P 3(X)Σ(ker(t)HG) \mathrm{curv}_\nabla : P_3(X) \to \Sigma(\mathrm{ker}(t) \to H \to G)

from tra\mathrm{tra} in the obvious way, namely by transporting with tra\mathrm{tra} around little cubes.

I haven’t checked in detail, but I’d think this pattern continues:

From every nn-transport tra \mathrm{tra}_\nabla we naturally obtain a (n+1)(n+1)-functor curv \mathrm{curv}_\nabla by applying tra\mathrm{tra} to bounbdaries.

Schematically

(4)curv (S)=tra (S). \mathrm{curv}_\nabla(S) = \mathrm{tra}_\nabla(\partial S) \,.

I can provide more details later. I am planning to post something on that anyway.

But there is more one should look at here.

When one translates the SuGra-FDA literature into transport 3-functors, one finds that it requires weakening the notion of vurvature a little.

The point is, roughly, that the SuGra equations of motion arise by imposing Bianchi-like equations on objects that look like curvatures, but that need not actually be curvatures of any given connection.

I can make this more precise, if desired.

So we need something like curv \mathrm{curv}_\nabla, but without there actually being a \nabla that it comes from.

Sorry, this is getting a little too vague to be of any use or interest, I am afraid. I will see if I can post more details over the weekend.

Let me just roughly say this.

Regard the 2-groupoid of surface elements P 2(X)P_2(X) as a double groupoid of little squares in the obvious way.

Then a 2-functor curv:P 2Σ(GG)\mathrm{curv} : P_2 \to \Sigma(G \to G) is in particular a 1-functor curv I\mathrm{curv}_I on every of the vertical 1-categories sitting inside the double category P 2(X)P_2(X), where II is the interval that all 2-cells in this 1-category project onto.

We may ask under which conditions these 1-functors curv I\mathrm{curv}_I are naturally isomorphic to the trivial 1-functor.

This is easily seen to be the case precisely if curv\mathrm{curv} is the curvature of a connection tra\mathrm{tra}! And in this case - by the funny translation from double categories to 1-categories that is used here - one sees that the functor tra\mathrm{tra} actually acts like the natural isomorphism establishing this.

So, it is pretty straightforward to generalize this to more general 2-functors on P 2(X)P_2(X), which need not have vertical restriction 1-functors that are naturally isomorphic to the trivial one.

These guys I call curvators. The point is that functors are morphism between curvators, just like natural transformations are morphisms between functors.

And I think the category-theoretic reformulation of the equtions of motion of 11D supergravity is not quite - as one might naively expect -

“There is a 3-connection tra\mathrm{tra} with values in the SuGra Lie 3-algebra.”

but

“There is a 3-curvator curv\mathrm{curv} with values in the SuGra Lie 3-algebra.”

Hm. I guess I should write an entry on that idea, providing some diagrams. It’s really quite trivial, but I am afraid my description here may sound a little opaque.

Actually, I should say that one reason why I haven’t written about this before is that I was still hoping that by adding a bunch of auxiliary fields, one can use fake flatness to reach the same result with an ordinary connection.

This does work nicely for Chern-Simons 2-gerbes, as I describe in the entry above. But I don’t see yet if it also works for that super-Poincaré Lie 3-algebra.

Posted by: urs on August 18, 2006 11:09 AM | Permalink | Reply to this

Re: Curvators

Deligne cohomology should fit in here somewhere (I haven’t time to read your entire reply yet - on a public library computer).

Recall that a U(1) bundle with connection is given by a Deligne 1-class with values in the complex

U(1) –dlog–> \Omega^1

it’s curvature is given by a sort of coboundary.

We get the same sort of thing for U(1)gerbes, where the complex is now

U(1) –dlog–> \Omega^1 –d–> \Omega^2

and the class is a 2-class. Isn’t this just the infinitesimal version of all the transport stuff? How would we describe the map giving us the curvature in terms of the transport pseudonatural tranformation given these facts? Can we think of these complexes as infinitesimal versions of (models of) n-groups?

Must dash.

Posted by: David Roberts on August 21, 2006 2:51 AM | Permalink | Reply to this

Re: Curvators

Isn’t this just the infinitesimal version of all the transport stuff?

Yes. Deligne cohomology is the same as equivalence classes of transitions of Σ n(U(1))\Sigma^n(U(1))-transport. I’ll post an entry on that.

How would we describe the map giving us the curvature in terms of the transport pseudonatural tranformation given these facts?

I agree. It would be best to first understand this simple case. I have thought more about it over the weekend. I’ll need to type some notes, then I get back to you.

Posted by: urs on August 21, 2006 12:06 PM | Permalink | Reply to this

Re: Curvators

Here’s a point which mystified me for a bit:

A G-gerbe is the geometric object representing a 2-cocycle (or a 1-cocycle with values in GAut(G)G \to Aut(G), let’s not go there for now). However, its curvature, some sort of characteristic class if you will, satisfies a 3-cocycle equation (Breen and Messing eqn (4.1.29)) a la Dedecker.

As for the curvature of a G-bundle if one considers it as a group-valued cocycle - it satisfies a 2-cocycle equation, wheras the transition functions, which really encode the bundle, satisfy a 1-cocycle equation.

So there is something slightly higher dimensional about curvatures than the geometry they come from.

Posted by: David Roberts on August 22, 2006 6:02 AM | Permalink | Reply to this

Curvature

So there is something slightly higher dimensional about curvatures than the geometry they come from.

Yes. It should be something like

The curvature curv tra\mathrm{curv}_\mathrm{tra} of a G nG_n-nn-bundle with connection tra\mathrm{tra} is a flat trivial Aut(G n)\mathrm{Aut}(G_n)-(n+1)(n+1)-bundle with connection.

Flatness of curv tra\mathrm{curv}_\mathrm{tra} should be the statement of the Bianchi identity.

I think it is clear how it works for n=1n=1. I am currently working on the case n=2n=2.

Let G 2G_2 be a 2-group. Let Aut(G 2)\mathrm{Aut}(G_2) be its automorphism 3-group.

I want to understand flat trivial Aut(G 2)\mathrm{Aut}(G_2) 3-bundles with 3-connection. These should be the curvatures of 2-bundles with connection.

I have started working out the details in the pdf provided here.

One interesting thing is that flat trivial Aut(G 2)\mathrm{Aut}(G_2)-3-transport can be seen to allow non-vanishing “fake” (=(3-1=2)-form) curvature, whereas any G 2G_2-2-transport itself is always fake flat.

This can be easily understood by looking at what Aut(G 2)\mathrm{Aut}(G_2) is like. It is almost (but not quite, only up to some 3-isomorphisms) a strict crossed module of crossed modules, where the 1-morphisms are labeled by GG, whereas the 2-morphisms are labeled by a semidirect product GHG \ltimes H with target homomorphism

(1)t˜(g,h)=g 1t(h). \tilde t(g,h) = g^{-1}t(h) \,.

Plugging this into the equation for transport over a small surface immediately tells you that now there is not just Lie(H)\mathrm{Lie}(H)-data associated to surfaces, but also Lie(G)\mathrm{Lie}(G)-data. That’s the fake curvature.

See the pdf for the details, if you like.

And thanks for all your comments. They helped pushing me in the right direction. I’d be very interested in any further comments you might have.

As for the curvature of a G-bundle if one considers it as a group-valued cocycle - it satisfies a 2-cocycle equation, wheras the transition functions, which really encode the bundle, satisfy a 1-cocycle equation.

While essentially I agree - as described above - I would say it is important that these cocycles satisfied by the curvature are “trivial”, with a suitable notion for triviality. No? If not, I am thinking of something else than you do. It would be great if you could write down some formula in detail, so that I see precisely what you have in mind.

Posted by: urs on August 22, 2006 9:37 AM | Permalink | Reply to this

Re: Curvature

I would say it is important that these cocycles satisfied by the curvature are “trivial”

We had a little private discussion on this point.

I should clarify that I mean (the nonabelian analog of) Deligne cocycles.

As an example, take a trivial bundle gerbe with flat connection. It is trivialized by some line bundle. It’s curving is precisely the curvature of that line bundle.

The fact that the bundle gerbe has a flat connection, i.e. a globally defined closed “curving” 2-form, is an expression of the Bianchi identity of the connection on the bundle that trivializes it.

So a line bundle with connection (a Deligne 2-cocycle) has a curvature gerbe which, as a gerbe, is trivial, hence given by a trivial Deligne 3-cocycle.

That’s what I meant.

And what I am saying is that this example generalizes.

An nn-gerbe with nn-connection can be thought of as trivializing an (n+1)(n+1)-gerbe with flat (n+1)(n+1)-connection.

That (n+1)(n+1)-connection is really the curvature of the nn-connection. Being flat expresses the nn-Bianchi identity.

The interesting point of this is that in order to make sense of passing from a G nG_{n}-nn-transport tra\mathrm{tra} to a (n+1)(n+1)-transport curv tra\mathrm{curv}_\mathrm{tra}, we need to pass from the structure nn-group G nG_{n} to some (n+1)(n+1)-group G n+1G_{n+1}, which contains G nG_{n} in a suitable sense. Only then can we define the curvature curv tra\mathrm{curv}_\mathrm{tra}, evidently.

But this implies that if you want to define a curvature for an nn-functor tra\mathrm{tra}, you have to conceive tra\mathrm{tra} as a pseudo nn-functor from nn-paths to the n+1n+1-group G n+1G_{n+1}, instead of just to G nG_n.

And, I claim, it is precisely this step which allows tra\mathrm{tra} to have non-vanishing fake curvature.

And, I believe, the right G n+1G_{n+1}-group to associate to a given G nG_n-group is Inn(G n)\mathrm{Inn}(G_n), the (n+1)(n+1)-group of inner automorphisms of G nG_n.

For n=1n=1 this reproduces the example that we already discussed here, because Inn(G)\mathrm{Inn}(G) is the 2-group coming from the crossed module (GIdG)(G \stackrel{\mathrm{Id}}{\to} G).

For n=2n=2 I work out the nature of Inn(G 2)\mathrm{Inn}(G_2) in those notes and check that Inn(G 2)\mathrm{Inn}(G_2) 2-transport indeed has arbitrary fake curvature.

I am in the process of writing down more of technical details. I need a little more time.

Posted by: urs on August 23, 2006 10:19 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Suppose we demand this 2-curvature to vanish. This is equivalent to B=F AB=-F_A. Hence a flat (GG)(G\to G)-2-connection is the same as an ordinary GG-connection.

Isn’t this – and I am sorry to use a cliché – obvious? Given an ordinary 1-form AA with curvature F AF_A, define a two-form BB by B=F A.B=-F_A. Then of course d AB=0d_A B = 0 and you have a flat GGG\rightarrow G 2-connection.

Posted by: Amitabha Lahiri on August 18, 2006 11:55 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Isn’t this […] obvious?

Yes.

It does requires a thought or two to check that this also makes sense globally, but it’s not hard either.

The reason I dared to bore the café readers with such obvious statements was that it provided a jumping-off point for a not quite as obvious - albeit still elementary - statement,

namely that there is a certain Lie 4-algebra such that flat connections with values in that guy are 3-connections – but with the special property that the 3-form is locally a Chern-Simons 3-form.

That gives a rather elegant way to talk about Chern-Simons 2-gerbes - I think. They are precisely certain flat 3-gerbes.

And that’s relevant here, because one part of the SuGra 3-form is a Chern-Simons 3-form.

Posted by: urs on August 18, 2006 12:03 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

1) Do you happen to know if all the massive IIA string states can be obtained as KK modes from compactifying 11d sugrav’s 3-form field, elfbein, and gravitino on a circle?

My gut instinct is only some, not all, though I haven’t checked.

It’d be a big step forward to understand the underpinnings of M theory, and if this algebraic structure suffices to define everything, then it’d be great!

If not, if there’s more structure out there, then do you have any ideas for what form that structure should take? For example, what’s known about representation theory of super Lie 3-groups?

Maybe the extra stuff appears in such representations.

2) Along the same lines, do you understand how to compute, say, graviton-graviton scattering in that algebraic language?

3) Can you formulate the entire low-energy effective action for 11d sugrav in algebraic terms? ie, is there something that looks like a kinetic term on super Lie 3-groups from which all the action’s terms can be derived? I gather this is true of some of the terms, but not clear about the rest.

If someone could find an algebraic structure that encodes

- massless states of 11d sugrav

- all states, massive & massless, of 10d IIA string theory, obtained as KK modes of 11d theory on S 1S^1,

- all scattering amplitudes,

- and maybe even, optionally, give some concrete understanding of why m(atrix) theory should have anything to do with this,

then you’d have plausibly gotten a handle on the structure defining quantum M theory, and that would be a really important step forward in the field.

Posted by: S on August 18, 2006 10:33 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Do you happen to know if all the massive IIA string states can be obtained

Sorry, I don’t. But there might be people reading this here who do.

if this algebraic structure suffices to define everything

The standard FDA-SuGra literature is just classical theory, without any quantization conditions. In the above discussion I was talking about how one could implement these conditions.

I describe how it should work for the E 8E_8 part of the 3-form. I don’t see yet how it works for the spin connection part.

what is known about representation theory of super Lie 3-groups?

Good question. I think I do know the class of reps of 2-groups that are relevant for string theory. Generalizing this to 3-groups is on my to-do list.

But that’s just me. Maybe somebody out there knows more about this.

do you understand how to compute, say, graviton-graviton scattering in that algebraic language?

Well, so far, what the formalism gives us is a convenient way to unify the entire filed content into a single object and to naturally obtain the classical equations of motion of supergravity from that.

Can you formulate the entire low-energy effective action for 11d sugrav in algebraic terms? ie, is there something that looks like a kinetic term on super Lie 3-groups from which all the action’s terms can be derived?

It is not the action that appears naturally here, but the equations of motion.

You can find this discussed in any of the papers by Castellani and coauthor’s that are linked to above.

Roughly, you have a 4-curvature with values in the Lie 3-algebra. Imposing the Bianchi identities on this curvature is equivalent to all the equations of motion of all supergravity fields.

There are two subtleties:

1) For an ordinary curvature, the Bianchi identity is an identity, not a condition. So more precisely we have something that looks like a curvature, but does not in fact come from any connection. See my remarks on that here.

2) In addition, one needs to require the so-called “rheonomy constraints” on the curvature. These look pretty natural, but are extra conditions. They require that all curvatures have components only in external degrees of freedom (vielbein), not internal ones (e.g. spinors).

This condition looks a lot like the so-called “fake-flatness” conditions that one finds for nn-connections. I was trying to make the relation between the two concepts explicit, but have not succeeded yet.

Posted by: urs on August 18, 2006 11:25 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

I said (repeatedly even) things like

Imposing the Bianchi identities on this curvature is equivalent to all the equations of motion of all supergravity fields.

That’s wrong! And even obviously so.

I misinterpreted something I read, and didn’t properly think about it.

For details, see Castellani: Group geometric methods in supergravity and superstring theories.

Posted by: urs on August 24, 2006 11:29 AM | Permalink | Reply to this
Read the post On n-Transport, Part I
Weblog: The n-Category Café
Excerpt: The concept of n-transport and its local trivialization and transition.
Tracked: August 18, 2006 5:46 PM
Read the post n-Curvature
Weblog: The n-Category Café
Excerpt: Definition of n-curvature and the nature of "fake curvature".
Tracked: August 19, 2006 1:38 PM
Read the post On n-Transport, Part II
Weblog: The n-Category Café
Excerpt: Smooth transport, differentials of transport, and nonabelian differential cocycles.
Tracked: August 21, 2006 8:43 PM

FDA lab

In order to discuss some of the more technical questions in this business, one needs a couple of detailed examples for semistrict Lie nn-algebras and the corresponding FDAs.

I have begun compiling some of the formulas that one needs frequently: \;\;\;FDA Laboratory

In particular, in this pdf I discuss the FDA version of the Lie 3-algebra of the Lie 3-group Inn(HG)\mathrm{Inn}(H\to G) of inner automorphisms of the strict 2-group coming from the crossed module HtGH \stackrel{t}{\to} G.

Recall that we said that the most general 2-connections should take values in this beast.

I show that, indeed, a 2-connection with values in this 3-algebra is given by a Lie(G)\mathrm{Lie}(G)-valued 1-form AA and a Lie(H)\mathrm{Lie}(H)-valued 2-form BB, giving rise to

(1)β=F A+t(B) \beta = F_A + t(B)

(known as the fake curvature) and satisfying the two “Bianchi identities

(2)d Aβ=t(H) d_A \beta = t(H)

and

(3)d AH+βB=0. d_A H + \beta \wedge B = 0 \,.

So this does correctly match the discussion of principal 2-transport with curvature which I described in those notes that I linked to before.

The FDA governing this is the following.

Let (htg)(h \stackrel{t}{\to} g) be the differential crossed module corresponding to (HG)(H \to G).

Choose a dual basis {q a}\{q^a\} of g *g^* and let the structure constants of gg in that basis be C a bcC^a{}_{bc}.

Choose also a basis of hh and let the action of gg on hh have components α i aj\alpha^i{}_{aj}. Similarly, let the morphism tt have components t a it^a{}_i.

Consider the graded-commutative free algebra

(4) (g *(g *h *)h *), \bigwedge^\bullet (g^* \otimes (g^* \otimes h^*) \otimes h^* ) \,,

where the first g *g^* is in degree 1, the bracket factor is in degree 2 and the last h *h^* is in degree 3.

Denote by {q a}\{q^a\} the basis of g *g^* which lives in degree 1, by {r i}\{r^i\} the basis of g *g^* which lives in degree 2, by {s i}\{s^i\} the basis of h *h^* in degree 2 and by {t i}\{t^i\} the basis of h *h^* in degree 3.

I claim that the following defines a nilpotent differential on the above free algebra, making it an FDA:

(5) dq a+12C a bcq bq c+t a is i+r a=0 dr a+C a bcq br c+t a it i=0 ds i+α i ajq as jt i=0 dt i+α i ajq at j+α i ajr as j=0. \begin{aligned} & d q^a + \frac{1}{2}C^a{}_{bc}q^b q^c + t^a{}_i s^i + r^a = 0 \\ & d r^a + C^a{}_{bc}q^b r^c + t^a{}_i t^i = 0 \\ & d s^i + \alpha^i{}_{aj}q^a s^j - t^i = 0 \\ &d t^i + \alpha^i{}_{aj}q^a t^j + \alpha^i{}_{aj}r^a s^j = 0 \end{aligned} \,.

This looks complicated, but is in fact the only thing (up to the constant factors) that you can write down using just the structure provided by the differential crossed module.

I found this by differentiating the Lie 3-group that I computed here.

In the FDA-Lab pdf I spell out the proof of the nilpotency of the above in full detail.

So, to recap what we are trying to accomplish:

the next step is to compute the 4-algebra inn(sugra(11))\mathrm{inn}(\mathrm{sugra}(11)) of inner automorphisms of the supergravity 3-algebra.

Then it should be true, unless I am missing something, that solutions of the equations of motion of 11D sugra are in bijection with 3-connections with values in inn(sugra(11))\mathrm{inn}(\mathrm{sugra}(11)).

The Bianchi identities, which give the equations of motion, should follow automatically, precisely as in the above n=2n=2 example (and as in the well-known n=1n=1-example).

Er, that is, maybe up to those rheonomy constraints. The sugra equations of motion are equivalent to imposing the Bianchi identity on a sugra(11)\mathrm{sugra}(11)-connection and imposing something called a “rheonomy constraint”. I still have to figure out what that latter constraint means in terms of the nn-transport formalism.

But it looks like we are on the right track.

Posted by: urs on August 23, 2006 4:33 PM | Permalink | Reply to this

Re: FDA lab

While it certainly looks as if I am having a conversation with myself here, there is indeed discussion going on by private email and I am just posting parts of my reactions to the blog.

Recall that the above fda Inn(hg)\mathrm{Inn}(h\to g), describing the Lie 3-algebra of inner derivation of the Lie 2-algebra coming from the differential crossed module hgh \to g, has the property that a flat 3-connection with values in this guy correctly reproduces the information and the Bianchi identity of a non-fake flat local connection of a nonabelian gerbe.

So the next question is if transition data is correctly reproduced.

Doing this properly involves computing differentials of diagrams describing morphisms of 3-functors, as indicated at the end of this. But that’s serious work, involving drawing 3-d diagrams. I’ll do that next.

For the moment, we can make a quick check up to first order in the transition data, by using the technique of differential morphisms of nn-connections described in detail here.

Using the fda-language, a connection is just a chain map Φ\Phi from the fda to the deRham complex.

(1) Φ(q a)=A a Φ(s i)=B i Φ(r a)=β a Φ(t i)=H i. \begin{aligned} & \Phi(q^a) = A^a \\ & \Phi(s^i) = B^i \\ & \Phi(r^a) = \beta^a \\ & \Phi(t^i) = H^i \,. \end{aligned}

A morphism ϵ\epsilon between Φ\Phi and Φ\Phi' is a chain homotopy which is Φ\Phi-Leibnitz. This means it’s a degree (-1)-map defined on basis elements by

(2) ϵ(r a)=lng a ϵ(s i)=a i ϵ(r a)=0 ϵ(t i)=d i. \begin{aligned} & \epsilon(r^a) = \ln g^a \\ & \epsilon(s^i) = a^i \\ & \epsilon(r^a) = 0 \\ & \epsilon(t^i) = d^i \end{aligned} \,.

On the right we have differential forms. For the moment I am restricting attention to the case where ϵ(r a)=0\epsilon(r^a) = 0.

This assignment extends to a full degree -1 map by demanding it to be Φ\Phi-Leibnitz. This means for example

(3)ϵ(r ar br c)=ϵ(r a)Φ(r br c)+Φ(r a)ϵ(r b)Φ(r c)+Φ(r ar b)ϵ(r c). \epsilon(r^a r^b r^c) = \epsilon(r^a)\Phi(r^b r^c) + \Phi(r^a)\epsilon(r^b)\Phi(r^c) + \Phi(r^a r^b)\epsilon(r^c) \,.

You get the idea.

Now, Φ\Phi' and Φ\Phi have such a chain homotopy going between them iff

(4)ΦΦ=[Q,ϵ], \Phi' - \Phi = [Q,\epsilon] \,,

where Q:=d inn(hg)d dRQ := d_{\mathrm{inn}(h\to g)} \oplus d_\mathrm{dR} is the differential on the direct sum complex inn(hg)Ω dR\inn(h\to g)\oplus \Omega_{dR}. Since ϵ\epsilon is of odd degree, the bracket denotes the anti-commutator.

Using this formalism the infinitesimal morphism between two 2-connections is obtained simply as

(5)(AA) a=[Q,ϵ](q q)=dlng a+C a bcA b(lng c)t a ia i (A'-A)^a = [Q,\epsilon](q^q) = d \ln g^a + C^a{}_{bc} A^b (\ln g^c) - t^a{}_i a^i

and

(6)(BB) i=[Q,ϵ](s i)=da iα i aj(lng a)B j+α i ajA aa j+d i. (B' - B)^i = [Q,\epsilon](s^i) = d a^i - \alpha^i{}_{aj}(\ln g^a)B^j + \alpha^i{}_{aj} A^a a^j + d^i \,.

This is indeed the first order expansion of

(7)A=g 1Ag+g 1dgt(a) A' = g^{-1}Ag + g^{-1} dg - t(a)

and

(8)B=g 1Bg+d Aa+aa+d. B' = g^{-1}Bg + d_A a + a\wedge a + d \,.

Compare equations (6.1.4)-(6.1.6) and (6.1.18) of page 58 in Breen&Messing.

The crucial difference to the situation in the fake flat case is the 2-form dd, which appears because we passed from a Lie 2-algebra to its Lie 3-algebra of inner derivations.

In the same manner, we find the first order transition for the 2-form and 3-form curvature to be

(9)(ββ) a=[Q,ϵ]ofr a=C a bc(lng b)β ct a id i (\beta'-\beta)^a = [Q,\epsilon]\of{r^a} = -C^a{}_{bc}(\ln g^b)\beta^c - t^a{}_i d^i

and

(10)(HH) i=[Q,ϵ]oft i=dd iα i aj(lng a)H i+α i ajA ad jα i ajβ aa j. (H' - H)^i = [Q,\epsilon]\of{t^i} = d d^i - \alpha^i{}_{aj}(\ln g^a)H^i + \alpha^i{}_{aj}A^a d^j - \alpha^i{}_{aj} \beta^a a^j \,.

This is indeed the first order expansion of

(11)β=g 1βgtofd \beta' = g^{-1}\beta g - t\of{d}

and

(12)H=g 1Hg+d Adβ(a)±[a,d]. H' = g^{-1}H g + d_A d - \beta(a) \pm [a,d] \,.

Compare with equations (6.1.12) and (6.1.23) in Breen&Messing.

So everything works as expected here.

Recall that I set one of the components of ϵ\epsilon to 0 by hand above. Removing that restriction gives rise to another 1-form appearing in these transition relations, a 1-form which does not seem to have an analog in the nonabelian gerbe literature. So what we have here seems to be even slightly more general than what has been considered so far.

Posted by: urs on August 25, 2006 11:35 AM | Permalink | Reply to this
Read the post 10D SuGra 2-Connection
Weblog: The n-Category Café
Excerpt: On the Lie 2-algebra governing 10-dimensional supergravity.
Tracked: August 28, 2006 3:53 PM
Read the post n-Transport and higher Schreier Theory
Weblog: The n-Category Café
Excerpt: Understanding n-transport in terms of Schreier theory for groupoids.
Tracked: September 5, 2006 3:24 PM

soft group manifolds

Is there a definition somewhere of a soft group manifold? Castellani andPerotto refer to it as a deformation of a Lie group GG, but it is unclear if that’s whjat they mean in the math sense i.e. deform the structure constants (so really a deformation of the Lie alg) OR and extension of GG??

the Poincaré example they give would probably tell me if I knew by heart the formulas for the Poincaré algebra.

Posted by: Jim Stasheff on September 26, 2006 6:09 PM | Permalink | Reply to this

Re: soft group manifolds

Is there a definition somewhere of a soft group manifold?

My impression is that “soft group manifold” is not a definition that you may find in any math text. (Maybe I am wrong, but that’s my impression.)

What these authors mean, when they refer to “soft group manifolds”, is a manifold equipped with a collection of pp-forms such that these pp-forms satisfy something like a Maurer-Cartan equation

(1)dσ+σσ=0, d\sigma + \sigma\wedge \sigma = 0 \,,

but with a nonvanishing term on the right

(2)dσ+σσ=R. d\sigma + \sigma\wedge \sigma = R \,.

I am sure that the following sentence expresses what is meant whenever people talk about a “soft group manifold”. They mean:

a morphism of free differential graded commutative algebras

(3)(Ω (X),d dR)( V *,d V *) (\Omega^\bullet(X),d_\mathrm{dR}) \to (\wedge^\bullet V^*,d_{V^*})

(where on the left we have the deRham complex of the base space XX) which is not required to be a chain map.

You should compare this directly to what we talked about here in the context of fake flatness.

For the reasons explained there, I think that “soft group manifold” is really a placeholder for

local nn-connection with values in some Lie nn-algebra.

Posted by: urs on September 26, 2006 6:28 PM | Permalink | Reply to this
Read the post Chern-Simons Lie-3-Algebra inside derivations of String Lie-2-Algebra
Weblog: The n-Category Café
Excerpt: The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
Tracked: November 7, 2006 8:58 PM
Read the post Nicolai on E10 and Supergravity
Weblog: The n-Category Café
Excerpt: H. Nicolai on further progress in checking the hypothesis that the dynamics of supergravity is encoded in geodesic motion on a Kac-Moody group coset.
Tracked: November 29, 2006 6:40 PM

Re: SuGra 3-Connection Reloaded

This is an attempt to take notes on D’Auria & Fre’ 1982 Geometric supergrav… directly to file rather than on paper. Comments in ( ) are mine

Were they the first to use the pp-form approach?

new concept of Cartan integrable system CIS

earlier Cremmer et al version - at most spin 2 particles

motivated in terms of search for the group - though it’s probably just the Lie alg right from the start: generators with indices - 1 or 2 or 5 !! and 3 index field AA and 6 form BB

3-form is not a group 1-form - (NOT a ghost !!)

CIS means a generalized Maura-Cartan

?reduction to ordianry supergroup? (to strict Lie alg?)

Fierz identities crucial (= ?)

(2.6) i.e. dd=0d d=0 \Leftrightarrow Jacobi for a Lie alg

(2.9) arbitrary dd=0d d = 0 on a free cga thought of as forms

(their own terminology e.g. for solutions of \infty-MCE)

(soft - essentially L L_\infty)

arbitrary set of forms can be used as a `potential’

deviation of non-solution of (2.9) called curvature etc etc

contraction as functional variation (oh, of course)

geometric action means integral of a certain form on some vaguely describe `surface’

cosmo-cocycle condition is on the fist 2 terms of the action

?reduction to ordianry Lie alg? in trying to identify the manifold!!

\Leftrightarrow are the forms just products of ghosts whihc reduces to some equations the hoped for coefficints must satisfy

Fierz identities = decomp of gravitino products !! meaning apparently decomposition into irreps

then they concentrate on D=11 with additional assumptions

spin <\lt 2 implies pp-forms !!

‘Hodge dualization is a meaningless operation in the group manifold approach’ ???

3-form AA has problems so introduce 6-form B to compensate

compute - grind, crank…

D=11 is assumed, not revealed by computation

Posted by: Jim Stasheff on March 16, 2007 7:23 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

I’ll try to reply to this in more detail over the weekend. Have to run to catch a train right now. (Well, I guess I already missed the one I should have taken by replying to your questions in another thread. But now I have to run to even get the next train…)

You remarked:

(soft - essentially L L_\infty)

I think that saying “soft group-manifold” is the wrong thing to do. What these authors address as a soft group manifold is exactly and precisely an nn-connection with values in a Lie nn-algebra on some manifold.

Supergravity is not a theory on a soft group manifold, but a gauge theory of a 3-connection.

See my previous reply here for more details.

The thing is that these authors realized the surprising usefulness of (quasi-) free differential algebras in the study of supergravity, but did not know what this means. Saying “soft group manifold” is an attempt at an interpretation. I don’t think it is successful attempt, to be honest.

Posted by: urs on March 16, 2007 7:32 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

I agree completely - soft group manifold
is not helpful at all. They are in search of a group but work entirely at the (generalized) Lie algebra level a.k.a. a generalized Maurer-Cartan system.

just one quibble: in terms of the action they seek, the generalized Maurer-Cartan system (CIS) corresponds to the gauge algebra. I think the connection comes in at the action/Lagrangian/field level.

Compare Ikeda’s approach and others:
hep-th/9312059 [abs, ps, pdf, other] :
Title: Two-Dimensional Gravity and Nonlinear Gauge Theory
Authors: Noriaki Ikeda
Comments: 35 pages, phyzzx, RIMS-953
Journal-ref: Annals Phys. 235 (1994) 435-464

18. hep-th/9304012 [abs, ps, pdf, other] :
Title: General Form of Dilaton Gravity and Nonlinear Gauge Theory
Authors: Noriaki Ikeda, Izawa K.-I

As I recall, Non-linear gauge theory
`soft group manifold’

jim
apologies for missing your train

Posted by: jim stasheff on March 17, 2007 1:30 AM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

D=11 is assumed, not revealed by computation

Well, one restricts attention from the outset at D=11, but it is revealed by computation that for 11 dimensions, the super-Poincaré Lie algebra does admit a certain cocycle, which then gives rise to the existence of this 3-form.

So this computation does not tell you why we want to look at eleven dimensions in the first place, but it does reveal some aspects of why eleven dimensions are special.

Posted by: urs on March 19, 2007 2:04 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

I’ve still never seen what’s special about D=11D = 11 in this computation. I know you’ve assured me there is something special here, Urs, and I really should go back and reread the paper — but to obtain the cocycle condition, all they seem to use are some Fierz identities. These look quite generic to me: it seems they should hold in lots of dimensions (e.g. every dimension for which spinors of a given sort exist, which is a period-4 or period-8 kind of thing).

Posted by: John Baez on March 19, 2007 7:14 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

I’ve still never seen what’s special about D=11 in this computation. I know you’ve assured me there is something special here

There is little I know about this which you have not also seen in D’Auria and Fré and mentioned some place in this discussion.

But apart from real technical arguments, I’d tend to doubt that similar phenomena occur in lots of other dimensions. Of course there are supergravity theories in lower dimensions, but they essentially all come more or less from the one in 11d.

Posted by: urs on March 19, 2007 10:25 PM | Permalink | Reply to this

Re: SuGra 3-Connection Reloaded

Were they the first to use the p-form approach?

As far as I am aware, they were the first to emphasize the usefulness of the systematic application of “Cartan Integrable Systems” or qFDAs or the like in the context of supergravity and superstrings.

This point of view is famously laid out in the big textbook “Supergravity and superstrings: a geometric perspective”.

Posted by: urs on March 19, 2007 10:28 PM | Permalink | Reply to this

George Sparling, 6-D, Triality; Re: SuGra 3-Connection Reloaded

Is this the right thread, given Cartan and Triality?

Mathematician suggests extra dimensions are time-like


Posted by: Jonathan Vos Post on April 18, 2007 2:54 PM | Permalink | Reply to this

Re: George Sparling, 6-D, Triality; Re: SuGra 3-Connection Reloaded

Is this the right thread […]?

Not really, I think.

And I’d be a little careful with the statements in the paper to whose discussion you are linking.

By the way, I have taken the liberty of removing, in your comment above, the entire text of that article which you posted, and just retained the link to the site where it can be found.

Posted by: urs on April 18, 2007 4:26 PM | Permalink | Reply to this

Re: George Sparling, 6-D, Triality; Re: SuGra 3-Connection Reloaded

Itzhak Bars has written several papers on “two-time physics.” This paper seems even less sensible.

Posted by: Jacques Distler on April 18, 2007 7:19 PM | Permalink | PGP Sig | Reply to this

Re: George Sparling, 6-D, Triality; Re: SuGra 3-Connection Reloaded

The current issue (4/2007) of Physik Journal, organ of the German Physical Society, has an article titled

The Dimension of Life

by

Harald Lesch und Josef M. Gassner

whose abstract says (my ad hoc translation):

Would life be possible in a universe with more than three spatial dimensions or more than one time dimension? Did god somehow have a choice when creating space and time? These questions have occupied the minds of philosophers and scientists since Aristotle. Intelligent life is subject to very restrictive constraints, which allows to demonstrate that our 4-dimensional spacetime is really the only possible one among all conceivable combinations.

:-)

Anyway, they argue that laws of nature that give rise to ultra-hyperbolic differential equations would not, in practice, allow to make predictions, the way we are used to from hyperbolic equations.

I am not sure I follow from there to the strong statement of the last sentence of their abstract, but it is certainly true that hyperbolic signature is “exceptional” among all possible signatures in various ways, and also that 3+1 dimensions are in many ways exceptional.

Recently I saw Warren Siegel being particularly bothered by that.

Posted by: urs on April 18, 2007 7:46 PM | Permalink | Reply to this

Re: George Sparling, 6-D, Triality; Re: SuGra 3-Connection Reloaded

Thank you, Dr. Schreiber, for removing the copyright-protected text, and othwerwise editing to everyone’s benefit.

I thought that the twenty-seven dimensional exceptional Jordan algebra and purported unification with twistor theory and sheaf cohomology gave a categorical flavor, as a mere hunch. Also, I recall Triality debated in the context of a wider discussion of Daulity on this blog.

I’ve read quite a bit by Penrose, but am not familar with Sparling.

I’ve read at least one of the Itzhak Bars papers, and my wife has a draft novel based on 2-time planck-length beings just after the Big Bang. hence I lean towards Jacques Distler’s suggestion that this is closer to Science Fiction than Science, whatever abstract Math is invoked.

I’ve written crazier papers, but try to keep an open mind.

I’ll be at Caltech in an hour or two, in part to see Scott Aaronson’s talk to the CS department on Quantum Gravity Computers, Complexity, and the like.

Sparling, George A. J. “Germ of a synthesis: space-time is spinorial, extra dimensions are time-like.” Proc. R. Soc. A. doi:10.1098/rspa.2007.1839

Posted by: Jonathan Vos Post on April 18, 2007 9:01 PM | Permalink | Reply to this
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