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March 11, 2011

A Categorified Supergroup for String Theory

Posted by John Baez

My student John Huerta is looking for a job. You should hire him! And not just because he’s a great guy. He’s also done some great work.

He recently gave a talk at the School on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon:

This has got to be the first talk that combines tricategories and the octonions in a mathematically rigorous way to shed light on the foundations of M-theory! It’s a preview of his thesis.

What’s the idea?

Very roughly, the idea is to see how these 3 facts fit together:

  • The only normed division algebras have dimensions 1, 2, 4, and 8.
  • There are classical superstring Lagrangians of the simplest sort only in dimensions 3, 4, 6 and 10.
  • There are classical super-2-brane Lagrangians of the simplest sort only in dimensions 4, 5, 7 and 11.

In particular, the 8-dimensional octonions lead to a special relation between vectors and spinors in 10-dimensional spacetime. John used this to construct a ‘Lie 2-supergroup’ extending the Poincaré group in 10 dimensions. A Lie 2-group is a categorified Lie group, but John is doing the supersymmetric case. Just as ordinary groups are important for describing the motion of point particles in gauge theory, this 2-supergroup is important for describing the motion of superstrings.

Similarly, in 11 dimensions John used the octonions to construct a ‘Lie 3-supergroup’ extending the Poincaré group. This is important for describing the parallel transport of super-2-branes in 11 dimensions. And that, in turn, is probably important for M-theory.

Actually, John used the same construction starting from all four normed division algebras to get Lie 2-supergroups extending the Poincaré group in 3, 4, 6 and 10 dimensions. He used a similar construction to get Lie 3-supergroups in dimensions 4, 5, 7 and 11. But curiously, the octonionic cases seem to be the ones that lead to sensible quantum theories of superstrings and super-2-branes. That’s the main reason I’m emphasizing those cases.

There are general procedures for integrating Lie nn-algebras to get ‘stacky’ Lie nn-groups, but John uses a special procedure that gives a very nice simple model of the Lie 2-supergroups and Lie 3-supergroups he wants.

In particular, his Lie 3-supergroup is a one-object tricategory with strict inverses for everything… in the category of affine superschemes. So, it’s actually what you might call an ‘affine algebraic 3-supergroup’, by analogy with the more familiar concept of ‘affine algebraic group’.

So, if you want to hire someone who knows some serious heavy-duty math and physics, he’s your man.

For a talk that focuses on the normed division algebra and Lie superalgebra aspects, not on the nn-supergroup aspects, try:

Posted at March 11, 2011 5:55 AM UTC

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26 Comments & 1 Trackback

Re: A Categorified Supergroup for String Theory

Hi, thanks for blogging about this! But that talk doesn’t mention tricategories at all. For now, that’s only in the thesis.

Posted by: John Huerta on March 11, 2011 11:16 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Okay, I guess my problem was that I know what you meant where your slides say:

There is also a Lie 3-supergroup 2-Brane(n; 1) associated with super-2-branes.

But of course anyone who hangs out here should know that a 3-group can be defined as a kind of tricategory.

Posted by: John Baez on March 12, 2011 12:41 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

In Lisbon a few weeks back – at the conference where the above slides are taken from – Hisham, John H. and I talked about this and its applications a bit.

I think I had promised to provide a reference (which in turn had been kindly provided by Hisham) for the fermion terms in the twisted Bianchi identity for the supersymmetric Green-Schwarz mechanism. It’s here:

  • L. Bonora, M. Bregola, R. D’Auria, P. Fré K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. Tonin and D. Zanon,

    Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in D=10D=10 N=1N=1 supergravity theories (pdf)

The relevant equation is (2) or rather its enhancement to (3), which displays the twisted Bianchi identity including the fermions:

H=dB+βCS(A)γCS(ω)12CS(ψ). H = d B + \beta CS(A) -\gamma CS(\omega) - \frac{1}{2}CS(\psi) \,.

Here the first two Chern-Simons terms are the standard bosonic ones that come from the Lie algebra 3-cocycles of the form ,[,]\langle -,[-,-]\rangle on 𝔰𝔬\mathfrak{so} and 𝔰𝔲\mathfrak{su} and classify the bosonic string Lie 2-algebra-extension.

The third one – which here I schematically denote CS(ψ)CS(\psi) – is the Chern-Simons term for the fermionic 3-cocycle (V,ψ,ϕ)g(V,[ψ,ϕ])(V,\psi, \phi) \mapsto g(V,[\psi,\phi]) that John H. displays on his slide 12, compare with equation (2) in Bonora et al.

(This full CS-element includes the XX-terms in equation (3) , whose expression is releganted there to references given below that).

This means that refining to the cohesive context of SmoothSuper∞Grpd the derivation of the GS-data as differential string structures, hence as the homotopy fibers

SuperString diff tw(X) H diff 4(X) H conn(X,B(SISO×U)) 12p^ 2 super+ch^ 2 super H diff(X,B 3U(1)) \array{ SuperString_{diff}^{tw}(X) &\to& H^4_{diff}(X) \\ \downarrow && \downarrow \\ \mathbf{H}_{conn}(X, \mathbf{B}(SISO\times U)) &\stackrel{\frac{1}{2}\hat \mathbf{p}_2^{super} + \hat \mathbf{ch}_2^{super}}{\to}& \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) }

of the super-geometric differential characteristic classes, makes the cocycles in SuperString diff tw(X)SuperString_{diff}^{tw}(X) (the 2-groupoid of twisted differential super string-structures on XX) be locally given by differential form data goverened by the above twisted super-Bianchi-identity. Here the homotopy fiber

BSString * BSISO 12p super B 3U(1) \array{ \mathbf{B} SString &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SISO &\stackrel{\frac{1}{2} \mathbf{p}^{super}}{\to}& \mathbf{B}^3 U(1) }

is presented by the super-extension of the string-2-group as in John H.’s work.

I shouldn’t be posting that late at night, because I guess I am rambling. What I want to say is:

the relative coefficients for the supersymmetric pairing of the bosonic 3-cocycle that we used to concentrate on and its fermionic counterpart that John H. is looking at now is the γ=132\gamma = - \frac{1}{32} in equation (2) from that article. Accordingly the full superstring Lie 2-algebra is the extension of 𝔰𝔦𝔰𝔬\mathfrak{siso} classified by the full 3-cocycle

γ,[,] bos+12,[,] fermCE(𝔰𝔦𝔰𝔬). \gamma\langle-,[-,-]\rangle_{bos} + \frac{1}{2} \langle-,[-,-]\rangle_{ferm} \in CE(\mathfrak{siso}) \,.

Posted by: Urs Schreiber on March 11, 2011 11:37 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Thanks! What’s the simplest way to see what makes γ=1/32\gamma = -1/32 special?

Posted by: John Baez on March 12, 2011 12:46 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

What’s the simplest way to see what makes γ=132\gamma = - \frac{1}{32} special?

As you know, these prefactors are generally a tremendous source of headaches, that come in two stages: first those related to the sorting out of all conventions (your own and those of all the references) and second those coming from the actual computation.

I don’t think I can give an easy way to see why it is that number and not another. It is fixed by the given fermion anomaly term in 10 D sugra and the way it has to be canceled in the GS mechanism. At the moment I am just trusting Bonora et al. that they can count correctly.

The main message for the moment is that both the bosonic and the fermionic cocycle need to be added together, in a uniquely fixed way. (Where “need to” is in the sense of: such that the corresponding super-Lie 2-group is precisely such that lifts of differential cocycles from SISOSISO to it correspond exactly to the obstructions/anomalies seen in string physics).

Posted by: Urs Schreiber on March 12, 2011 1:40 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Okay, thanks. Someday there should be a really nice explanation of this particular linear combination of cocycles… some intrinsic sense in which the resulting 2-supergroup is ‘better’ than other 2-supergroups extending the Poincaré supergroup.

Posted by: John Baez on March 12, 2011 6:07 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

I have just discovered the existence of this recent article

  • Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries arXiv:1011.4731

and the series of article by Brandt cited there (and now in the nnLab entries on super Poincaré Lie algebra and Lie algebra cohomology).

I haven’t had a chance to have a closer look. Can you say how this would relate to your and John H.’s work?

Posted by: Urs Schreiber on March 12, 2011 1:24 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

I’ve seen that article, but John Huerta has read it more carefully than I have.

Part of the difference is that they use the computer algebra system LiE to compute the cohomology of superPoincaré Lie algebras for spacetimes of dimension 6 and 10, while in SUSY2 we used normed division algebras to give a unified proof that certain 3-cocycles in dimensions 3, 4, 6, and 10 and 4-cocycles in dimensions 4, 5, 7 and 11 are nontrivial.

Of course, I don’t claim that this is a complete account of what they do!

Posted by: John Baez on March 12, 2011 6:14 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Hi John

As the split octonions and bioctonions are also simple alternative algebras, the 3ψ3-\psi’s rule should also hold for two-component spinors over these algebras, hence any composition algebra over 𝕂=,\mathbb{K}=\mathbb{R},\mathbb{C}. This generalization might even be necessary to properly represent the constant two-component Majorana spinors used in supersymmetry transformations on the worldsheet. Such Majorana spinors have Grassmann number components, so any composition algebra representation of the components must necessarily contain zero divisors.

Posted by: Mike Rios on March 12, 2011 11:42 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Hi Mike,

The way to incorporate Grassmann-valued spinors into this formalism is simply to tensor the spinors SS with a Grassmann algebra:

Λ NS\Lambda_N \otimes S.

But your observation is certainly correct: the 3-ψ\psi’s rule does work over the octonions, 𝕆\mathbb{O}, the split-octonions 𝕆\mathbb{O}', and the bioctonions, 𝕆\mathbb{C} \otimes \mathbb{O}. For the later case, the spinors are the complexification of the Majorana-Weyl spinors in 9+1 spacetime, so it’s just the Weyl spinors. For the split octonions, the spinors are for 5+5 spacetime thanks to the signature of the quadratic form on 𝕆\mathbb{O}' being 4+4. I can explain this more carefully if you are interested.

Posted by: John Huerta on March 13, 2011 12:57 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Sounds great. Indeed, the (9,1) and (5,5) signatures are quickly recovered from the determinant over the degree two Jordan algebras J(2,𝕆)J(2,\mathbb{O}) and J(2,𝕆 s)J(2,\mathbb{O}_s), which are subalgebras of J(2,𝕆 )J(2,\mathbb{O}_{\mathbb{C}}). Instead of tensoring, have you considered using the bioctonions to represent Grassmann numbers? For example, using the octonion imaginary units, one can define generators of the form θ=e n+ie m\theta=e_n+i e_m (nmn\neq m), which satisfy θ iθ j=θ jθ i\theta_i\theta_j=-\theta_j\theta_i, θ 2=0\theta^2=0 and cθ=θcc\theta=\theta c, for cc\in\mathbb{R}. This would provide an ‘in-house’ construction of two-component Grassmann valued spinors that might be useful for supersymmetry transformations.

Posted by: Mike Rios on March 13, 2011 2:26 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Right. Movshev et al. use the LiE software to decompose the cochain groups of the supertranslation algebras into irreps of 𝔰𝔬(n)\mathfrak{so}(n), for n=6n=6 and n=10n=10. They use this information to prove the cohomology rings are as stated on p. 4 and p. 6.

The relevant piece of cohomology is, in both cases:

H 4,1H^{4,1}

This contains an 𝔰𝔬(n)\mathfrak{so}(n)-invariant piece. This is generated by the fermionic Chern-Simons term you mention above, and it is the only part of the supertranslation algebra cohomology to contribute to 𝔰𝔦𝔰𝔬(n)\mathfrak{siso}(n)’s cohomology.

Posted by: John Huerta on March 13, 2011 12:38 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Thanks. I suppose I should have known this, either from reading your articles or from what you had told me.

Okay, so I gather the situation can be summarized as follows: the computation with the normed division algebras shows how some exceptional cocycles on 𝔰𝔦𝔰𝔬\mathfrak{siso} arise naturally, while the calculuation by Movshev et al. serves to demonstrate in dim 4 and 6 that this construction indeed already exhausts all the exceptional cocycles that there are.

Right?

What about the computations by Friedemann Brandt cited by Movshev et al.?

Posted by: Urs Schreiber on March 14, 2011 10:03 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

John H. wrote:

H 4,1H^{4,1}

What does the 5th cohomology have to do with this business?

Or are you using H p,qH^{p,q} to mean cohomology classes of cocycles that have pqp-q even arguments and qq odd ones? I’d have used it to mean cohomology classes of cocycles with pp even arguments and qq odd ones.

Posted by: John Baez on March 14, 2011 2:40 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

What does the 5th cohomology have to do with this business?

It’s secretly really 3rd cohomology, just that these authors work in a filtration they find more conveniently adapted to studying the classification:

for ωCE(𝔰𝔦𝔰𝔬)\omega \in CE(\mathfrak{siso}) a homogeneous element in the Chevalley-Eilenberg algebra (such as the 3-cocycle (,[,])(-,[-,-]) that we are discussing ) they write

  • nn for the number of ordinary Lie algebra generators it is dual to;

  • mm for the number of odd Lie algebra generators;

  • and then set k:=m+2nk := m + 2n

and then list the cohomology groups H k,nH^{k,n}.

So the degree n+mn + m that you expect to see is

n+m=kn n + m = k - n

and hence their H 4,1H^{4,1} does sit in the total H 3H^3.

Posted by: Urs Schreiber on March 14, 2011 4:47 PM | Permalink | Reply to this
Read the post Higher Gauge Theory, Division Algebras and Superstrings
Weblog: The n-Category Café
Excerpt: See the slides for a talk explaining John Huerta's work on higher gauge theory, division algebras and superstrings.
Tracked: March 22, 2011 8:42 AM

Re: A Categorified Supergroup for String Theory

Some thoughts about immediate next questions, with straightforward but rewarding albeit maybe somewhat tedious answers:

Leonardo Castellani once observed (see the discussion on the nnLab here) that (paragraphed) the inner automorphism \infty-Lie algebra of the supergravity Lie 3-algebra is, in degree 0, the polyvector extension of the super-Poincaré algebra known as the “M-algebra” with polyvector 2-brane charge.

I think it is clear that analogously the automorphism \infty-Lie algebra of the full supergravity Lie 6-algebra gives the full M-algebra including the polyvector 5-brane charge. It’s clear, but writing out all the prefactors correctly is tedious.

And by the same reasoning, it follows that all the higher central extensions of the super Poincaré Lie algebra induced by the exceptional super Lie algebra cocycles that you describe in terms of division algebras have automorphism \infty-Lie algebras that in degree 0 are polyvector extensions of the super-Poincaré Lie algebra (see the references here). It is a natural guess that actually the full classification of the latter (in the relevant cases) is reobtained this way from the classification of the exceptional super Lie algebra cocycles. Somebody should check this, it would considerably clarify the role of polyvector extensions of the super Poincaré-Lie algebra.

I am not sure if Castellani’s observation has found due recognition as for what it really is, maybe not even in his article.

Posted by: Urs Schreiber on March 25, 2011 1:38 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

I am currently trying to fine-tune my ambient cohesive topos for supergeometry (eventually to be discussed here).

My current impression is that the most clear-sighted account of what the standard lore of superalgebra and supergeometry is about is that exposed by Christoph Sachse in his thesis

  • A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

following directly Molotkov, but also Schwarz and Voronov (I have tried to collect the relevant refrences here).

Paraphrased in my words, what these authors argue is that superalgebra and supergeometry is most naturally thought of as mathematics over the base topos on the site of superpoints.

(The article by Balduzzi-Carmeli-Fioresi that I think John H. told me he is following is also in this school of thought, they reconsider the story for superpoints with possibly higher order nilpotent ideals in their function algebras)

I have collected an indication of how this perspective is useful at superalgebra, effectively by highlighting some aspects from Christoph Sachse’s thesis.

But I think if we accept this perspective, then it is also clear that we can and should go a bit further. For instance we want to answer the question

What is a smooth super nn-group ?

by saying:

An nn-truncated \infty-group object in the cohesive \infty-topos SmoothSuperGrpdSmoothSuper\infty Grpd.

for some sensible definition of the latter.

An obvious guess is that we want to take SmoothSuperGrpdSmoothSuper\infty Grpd to be \infty-sheaves over the large site of smooth supermanifolds, or equivalently over its dense small sub-site of super-Cartesian spaces { p|q}\{\mathbb{R}^{p|q}\}. But if there is something to the Molotkov-Schwarz-Voronov school as nicely exposed by Sachse, then this suggests that we should be doing something slightly different: work over the base \infty-topos of bare super \infty-groupoids. This would imply that smooth super \infty-groupoids would live in

SmoothSuperGrpd :=Sh (CartSp,SuperGrpd) Sh (CartSp×SuperPoint), \begin{aligned} Smooth Super \infty Grpd &:= Sh_{\infty}(CartSp, Super\infty Grpd) \\ & \simeq Sh_\infty(CartSp \times SuperPoint) \end{aligned} \,,

where the Cartesian product site of Cartesian spaces with superpoints appears, instead of the “semidirect product category” of Cartesian superspaces regarded as a full subcategory of supermanifolds.

I have been trying to make up my mind about which setup to consider. Currently I am tending towards the above definition. On top of this being in line with the above, what motivates me is that if SmoothSuperGrpdSmoothSuper\infty Grpd is defined this way, then the intrinsic notion of \infty-connection does automatically reproduce the rheonomy constraint in D’Auria-Fré formulation of supergravity as the “second Ehresmann condition”.

So this makes me tend towards the perspective of “supergeometry as geometry over the base topos on superpoints”. But I still feel I need to better understand some subtleties. For instance the formal Lie integration of a super Lie algebra in this context yield in general not a delooped group object, but a genuine groupoid object: in degree 0 it has not the point, but a superpoint.

Now, over the base topos on superpoints that is not too shocking, but it makes me wonder how the Lie integration process here matches with other constructions such as John H. is considering.

Does John H. have (or if he is reading here: do you have) an abstract definition of the integration of a super Lie nn-algebra such that with your construction you can show something like “My construction does model this abstract definition.”?

Posted by: Urs Schreiber on March 25, 2011 3:20 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Hi, Urs. I’m reading!

I do not have a nice abstract definition of integration, just this funny procedure that works for the special case that I consider. The special case is this:

Integrate a slim, nilpotent Lie nn-superalgebra to a slim, nilpotent Lie nn-supergroup.

A slim, nilpotent Lie nn-superalgebra is an L L_\infty-superalgebra with zero differential that is trivial except for degrees 0 and n1n-1, and which is nilpotent in the usual sense (nested brackets are eventually zero).

This reduces to the following data: a nilpotent Lie superalgebra 𝔫\mathfrak{n} with a representation 𝔥\mathfrak{h} and an 𝔥\mathfrak{h}-valued Lie superalgebra (n+1)(n+1)-cocycle on 𝔫\mathfrak{n}.

A slim, Lie nn-supergroup is a skeletal, globular (n+1)(n+1)-groupoid internal to supermanifolds, with the one-point supermanifold as objects, trivial morphisms except at level 1 and level nn, and no coherence ismorphisms besides the associatorator…ator. It is nilpotent in the sense that the Lie supergroups of 1-morphisms and nn-morphisms are both nilpotent.

This reduces to the following data: A nilpotent Lie supergroup NN which acts by automorphism on an abelian Lie supergroup HH, and an HH-valued Lie supergroup (n+1)(n+1)-cocycle on NN.

I like Sachse’s article about supergeometry a lot. It’s certainly the clearest one I have read, though I have been thinking about the same kind of thing a lot to work on my thesis, and that helps to make it clear. The equivalence between superalgebra over \mathbb{R} and ordinary algebra over what Sachse calls ¯\overline{\mathbb{R}} is a cornerstone of my thesis which I have never seen quantified so clearly elsewhere.

Now, a question for you. Let GG be a Lie supergroup with Lie superalgebra 𝔤\mathfrak{g}. You talk of applying the general procedure for integrating Lie \infty-algebras to 𝔤\mathfrak{g}. Let’s call the \infty-supergroup we get out of this 𝒢\mathcal{G}.

You say:

For instance the formal Lie integration of a super Lie algebra in this context yield in general not a delooped group object, but a genuine groupoid object: in degree 0 it has not the point, but a superpoint.

To me, this means that 𝒢\mathcal{G} differs from the shifted supergroup BG\mathbf{B}G in that BG\mathbf{B}G has a single point as its objects, while 𝒢\mathcal{G} has some superpoint as objects. But if I hand you a Lie superalgebra 𝔤\mathfrak{g}, can you tell me which superpoint I’ll get as the objects of 𝒢\mathcal{G}?

Posted by: John Huerta on March 27, 2011 1:56 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

John H. wrote:

I do not have a nice abstract definition of integration, just this funny procedure that works for the special case that I consider.

Public displays of modesty are risky: sometimes people fail to see through them.

It’s not really a ‘funny’ procedure: it’s a very beautiful and sensible procedure. For starters, it’s a very concrete and geometrically well-motivated way to get smooth Lie group nn-cocycles from Lie algebra nn-cocycles, which works in the special case when the exponential map is a diffeomorphism. This is something that people can enjoy even if they don’t enjoy nn-categories.

It might be nice to prove a theorem of the sort ‘this construction realizes some nice abstract definition of integration’. I bet Urs could provide the nice abstract definition.

It might also be nice, and perhaps easy, to prove a theorem of this sort:

For Lie groups whose exponential map is a diffeomorphism, you have a map from the cochain complex for Lie algebra cohomology to the cochain complex for smooth Lie group cohomology that is an inverse, up to chain homotopy, of some obvious map going the other way.

Hmm. Maybe you’ve already proved this and I’ve just forgotten. But anyway, then you could try to translate such a result into something like this:

For Lie groups GG whose exponential map is a diffeomorphism, you have a map from slim Lie nn-algebras having 𝔤\mathfrak{g} as their Lie algebra of objects to slim Lie nn-groups having GG as their Lie group of objects that is an inverse, up to equivalence, to some obvious map going the other way.

(If proving this for arbitrary nn seems tiresome, you could prove it for the cases you’re focused on, namely n=2n = 2 and n=3n = 3.)

This would be another way of making precise the sense in which your construction ‘works’.

And of course you can stick ‘super’ everywhere, too.

Posted by: John Baez on March 27, 2011 2:33 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

This brings to mind the van Est triumvirate - comparing H*Lieg with H*topG with H*continuous group for a Lie group G.

?? Is the restriction to thin just because otherwise the technicalities become unbearable?

Posted by: jim stasheff on March 27, 2011 4:12 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Jim wrote:

This brings to mind the van Est triumvirate…

Yes, that’s exactly what it’s about! I started talking about that triumvirate back in Theorem 59 of HDA6, where the fact that

H smoothgroup 3(G,U(1))=0 H^3_{\mathrm{smooth} \mathrm{group}}(G,\mathrm{U}(1)) = 0

for GG connected and compact was used to show that we can’t find a model of a certain simple-minded sort for the string 2-group. This problem was solved in various ways later, basically by considering more sophisticated notions of Lie 2-group. But Huerta is considering situations where the smooth (super)group cohomology is nonzero and indeed isomorphic to the Lie (super)algebra cohomology, so these more sophisticated notions aren’t required.

Is the restriction to slim just because otherwise the technicalities become unbearable?

John H. is only constructing Lie nn-supergroups from slim Lie nn-superalgebras, meaning those with two nonzero terms, L 0L_0 and L n1L_{n-1}, and vanishing differential. This keeps the technicalities down to a bare minimum… but it still includes the examples he actually cares about.

Posted by: John Baez on March 28, 2011 9:23 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

But if I hand you a Lie superalgebra 𝔤\mathfrak{g}, can you tell me […]

Right, let me spell that out. In fact I believe what I said above was misled, and that what I say now will harmonize with what you have:

First briefly recall the story of Lie integration for ordinary L L_\infty-algebras: given an L L_\infty-algebra 𝔤\mathfrak{g} with Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}), the bare \infty-groupoid (not remembering mooth structure) that it universally integrates to (“universally” in the sense of “universal cover”, the actual nn-groupoids that we are interested in are truncations of this) is the Kan complex

exp(𝔤):[k]{Ω (Δ k)CE(𝔤)}. \exp(\mathfrak{g}) : [k] \mapsto \{\Omega^\bullet(\Delta^k) \leftarrow CE(\mathfrak{g})\} \,.

This becomes a smooth \infty-groupoid by remembering what the UU-parameterized smooth families of kk-morphisms are, and these we take to be

exp(𝔤):(U,[k]){Ω vert (U×Δ n)CE(𝔤)}, \exp(\mathfrak{g}) : (U, [k]) \mapsto \{\Omega^\bullet_{vert}(U \times \Delta^n) \leftarrow CE(\mathfrak{g})\} \,,

where on the left now we have the complex of differential forms on U×Δ nU \times \Delta^n that 1. have sitting instants towards the boundaries of the simplex and 2. are vertical with respect to the projection U×Δ nUU \times \Delta^n \to U.

(For all details see the above link.)

Now we generalize this to super-L L_\infty-algebras 𝔤\mathfrak{g}. First consider again the bare case, without smooth, but with superstructure.

Then by Molotkov-Schwarz and others we are to produce a simplicial presheaf on super points to get the bare super \infty-groupoid

exp(𝔤):( 0|q,[k]){Ω vert ( 0|q×Δ n)CE(𝔤)}, \exp(\mathfrak{g}) : (\mathbb{R}^{0|q}, [k]) \mapsto \{\Omega^\bullet_{vert}(\mathbb{R}^{0|q} \times \Delta^n) \leftarrow CE(\mathfrak{g}) \} \,,

whre again the “vertical” is with respect to the evident projection. So if we write Λ q:=C ( 0|q)\Lambda_q := C^\infty(\mathbb{R}^{0|q}) for the Grassmann algebra of functions on this superpoint, this is

exp(𝔤):( 0|q,[k]){Λ q Ω (Δ n)CE(𝔤)}. \exp(\mathfrak{g}) : (\mathbb{R}^{0|q}, [k]) \mapsto \{\Lambda_q \otimes_{\mathbb{R}}\Omega^\bullet(\Delta^n) \leftarrow CE(\mathfrak{g}) \} \,.

Finally, to make this super simplicial set a smooth super simplicial set, hence a smoth super \infty-groupoid we add again the probes by smooth test spaces U= nU = \mathbb{R}^n to get in total

exp(𝔤):( n, 0|q,[k]){Ω vert ( n|q×Δ n)CE(𝔤)}, \exp(\mathfrak{g}) : (\mathbb{R}^n , \mathbb{R}^{0|q}, [k]) \mapsto \{\Omega^\bullet_{vert}(\mathbb{R}^{n|q} \times \Delta^n) \leftarrow CE(\mathfrak{g}) \} \,,

where the verticality of the forms is with respect to the full projection n|q×Δ k p|q\mathbb{R}^{n|q} \times \Delta^k \to \mathbb{R}^{p|q}.

This way, we indeed get that this thing is connected, hence is of the form

=BG \cdots = \mathbf{B} G

for GG a smooth super \infty-group integrating 𝔤\mathfrak{g}.

To give an example: for 𝔤\mathfrak{g} the super Poincaré Lie algebra for which CE(𝔤)CE(\mathfrak{g}) is generated from (e a)(e^a) and (ω ab)(\omega^{a b}) in degree (1,even)(1,even) and from (ψ)(\psi) in degree (1,odd)(1,odd) with the differential defined by

dω ab=ω a cω cb d \omega^{a b} = \omega^{a}{}_c \wedge \omega^{c b}

de a=ω a be b+ψ¯Γ aψ d e^a = \omega^{a}{}_b \wedge e^b + \bar \psi \Gamma^a \wedge \psi

dψ=14ω abΓ abψ d \psi = \frac{1}{4} \omega^{a b}\Gamma_{a b} \wedge \psi

we have over the trivial superpoint q=0q = 0 that exp(𝔦𝔰𝔬)\exp(\mathfrak{iso}) is the ordinary universal Lie integration of the Poisson Lie algebra (all ψ\psis always have to map to 0-forms), whereas over the odd line we find in

exp(𝔦𝔰𝔬) n|1,[1] \exp(\mathfrak{iso})_{\mathbb{R}^{n|1}, [1]}

also contributions Ω vert ( n|1×Δ 1)\Omega^\bullet_{vert}(\mathbb{R}^{n|1} \times \Delta^1 ) of the form

ψθfdt, \psi \mapsto \theta f d t \,,

where fC ( n)f \in C^\infty(\mathbb{R}^n), where θ\theta is the canonical coordinate on 0|1\mathbb{R}^{0|1} and tt that on Δ 1\Delta^1.

In other words, for fixed qq the remaining simplicial presheaf on n\mathbb{R}^ns is that which integrates the ordinary (bosonic) Lie algebra (𝔤Λ q) even(\mathfrak{g} \otimes \Lambda_q)_{even}.

(All this should be obvious, I am just saying it so that we all know what I mean. )

Posted by: Urs Schreiber on March 28, 2011 5:27 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

In other words, for fixed qq the remaining [\infty-groupoid] is that which integrates the ordinary (bosonic) Lie algebra (𝔤 kΛ q) even(\mathfrak{g} \otimes_k \Lambda_q)_{even}.

In particular it follows from this that your definition of Lie integration of super L L_\infty-algebras (assumed to be concentrated in lowest degree on the super Lie algebra of a contractible super Lie group and in one single more degree) coincides with the universal one that I just recalled: because each (𝔤 kΛ q)(\mathfrak{g}\otimes_k \Lambda_q) is an ordinary L L_\infty-algebra with these properties, and hence André’s theorem 6.4 applies to each exp((𝔤 kΛ q) even)\exp((\mathfrak{g}\otimes_k \Lambda_q)_{even}).

Posted by: Urs Schreiber on March 28, 2011 9:30 PM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

Your last sentence sheds the most light on this for me: you’re saying that, for fixed qq, we get the ordinary smooth \infty-groupoid integrating (𝔤Λ q) even(\mathfrak{g} \otimes \Lambda_q)_\mathrm{even}. This is exactly what I would expect.

Posted by: John Huerta on March 29, 2011 4:25 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

This is exactly what I would expect.

Yes, it looks right. But – unless I am mixed up here – it is different from something else that one might also expect : saying that a smooth super \infty-groupoid is a presheaf on smooth manifolds (or Cartesian spaces) with values in simplicial presheaves on superpoints – for instance a diffeological super-\infty-groupoid – is different from saying that a smooth super \infty-groupoid is presented by a simplicial presheaf on the category of supermanifolds, or super-Cartesian spaces, such as super-diffeological \infty-groupoids.

In the first case the site of probes that test the smooth and super structure is the Cartesian product category CartSp×SuperpointCartSp \times Superpoint whose objects are p|q\mathbb{R}^{p|q}s but whose morphisms are restricted to be products of a morphism p|0 p|0\mathbb{R}^{p|0} \to \mathbb{R}^{p'|0} and a morphism 0|q 0|q\mathbb{R}^{0|q} \to \mathbb{R}^{0|q'}.

In the second case, which might seem like a more natural guess from this point of view, the site is the “semidirect product category” CartSpSuperPointCartSp \ltimes SuperPoint: the full subcategory of supermanifolds on p|q\mathbb{R}^{p|q}s.

This is why I said “I have been trying to make up my mind” on what to expect here.

Coming back to practical applications: as we have seen now, the Lie integration of the full 𝔰𝔲𝔭𝔢𝔯𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{superstring} Lie 2-algebra coming from the full 3-cocycle μ bos+μ ferm\mu_{bos} + \mu_{ferm} on 𝔰𝔦𝔰𝔬\mathfrak{siso} is super-point-wise an ordinary String-like 2-group extension of the ordinary simply connected Lie group G q:=Ωτ 1exp((𝔰𝔦𝔰𝔬Λ q) even)G_q := \Omega \tau_{\leq 1} \exp((\mathfrak{siso}\otimes \Lambda_q)_{even}) integrating the ordinary Lie algebra (𝔰𝔦𝔰𝔬Λ q) even(\mathfrak{siso}\otimes \Lambda_q)_{even}.

Hm, now what does this give? What are the periods of the 3-form μ bos( 0|q)+μ ferm( 0|q)\mu_{bos}(\mathbb{R}^{0|q}) + \mu_{ferm}(\mathbb{R}^{0|q}) in G qG_q? Those of μ bos\mu_{bos} still span the lattice of integers, and μ ferm\mu_{ferm} should have vanishing periods, right?

If so, we get super-point-wise an ordinary String-extension

BU(1)SuperString qG q \mathbf{B}U(1) \to SuperString_q \to G_q

and thus find the smooth super 2-group SuperString:SuperPoint opSmooth2GrpSuperString : SuperPoint^{op} \to Smooth2Grp.

Now, each SuperString qSuperString_q should be a semidirect product of sorts of the ordinary StringString-2-group with a topologically contractible piece. My understanding is that what you discuss currently is that topologically contractible piece. Therefore probably to get the full SuperstringSuperstring-2-group from this it would be sufficient to make the action of the ordinary StringString-2-group on that contractible piece explicit and then form the corresponding semidirect product 2-group.

But that’s just a rough idea, I haven’t thought this through.

Posted by: Urs Schreiber on March 29, 2011 11:37 AM | Permalink | Reply to this

Re: A Categorified Supergroup for String Theory

John Huerta’s thesis is now on the arXiv:

Division Algebras, Supersymmetry and Higher Gauge Theory (arXiv:1106.3385)

Posted by: Urs Schreiber on June 21, 2011 8:36 AM | Permalink | Reply to this

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