AKSZ-Models in Higher Chern-Weil Theory

August 6, 2011

AKSZ-Models in Higher Chern-Weil Theory

Posted by Urs Schreiber

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We would like to ask for comments on an early version of an article that we are writing:

Domenico Fiorenza, Chris Rogers, U.S., A higher Chern-Weil derivation of AKSZ $\sigma$ -models (pdf)

but before I say what this is about (below the fold) here some background meant to put our theorem into perspective.

In the previous entry I gave a rough indication of the original definition of the class of topological sigma-model quantum field theories called AKSZ models .

This class coincides in dimension 2 with the class of Poisson sigma-models – which in turn contains the A-model and the B-model – and in dimension 3 with the class of Courant sigma-models – which in turn contains the class of ordinary Chern-Simons theory as the special case where the base of target space is the point.

Therefore it is clear that the AKSZ models are some noteworthy type of generalization of Chern-Simons theory. Here I want to discuss a precise sense in which this is true systematically and give an alternative definition of the AKSZ models that identifies them as a canonical construction in abstract higher Chern-Weil theory. In fact, the claim is that the action functional that defines the AKSZ models is precisely the value of the higher Chern-Weil homomorphism with values in”secondary characteristic classes” and applied to a binary and non-degenerate invariant polynomial on any L-infinity algebroid.

This in turn shows that the class of AKSZ models itself is only a special case of something more general which exists on very general abstract grounds, and which we call infinity-Chern-Simons theory : this is defined for every invariant polynomial on every $L_\infty$ -algebroid. Aspects of this I had mentioned before: this larger class contains of course higher dimensional abelian Chern-Simons theories (these come from the canonical invariant polynomial on line Lie n-algebras) but for instance also the class of infinity-Dijkgraaf-Witten theories with sub-classes such as ordinary Dijkgraaf-Witten theory and the Yetter models, and also for instace higher Chern-Simons supergravity.

Therefore all these topological $\sigma$ -models (and many more that haven’t been given names yet) are incarnations of one single phenomenon: the higher Chern-Weil homomorphism. This exists on entirely abstract grounds in every cohesive ∞-topos. Therefore, in a sense, all these types of $\sigma$ -models have an existence from “first principles”.

This is maybe noteworthy, since many of these topological QFTs (maybe all of them?) play a role in the description of genuine physics via the holographic principle: for instance the 2d Poisson $\sigma$ -model as well as the A-model holographically encode ordinary quantum mechanics of particles (= 1-dimensional non-topological QFT), then 3-dimensional Chern-Simons theory holographically encodes the quantum mechanics of non-topological strings and generally higher dimensional Chern-Simons theory in dimension $D = 4k+3$ (for $k \in \mathbb{N}$ ) holographically encodes self-dual higher gauge theory in dimension $d = 4k+2$ (at least in the abelian case), such as the effective type II-superstring QFT in $d = 10$ – which in turn is famously thought to have vacua that look like the standard model of observed particle physics.

Due to all these relations it should be interesting to see that and how AKSZ $\sigma$ -models are a special class of $\infty$ -Chern-Simons theories, too. This I have tried to work out with Domenico Fiorenza and Chris Rogers. We now have an early writeup and would enjoy to hear whatever comments you might have:

A higher Chern-Weil derivation of AKSZ $\sigma$ -models (pdf)

The essence of our main theorem is easily stated. See below.

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Our main observation comes down to the following simple statement.

After all the dust has settled…

(… after the abstract infty-Chern-Weil homomorphism has been presented in a model structure on simplicial-presheaves, after the objects appearing there have in turn be constructed by Lie integration of $L_\infty$ -algebroid valued forms, after their resolutions have been constructed and after the correspondences/infinity-anafunctors modelling the CW morphim have been built… )

…after all this dust has settled, one arrives at the following differential-geometric statement:

Let $\mathfrak{a}$ be an L-infinity algebroid (which you may think of as a dg-manifold, if that helps). There are two dg-algebras canonically associated with this, called the Chevalley-Eilenberg algebra $CE(\mathfrak{a})$ and the Weil algebra $W(\mathfrak{a})$ . You may think of this, respectively, as the function algebra and as a twisted version of the de Rham complex on the corresponding dg-manifold, if that helps, but I suggest to think of it as follows:

there is a canonical morphism

$CE(\mathfrak{a}) \leftarrow W(\mathfrak{a}) : i^*$

and this is the (dual of the) infinitesimal approximation to the inclusion

$A \to \mathbf{E}A$

of the Lie integration $A$ of $\mathfrak{a}$ (a smooth infinity-groupoid) into the $A$ -principal universal infinity-bundle.

We need the following definitions:

An L-infinity cocycle $\mu$ of degree $n+1$ on $\mathfrak{a}$ is a closed element of degree $n+1$ in $\mathrm{CE}(\mathfrak{a})$ . This is equivalently a morphism of $L_\infty$ -algebroids

$\mu : \mathfrak{a} \to b^{n+1} \mathbb{R}$

or dually

$CE(\mathfrak{a}) \leftarrow CE(b^{n+1} \mathbb{R}) : \mu \,,$

where $b^{n+1} \mathbb{R}$ is the line Lie (n+1)-algebra : the $(n+1)$ -fold delooping of the ordinary Lie algebra $\mathbb{R}$ .

An invariant polynomial on $\mathfrak{a}$ is an element $\langle-\rangle \in W(\mathfrak{a})$ in its Weil algebra which is

closed: $d_{W(\mathfrak{a})} \langle-,-\rangle = 0$
horizontal: it sits in the subalgebra generated from the shifted generators (the horizontal generators).

A Chern-Simons element witnessing transgression of an invariant polynomial $\langle-\rangle$ to a cocycle $\mu$ is an element $cs \in W(\mathfrak{a})$ such that

$i^* cs = \mu$
$d_{W(\mathfrak{a})} \,\, cs = \langle - \rangle$ .

This is in other terms a morphism

$W(\mathfrak{a}) \leftarrow W(b^{n+1} \mathbb{R}) \,\, : \,\, cs$

that restricts to $\langle-\rangle$ along the inclusion $inv(\mathfrak{a}) \to W(\mathfrak{a})$ and corestricts to $\mu$ along the surjection $W(\mathfrak{a}) \to CE(\mathfrak{a})$ .

More generally, the datum of $\mathfrak{a}$ -valued differential forms on a manifold $\Sigma$ is a morphism

$\Omega^\bullet(\Sigma) \leftarrow W(\mathfrak{a}) : \phi \,.$

These abstract infinity-Lie theoretic concepts define a $\sigma$ -model quantum field theory as follows (here ignoring global phenomena and concentrating only on the local differential form data, for simplicity of the presentation).

Morphisms $\phi$ as above are the fields of the $\sigma$ -model on $\Sigma$ with target space $\mathfrak{a}$ .

The composite

$\Omega^\bullet(\Sigma) \stackrel{\phi}{\leftarrow} W(\mathfrak{a}) \stackrel{cs}{\leftarrow} W(b^{n+1} \mathbb{R}) \,\,\, : \,\,\, cs(\phi) \,,$

which is an ordinary differential $(n+1)$ -form on $\Sigma$ – is the Lagrangian of the $\sigma$ -model evaluated on the field $\phi$ .

Hence the action functional is (locally)

$S_\omega : \phi \mapsto \int_\Sigma cs_\omega(\phi) \,.$

In the above article we mean to demonstrate the following theorem.

Theorem For $\mathfrak{P}$ an $L_\infty$ -algebroid equipped with a quadratic and non-degenerate invariant polynomal $\langle -,- \rangle = \omega(-,-)$ , we have that

$S_{\omega} = S_{AKSZ}$

is the action functional of the AKSZ $\sigma$ -model whose target space is the symplectic dg-manifold corresponding to $(\mathfrak{P}, \omega)$ .

Moreover, the Lagrangians differ only by at most an exact term (hence are equivalent as Lagrangians)

$cs_\omega \simeq L_{AKSZ} \,.$

Posted at August 6, 2011 10:29 AM UTC

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Re: AKSZ-Models in Higher Chern-Weil Theory

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Are all aspects of ordinary Chern-Simons theory and its applications captured as part of the machinery of the big abstract theory? If so, will there be suggestions of parallels to the use of quantum Chern-Simons in knot theory, Chern-Simons in number theory, etc.?

Posted by: David Corfield on August 8, 2011 12:01 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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Hi David,

just a very brief reply for the moment, because I have to rush off:

Are all aspects of ordinary Chern-Simons theory and its applications captured as part of the machinery of the big abstract theory?

In principle, I think so, yes. But many aspects still need considerably more work. For instance…

will there be suggestions of parallels to the use of quantum Chern-Simons in knot theory

The literature knows a little bit about attempts to consider topological invariants given by higher dimensional Chern-Simons theories. I have collected some references here. In principle it is clear what one has to do and to study. But in general it is hard, I think.

Also the Yetter model (higher Chern-Simons for discrete 2-groups) has been studied a little bit with respect to higher invariants.

Posted by: Urs Schreiber on August 8, 2011 7:14 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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There’s some vague question that occurs to me from your answer to do with the difficulty in finding specific constructions expected by general considerations, and whether there’s anything in the specific of value which escapes the general. I’ll try to formulate it some time.

Posted by: David Corfield on August 9, 2011 1:49 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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whether there’s anything in the specific of value which escapes the general.

Oh yes, I certainly think so. “The general” serves as connective tissue that relates “the various specifics” and thus allows to transport insight on one specific into insight into another.

Your question about possible generalized link invariants in systems other than standard Chern-Simons theory is a very example of this.

And there are more such:

for instance twisted differential string structures are nothing – as discussed there – but the homotopy fibers of the action functional of ordinary $\mathrm{Spin}$ -Chern-Simons theory in 3d – if “action functional” is identified with “Chern-Weil homomorphism” as indicated in part 5 of our notes. Therefore one is prompted to transport this general concept over to all the other special cases of $\infty$ -Chern-Simons theory: not only are there twisted differential Fivebrane structures – the analog for “ordinary but 7-dimensional” Chern-Simons theory – but we see now that just as naturally and fundamentally we ought to be considering “twisted differential Poisson structures” and “twisted differential Courant structures”, etc. Next one wants to think now about “twisted differential 11-d supergravity structures”, and so forth.

In particular (looking at the homotopy fibers over the trivial twisting cocycles) we see that we ought to be considering the analogs of the string Lie 2-algebra: this is the extension of a semisimple Lie algebra that is classifies by the $L_\infty$ -cocycle that is in transgressin with the canonical quadratinc invariant polynomial. This immediately teaches us in which way the analogous constructions will be interesting, too: there is analogously a canonical $b\mathbb{R}$ -extension of any Poisson Lie algebroid, a canonical $b^2 \mathbb{R}$ -extension of every Courant Lie 2-algebroid, etc. together with the extensions of smooth $\infty$ -groupoids that these integrate to.

Of course one could have seen this also without the big picture, but with the big picture it is more compelling and it is clearer where one will be headed. The more systems are examples of a single principle, the clearer everything becomes. I think.

Posted by: Urs Schreiber on August 10, 2011 3:32 AM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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There is now something on the question

“What do we leanr from the fact that AKSZ theory is a special case of higher Chern-Weil theory?”

in chapter 6 of the latest version of our notes.

The four points currently discussed there (briefly) are

1) Symplectic $n$ -groupoids and nontrivial topology;

2) $\infty$ -Connections on nontrivial $\mathfrak{a}$ -principal $\infty$ -bundles: AKSZ instantons;

3) Twisted AKSZ structures and higher extensions of symplectic $L_\infty$ -algebroids;

4) Relation to higher dimensional supergravity.

Posted by: Urs Schreiber on August 20, 2011 6:05 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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Thanks for this. Relating specifics by general principles seems a way to reduce the total amount of specificity, chasing back the specificity of certain good action functionals to the specificity of invariant polynomials via higher Chern-Weil theory which itself works on “very general abstract grounds”.

Can we chase further? Can we see where invariant polynomials come from? And are there morphisms between invariant polynomials?

Typos:

p. 4 You have $\mathfrak{g}$ instead of $\mathfrak{a}$ twice.

p. 23 ‘Simplicial preshaves’.

Posted by: David Corfield on August 12, 2011 9:36 AM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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Typos

Thanks! I think I have fixed this in the latest version.

We have also polished the exposition here and there and expanded the discussion of $L_\infty$ -algebra valued forms.

Posted by: Urs Schreiber on August 13, 2011 1:02 AM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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Can we see where invariant polynomials come from?

Yes. This question has been a central part of my motivation for fomulating higher Chern-Weil theory in cohesive $\infty$ -toposes, and there it has the following answer:

In every cohesive ∞-topos for $A$ an abelian group there is for all $n$ a canonical morphism

$\theta : \mathbf{B}^n A \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}A$

that represents a differential $n$ -form

$[\theta] \in H_{dR}^{n+1}(\mathbf{B}^n A, A) \,.$

This is the higher Maurer-Cartan form on the $(n+1)$ -group $\mathbf{B}^n A$ , and it plays the role of the universal curvature characteristic form:

for

$\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n A$

any characteristic map representing a class in $H^n(\mathbf{B}A, A)$ the composite

$\mathbf{c}^* \theta : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A \stackrel{\theta}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1} A$

is a canonical curvature form on $\mathbf{B}G$ . This is the (unrefined) Chern-Weil homomorphism in the sense that for any object $X$ postcomposition with this gives a map

$\mathbf{c}^* \theta : H^1(X,G) \stackrel{}{\to} H_{dR}^{n+1}(X,A)$

that sends $G$ -principal $\infty$ -bundles on $X$ to sifferential $(n+1)$ -forms on $X$ : namely to the de Rham refinement of the class $[\mathbf{c}] \in H^n(X,A)$ .

This construction exists on purely abstract grounds in every cohesive $\infty$ -topos.

But now one can ask how to present this in the special context of smooth ∞-groupoids: suppose that the smooth $\infty$ -group $G$ arises from Lie integration of an $L_\infty$ -algebra $\mathfrak{g}$ and that the characteristic class $\mathbf{c}$ arises from Lie integration of an $L_\infty$ -cocycle on $\mathfrak{g}$ : which $L_\infty$ -algebraic data does then model the Chern-Weil homomorphism $\mathbf{c}^* \theta$ ?

The notion of invariant polynomials is one part of the answer to this: they arise as part of the $L_\infty$ -algebraic presentation of the abstractly defined Chern-Weil homomorphism.

This story appears in my pdf document in section 3.3.11.

Posted by: Urs Schreiber on August 12, 2011 5:32 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

David,

Is that the sort of answer you wanted or were you asking something more conceptual, even ‘philosophical’?

Posted by: jim stasheff on August 13, 2011 12:47 PM | Permalink | Reply to this

Re: AKSZ-Models in Higher Chern-Weil Theory

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Behind my question is an interest in the issue of why specific mathematical entities or constructions are studied. Leaving aside the more contingent historical and psychological factors, sometimes there’s the possibility of a mathematical account of why something is studied.

Putting things very naively, one effect of showing that a set of constructions arises by general principles from another set is a reduction in the total ‘arbitrariness’. E.g., we pass from the arbitrariness of the selection of certain action functionals to that of the selection of the concept of invariant polynomial and of Chern-Weil theory. The arbitrariness of these in turn can be reduced.

Other reductions can involve showing that a choice of basis or a choice to truncate at a certain point need not be made.

Another move is to show something holds a privileged position within a collection of entities of the same kind. Back here I was looking into characterization via universal properties as an explanation for why certain entities are studied, e.g., the integers, the reals, the category $Set$ , after Hazewinkel’s article on why constructions may have more nice properties than expected from their definition.

An intriguing extra ingredient is when the specificity of a mathematical construction appears to be down to the fact that it has arisen from our attempts to understand our specific world. Sometimes it seems we can reduce this by mathematical classification, e.g., of infinite-dimensional Lie pseudogroups to be able to say that only certain kinds of dynamics are mathematically possible.

Posted by: David Corfield on August 15, 2011 10:58 AM | Permalink | Reply to this

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