### AKSZ-Models in Higher Chern-Weil Theory

#### Posted by Urs Schreiber

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.

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)

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

## Re: AKSZ-Models in Higher Chern-Weil Theory

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.?