### AKSZ Sigma-Models

#### Posted by Urs Schreiber

This is a continuation of the series of posts on sigma-model quantum field theories. It had started as a series of comments in

and continued in

String Topology Operations as a Sigma-Model.

Here I indicate the original definition of the class of models called **AKSZ sigma-models** (see there for a hyperlinked version of the following text).

In a previous post on *exposition of higher gauge theories as sigma-models* I had discussed how ordinary *Chern-Simons theory* is a $\sigma$-model. Indeed this is also a special case of the class of AKSZ $\sigma$-models.

In a followup post I will explain that AKSZ sigma-models are characterized as precisely those ∞-Chern-Simons theories that are induced from invariant polynomials which are both binary and non-degenerate. (Which is incidentally precisely the case in which all diffeomorphisms of the worldvolume can be absorbed into gauge transformations.)

Recall that a sigma-model quantum field theory is, roughly, one

whose fields are maps $\phi : \Sigma \to X$ to some space $X$;

whose action functional is, apart from a kinetic term, the transgression of some kind of cocycle on $X$ to the mapping space $\mathrm{Map}(\Sigma,X)$.

Here the terms “space”, “maps” and “cocycles”
are to be made precise in a suitable context. One says that $\Sigma$ is the *worldvolume* , $X$ is the *target space* and the cocycle is
the *background gauge field* .

For instance an ordinary charged particle
(such as an electron) is described by a $\sigma$-model
where $\Sigma = (0,t) \subset \mathbb{R}$ is the abstract
*worldline*, where $X$ is a smooth
(pseudo-)Riemannian manifold
(for instance our spacetime) and where the background cocycle
is a circle bundle with connection
on $X$ (a degree-2 cocycle in ordinary differential cohomology of $X$, representing a background *electromagnetic field* :
up to a kinetic term the action functional is the holonomy of the connection over a given curve $\phi : \Sigma \to X$.

The $\sigma$-models to be considered here are
*higher* generalizations of this example,
where the background gauge field is a cocycle of higher degree
(a higher bundle with connection)
and where the worldvolume is
accordingly higher dimensional – and where $X$ is allowed to be
not just a manifold but an approximation to a
_higher orbifold (a smooth ∞-groupoid).

More precisely, here we take the category of spaces to be smooth dg-manifolds. One may imagine that we can equip this with an internal hom $\mathrm{Maps}(\Sigma,X)$ given by $\mathbb{Z}$-graded objects. Given dg-manifolds $\Sigma$ and $X$ , their canonical degree-1 vector fields $v_\Sigma$ and $v_X$ act on the mapping space from the left and right. In this sense their linear combination $v_\Sigma + k \, v_X$ for some $k \in \mathbb{R}$ equips also $\mathrm{Maps}(\Sigma,X)$ with the structure of a differential graded smooth manifold.

Moreover, we take the “cocycle” on $X$ to be a graded symplectic structure $\omega$, and assume that there is a kind of Riemannian structure on $\Sigma$ that allows to form the transgression

$\int_\Sigma \mathrm{ev}^* \omega := p_! \mathrm{ev}^* \omega$

by pull-push through the canonical correspondence

$\mathrm{Maps}(\Sigma,X) \stackrel{p}{\leftarrow} \mathrm{Maps}(\Sigma,X) \times \Sigma \stackrel{ev}{\to} X \,,$

where on the right we have the evaluation map.

Assuming that one succeeds in making precise sense of all this
one expects to
find that $\int_\Sigma \mathrm{ev}^* \omega$ is in turn a symplectic structure on the mapping space. This implies that the vector field $v_\Sigma + k\, v_X$ on mapping space has a Hamiltonian $\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X))$.
The grade-0 components $S_{\mathrm{AKSZ}}$ of $\mathbf{S}$
then constitute a functional on the space of maps of graded manifolds $\Sigma \to X$. This is the **AKSZ action functional** defining the AKSZ $\sigma$-model
with target space $X$ and background field/cocycle $\omega$.

In the original article this procedure is indicated only somewhat vaguely. The focus of attention there is a discussion, from this perspective, of the action functionals
of the 2-dimensional $\sigma$-models called the *A-model* and the *B-model* .

In a review by Dmitry Roytenberg, a more detailed discussion of the general construction is given, including an explicit and general formula for $\mathbf{S}$ and hence for $S_{\mathrm{AKSZ}}$ . For $\{x^a\}$ a coordinate chart on $X$ that formula is the following.

**Definition** For $(X,\omega)$ a symplectic dg-manifold of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$ and $k \in \mathbb{R}$, the
**AKSZ action functional**

$S_{\mathrm{AKSZ},k} : \mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X) \to \mathbb{R}$

(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)

is

$S_{\mathrm{AKSZ},k} : \phi \mapsto \int_\Sigma \left( \frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b + k \, \phi^* \pi \right) \,,$

where $\pi$ is the Hamiltonian for $v_X$ with respect to $\omega$ and where on the right we are interpreting fields as forms on $\Sigma$.

This formula hence defines an infinite class of $\sigma$-models
depending on the target space structure $(X, \omega)$, and on the
relative factor $k \in \mathbb{R}$. It was
noticed from the beginning that ordinary Chern-Simons theory is a special
case of this for $\omega$ of grade 2, as is the
Poisson sigma-model
for $\omega$ of grade 1 (and hence, as shown there, also the A-model
and the B-model).
The general case for $\omega$ of grade 2 has been called the
*Courant sigma-model* .

One nice aspect of this construction is that it follows immediately that the full Hamiltonian $\mathbf{S}$ on mapping space satisfies $\{\mathbf{S}, \mathbf{S}\} = 0$. Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of $(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X)$ in degree 0 is the space of functions on those fields that satisfy the Euler-Lagrange equations of $S_{\mathrm{AKSZ}}$. Taken together this implies that $\mathbf{S}$ is a solution of the “master equation” of a BV-BRST complex for the quantum field theory defined by $S_{\mathrm{AKSZ}}$. This is a crucial ingredient for the quantization of the model, and this is what the AKSZ construction is mostly used for in the literature.

## Re: AKSZ Sigma-Models

in re: binary and non-degenerate.

why binary? you have in mind symplectic as later?

additional generality?

(Which is incidentally precisely the case in which all diffeomorphisms of the worldvolume can be absorbed into gauge transformations.)

assuming something about the dim of the world volume?