For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
What is called the AKSZ formalism – after the initials of its four authors – Alexandrov, Maxim Kontsevich, Albert Schwarz, Oleg Zaboronsky – is a technique for constructing action functionals in BV-BRST formalism for sigma model quantum field theories whose target space is an symplectic Lie n-algebroid $(\mathfrak{P}, \omega)$.
The action functional of AKSZ theory is that of ∞-Chern-Simons theory induced from the Chern-Simons element that correspondonds to the invariant polynomial $\omega$. Details on this are at ∞-Chern-Simons theory – Examples – AKSZ theory.
to a Poisson Lie algebroid corresponds the Poisson sigma-model;
the a Courant algebroid corresponds the Courant sigma-model;
in particular to a semisimple Lie algebra corresponds Chern-Simons theory.
Als the A-model and the B-model topological 2d sigma-models are examples.
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 the ordinary charged particle (for instance 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 $v_X$ acting 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
by pull-push through the canonical correspondence
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 (AKSZ) 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 (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.
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
(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)
is
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 $k \in \mathbb{R}$. In (AKSZ) it was already noticed 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 main example in (Roytenberg) is spelling out the general case for $\omega$ of grade 2, which is called the Courant sigma-model there.
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.
higher dimensional Chern-Simons theory
AKSZ $\sigma$-model
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
The original reference is
Dmitry Roytenberg wrote a useful exposition of the central idea of the original work and studied the case of the Courant sigma-model in
Other reviews include
Noriaki Ikeda, Deformation of graded (Batalin-Volkvisky) Structures in Dito, Lu, Maeda, Weinstein (eds.) Poisson geometry in mathematics and physics Contemp. Math. 450, AMS (2008)
Noriaki Ikeda, Lectures on AKSZ Topological Field Theories for Physicists (arXiv:1204.3714)
A cohomological reduction of the formalism is described in
That the AKSZ action on bounding manifolds $\partial \hat \Sigma$ is the integral of the graded symplectic form over $\hat \Sigma$ is theorem 4.4 in
The discussion of the AKSZ action functional as the ∞-Chern-Simons theory-functional induced from a symplectic Lie n-algebroid in ∞-Chern-Weil theory is due discussed in
In the broader context of smooth higher geometry this is discussed in section 4.3 of
Discussion of boundary conditions for the AKSZ sigma model includes
Peter Bouwknegt, Branislav Jurčo, AKSZ construction of topological open $p$-brane action and Nambu brackets, arxiv/1110.0134
Noriaki Ikeda, Xiaomeng Xu, Canonical Functions and Differential Graded Symplectic Pairs in Supergeometry and AKSZ Sigma Models with Boundary (arXiv:1301.4805)
An example in higher spin gauge theory is discussed in
See also