whose fields are maps ϕ:Σ→X\phi : \Sigma \to X to some space XX;
whose action functional is, apart from a kinetic term, the transgression of some kind of cocycle on XX to the mapping space Map(Σ,X)\mathrm{Map}(\Sigma,X).

Here the terms “space”, “maps” and “cocycles” are to be made precise in a suitable context. One says that Σ\Sigma is the worldvolume, XX 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 σ\sigma-model where Σ=(0,t)⊂ℝ\Sigma = (0,t) \subset \mathbb{R} is the abstract worldline, where XX is a smooth (pseudo-)Riemannian manifold (for instance our spacetime) and where the background cocycle is a circle bundle with connection on XX (a degree-2 cocycle in ordinary differential cohomology of XX, representing a background electromagnetic field : up to a kinetic term the action functional is the holonomy of the connection over a given curve ϕ:Σ→X\phi : \Sigma \to X.

The σ\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 XX 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 Maps(Σ,X)\mathrm{Maps}(\Sigma,X) given by ℤ\mathbb{Z}-graded objects. Given dg-manifolds Σ\Sigma and XX their canonical degree-1 vector fields v Σv_\Sigma and v Xv_X acting on the mapping space from the left and right. In this sense their linear combination v Σ+kv Xv_\Sigma + k \, v_X for some k∈ℝk \in \mathbb{R} equips also Maps(Σ,X)\mathrm{Maps}(\Sigma,X) with the structure of a differential graded smooth manifold.

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

∫ Σev *ω:=p !ev *ω \int_\Sigma \mathrm{ev}^* \omega := p_! \mathrm{ev}^* \omega

by pull-push through the canonical correspondence

Maps(Σ,X)←pMaps(Σ,X)×Σ→evX, \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 ∫ Σev *ω\int_\Sigma \mathrm{ev}^* \omega is in turn a symplectic structure on the mapping space. This implies that the vector field v Σ+kv Xv_\Sigma + k\, v_X on mapping space has a Hamiltonian S∈C ∞(Maps(Σ,X))\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X)). The grade-0 components S AKSZS_{\mathrm{AKSZ}} of S\mathbf{S} then constitute a functional on the space of maps of graded manifolds Σ→X\Sigma \to X. This is the AKSZ action functional defining the AKSZ σ\sigma-model with target space XX and background field/cocycle ω\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 σ\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 S\mathbf{S} and hence for S AKSZS_{\mathrm{AKSZ}} . For {x a}\{x^a\} a coordinate chart on XX that formula is the following.

Definition

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

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

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

S AKSZ,k:ϕ↦∫ Σ(12ω abϕ a∧d dRϕ b+kϕ *π), 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 π\pi is the Hamiltonian for v Xv_X with respect to ω\omega and where on the right we are interpreting fields as forms on Σ\Sigma.

This formula hence defines an infinite class of σ\sigma-models depending on the target space structure (X,ω)(X, \omega), and on the relative factor k∈ℝk \in \mathbb{R}. In (AKSZ) it was already noticed that ordinary Chern-Simons theory is a special case of this for ω\omega of grade 2, as is the Poisson sigma-model for ω\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 ω\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 S\mathbf{S} on mapping space satisfies {S,S}=0\{\mathbf{S}, \mathbf{S}\} = 0. Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of (Maps(Σ,X),v Σ+kv X)(\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 AKSZS_{\mathrm{AKSZ}}. Taken together this implies that S\mathbf{S} is a solution of the “master equation” of a BV-BRST complex for the quantum field theory defined by S AKSZS_{\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.

References

The original reference is

M. Alexandrov, Maxim Kontsevich, Albert Schwarz, Oleg Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997 (arXiv:hep-th/9502010)

Dmitry Roytenberg wrote a useful exposition of the central idea of the original work and studied the case of the Courant sigma-model in

Dmitry Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories Lett.Math.Phys.79:143-159,2007 (arXiv:hep-th/0608150).

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

F. Bonechi, P. Mnëv, Maxim Zabzine, Finite dimensional AKSZ-BV-theories (arXiv)

That the AKSZ action on bounding manifolds ∂Σ^\partial \hat \Sigma is the integral of the graded symplectic form over Σ^\hat \Sigma is theorem 4.4 in

A. Kotov, T. Strobl, Characteristic classes associated to Q-bundles (arXiv:0711.4106v1)

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

Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (arXiv:1108.4378)

In the broader context of smooth higher geometry this is discussed in section 4.3 of

Urs Schreiber, differential cohomology in a cohesive topos

Discussion of boundary conditions for the AKSZ sigma model includes

Peter Bouwknegt, Branislav Jurco, AKSZ construction of topological open pp-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)

The AKSZ model is extended to coisotropic boundary conditions in

Theo Johnson-Freyd, Exact triangles, Koszul duality, and coisotropic boundary conditions (arxiv/1608.08598)

An example in higher spin gauge theory is discussed in

K.B. Alkalaev, Maxim Grigoriev, E.D. Skvortsov, Uniformizing higher-spin equations (arXiv:1409.6507)