nLab AKSZ sigma-model

Idea

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.

Examples

• 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.

• BF-theory+topological Yang-Mills theory,

Also, the A-model and the B-model topological 2d sigma-models are examples.

Definition

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

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

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 (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.

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 $k \in \mathbb{R}$. 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.

Related concepts

• sigma-model

• infinity-Chern-Simons theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

  • 2d Chern-Simons theory

  • 3d Chern-Simons theory

  • 4d Chern-Simons theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

  • 11d Chern-Simons theory

  • string field theory

  • infinite-dimensional Chern-Simons theory

  • AKSZ $\sigma$-model

    • Poisson sigma-model

      • A-model, B-model

    • Courant sigma-model

      • Chern-Simons theory

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

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 $\partial \hat \Sigma$ is the integral of the graded symplectic form over $\hat \Sigma$ 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 $p$-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)

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)

See also

• Theodore Th. Voronov, Vector fields on mapping spaces and a converse to the AKSZ construction, arxiv/1211.6319