# nLab AKSZ sigma-model

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Examples

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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

Als 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.

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

$n \in \mathbb{N}$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-modelhigher symplectic geometry$(n+1)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $(n+1)$discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

## References

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

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

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

• 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

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