AKSZ sigma-model


\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory

Symplectic geometry



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 (𝔓,ω)(\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 ω\omega. Details on this are at ∞-Chern-Simons theory – Examples – AKSZ theory.


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


A sigma-model quantum field theory is, roughly, one

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)\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:ΣX\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 \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 kk \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)×ΣevX, \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 SC (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 ΣX\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.


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 kk \in \mathbb{R}, the AKSZ action functional

S AKSZ,k:SmoothGrMfd(𝔗Σ,X) 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 ad 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 kk \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.

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(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
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(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

  • 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 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

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

See also

  • Peter Bouwknegt, Branislav Jurčo, AKSZ construction of topological open p-brane action and Nambu brackets, arxiv/1110.0134

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

Revised on February 8, 2013 02:10:43 by Urs Schreiber (