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
for higher abelian targets
for symplectic Lie n-algebroid targets
FQFT and cohomology
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 .
The action functional of AKSZ theory is that of ∞-Chern-Simons theory induced from the Chern-Simons element that correspondonds to the invariant polynomial . Details on this are at ∞-Chern-Simons theory – Examples – AKSZ theory.
whose fields are maps to some space ;
For instance the ordinary charged particle (for instance an electron) is described by a -model where is the abstract worldline, where is a smooth (pseudo-)Riemannian manifold (for instance our spacetime) and where the background cocycle is a circle bundle with connection on (a degree-2 cocycle in ordinary differential cohomology of , representing a background electromagnetic field : up to a kinetic term the action functional is the holonomy of the connection over a given curve .
The -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 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 given by -graded objects. Given dg-manifolds and their canonical degree-1 vector fields and acting on the mapping space from the left and right. In this sense their linear combination for some equips also with the structure of a differential graded smooth manifold.
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 is in turn a symplectic structure on the mapping space. This implies that the vector field on mapping space has a Hamiltonian . The grade-0 components of then constitute a functional on the space of maps of graded manifolds . This is the AKSZ action functional defining the AKSZ -model with target space and background field/cocycle .
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 -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 and hence for . For a coordinate chart on that formula is the following.
For a symplectic dg-manifold of grade , a smooth compact manifold of dimension and , the AKSZ action functional
(where is the shifted tangent bundle)
where is the Hamiltonian for with respect to and where on the right we are interpreting fields as forms on .
This formula hence defines an infinite class of -models depending on the target space structure , and on the relative factor . In (AKSZ) it was already noticed that ordinary Chern-Simons theory is a special case of this for of grade 2, as is the Poisson sigma-model for 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 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 on mapping space satisfies . Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of in degree 0 is the space of functions on those fields that satisfy the Euler-Lagrange equations of . Taken together this implies that is a solution of the “master equation” of a BV-BRST complex for the quantum field theory defined by . 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.
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||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|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||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
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)
A cohomological reduction of the formalism is described in
That the AKSZ action on bounding manifolds is the integral of the graded symplectic form over is theorem 4.4 in
In the broader context of smooth higher geometry this is discussed in section 4.3 of
Theodore Th. Voronov, Vector fields on mapping spaces and a converse to the AKSZ construction, arxiv/1211.6319