nLab
AKSZ theory

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 $(𝔓,\omega )$.

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,

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

References

The original reference is

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. 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:

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

Another review is

• 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

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

The discussion of the AKSZ action functional as the ∞-Chern-Simons theory-functionl induced from a symplectic Lie n-algebroid in ∞-Chern-Weil theory is due to

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, The AKSZ-functional is the $\mathrm{\infty }$-Chern-Weil functional

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

• Urs Schreiber, differential cohomology in a cohesive topos