nLab
AKSZ theory

Idea

What is called the AKSZ formalism – after the initials of its four authors – Alexandrov, Kontsevich, Schwarz, Zaboronsky – is a a special technique for constructing action functionals in gauge theory.

More precisely, it is a method for constructing a BV Laplacian for a sigma-model with a given symplectic NQ-supermanifold $X$ target space which describes a generalization of the Poisson sigma-model.

details

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conceptual context and background

The AKSZ construction assumes parameter space and target space of the sigma-model to be quantized to be modeled in terms of NQ-supermanifolds. These may conceptually be thought of as representing Lie-∞ algebroids. In a general context of sigma-model quantization, such as described for instance at

• exercise in groupoidification - the path integral

• geometric ∞-function theory

target space would instead be represented by a Lie ∞-groupoid. In a complete picture of sigma-model quantization the ASZ method should therefore be a means to handle a linearized or infinitesimal approximation to the full theory.

References

The original reference is

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaborosnky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997

Dmitry Roytenberg’s useful review of the original work:

• Dmitry Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories (arXiv).

A cohomological reduction of the formalism is described in

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