On the AKSZ Formulation of the Poisson Sigma Model

By Alberto S. Cattaneo, Giovanni Felder. 2001

150 citations recorded by the source corpus.

We review and extend the Alexandrov–Kontsevich–Schwarz–Zaboronsky construction of solutions of the Batalin–Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin–Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6f0534fd89e99c72eccb428f9695e6ee0268d45b

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In Algebra and Number Theory

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism (2018, background): “omological vector elds Q and Q M induce vector elds Q^ and Q^ M on F BV and we can choose any linear combination of these to form Q BV. The compatibility between Q BV and ! BV is readily checked, cf. [102,103]. In particular, one can show that the map pdened in (5.120) is a symplectomorphism. Thus, ”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, methodology): “32), which involves only Lie algebroid gauge symmetries, and whose quantization on the disk is described in [17, 18].”
Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model (2003, background): “The boundary conditions for the case C =M were discussed in [4,6].”
AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories (2006, unknown): “Cattaneo-Felder [4] and Park [8] further refined the AKSZ procedure by generalizing it to the case of manifolds with boundary, and produced new examples: Cattaneo and Felder studied the Poisson sigma-model [11] on the disk [3] [4], while Park considered its higher-dimensional generalizations, the to”
Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “also [7] and [31]) for the case that the target carries also a compatible symplectic form, i.”
Classical BV Theories on Manifolds with Boundary (2011, methodology): “As shown in [18], in the case of AKSZ theories, one can obtain an adapted Lagrangian submanifold L by choosing an adapted Lagrangian submanifold Li of the target manifold M for each boundary component ∂iN of the spacetime N .”
Sheaves of categories and the notion of 1-affineness (2013, methodology): “d the PTVV construction to derived mapping stacks with boundary conditions, which is requiredin most applications to quantum ﬁeld theories(see e.g. the workofCattaneoFelder on the Poisson sigma model [8, 9], and the recent work of Cattaneo-Mnev-Reshetikhin [10, 11]). We provide many examples of Lagr”