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Supergeometry of gauge PDE and (presymplectic) AKSZ models

QMAP Seminar

Speaker: Maxim Grigoriev, Lebedev
Location: 1147 MSB
Start time: Wed, Nov 9 2016, 4:10PM

Alexandrov-Kontsevich-Schwartz-Zaboronsky (AKSZ) sigma models were originally proposed to describe topological systems. In fact, an AKSZ model with finite number of fields and space-time dimension higher than 1 is necessarily topological. These models are quite distinguished in the sense that the geometry of the target space manifold encodes not only its Lagrangian or equations of motion but also the complete BV-BRST differential. Moreover, the Hamiltonian BRST formulation is also naturally contained in the AKSZ one. In this sense AKSZ naturally unifies both the Lagrangian and the Hamiltonian formalism of gauge systems. Moreover one can employ AKSZ formulation to study (asymptotic) boundary values of fields. This technique appears promising in the context of AdS/CFT correspondence for higher-spin fields. By allowing the target space to be infinite-dimensional any gauge PDE can be reformulated as an AKSZ model. More precisely, as a target space one takes the equation manifold itself (seen as a surface in the respective jet-space). In the case of Lagrangian PDE, the target space is naturally equipped with the presymplectic structure so that the associated presymplectic AKSZ model gives an equivalent Lagrangian formulation, giving an explicit way to encode the Lagrangian in terms of the intrinsic geometry of the equation manifold.

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