Branes in the Poisson sigma model and deformation quantization of constrained systems
Alberto Cattaneo, Institut fuer Mathematik, Universitaet Zuerich
Room DC/1/26
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The Poisson sigma model is a topological string theory with target a Poisson manifold (i.e., a manifold whose algebra of smooth functions is endowed with a Lie bracket that is also a derivation; e.g., a symplectic manifold). Particularly interesting is the case of open strings in genus zero (i.e., the worldsheet is a disk) as its perturbative functional integral quantization with "free" boundary conditions yields Kontsevich's formula for the deformation quantization of the target (i.e., an associative algebra structure on formal power series in Plank's constant of smooth functions such that the commutator yields the Poisson bracket in the classical limit). In this talk we address the problem of changing the boundary conditions ("branes"). It turns out that the possible branes are coisotropic submanifolds (the global version of Dirac's first class constraints, which appear when one has to do with systems with symmetry - from Lagrange's top to, in infinite dimensions, Yang-Mills theory). The functional integral quantization of the Poisson sigma model with branes yields interesting formulae including (when obstruction are absent) the deformation quantization of the reduced phase space of a given coisotropic submanifold.