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Motivic quantization of local prequantum field theory
talk at GAP XI – Higher Geometry and Quantum Field Theory , Pittsburgh August 2013
and at Twists, generalised cohomology and applications, Münster, October 2013
about quantization of local prequantum field theory in the context of differential cohomology in a cohesive topos.
This is based on Local prequantum field theory. A more formalized discussion at at Quantization via Linear homotopy types (arXiv:1402.7041). For more background and context see at Synthetic Quantum Field Theory.
The first part of the talk indicates how topological local prequantum field theory is naturally presented by higher correspondences in the slice of a cohesive (∞,1)-topos over an ∞-group of units of some E-∞ ring $E$. These are correspondences of moduli stacks of fields (spaces of field trajectories) where the correspondence space is equipped with a cocycle in bivariant twisted E-cohomology. The second part of the talk indicates how quantization of such prequantum data is given by pull-push in twisted E-cohomology, sending correspondences to morphisms of $E$-∞-modules. This is analogous to how Chow motives yield cocycles in motivic cohomology. The existence condition on the required orientation in generalized cohomology are the quantum anomaly cancellation conditions. Finally we indicate a list of examples of holographic quantization of boundary field theories this way: the Poisson manifold at the boundary of the non-perturbative Poisson sigma-model (2d Chern-Simons theory), the charged particle at the boundary of the open string and the heterotic string at the boundary of the M2-brane.
The first part is joint work with Domenico Fiorenza and Hisham Sati. The second part is joint work with Joost Nuiten.
For details see
For background see also
differential cohomology in a cohesive topos
Synthetic Quantum Field Theory
Motivic quantization of local prequantum field theory
Motivic quantization of local prequantum field theory