Abstract
Factorization spaces give a nice geometric way to formalize the notion of operator product expansion, which is used by physicists to formalize quantum field theory in a locally covariant way. This approach was grounded by Beilinson-Drinfeld, and generalized to higher dimension by Francis-Gaitsgory and Rozenblyum. A related approach is given by Costello and Gwilliam’s quantization program. We start this chapter by recalling some basic facts about \(\mathcal{D}\)-modules over the Ran space and their monoidal structures, and then explain the relation, given by chiral Koszul duality, between chiral Lie algebras and factorization coalgebras. We illustrate these notions by various simple examples. We then discuss the derived \(\mathcal{D}\)-geometry of factorization spaces, and in particular the theory of multi-jets for systems of non-linear partial differential equations. We then give an introduction to the use of the generalized affine grassmanian in perturbative quantization. We finish by describing a quite general categorical quantization problem for factorization spaces.
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Paugam, F. (2014). Factorization Spaces and Quantization. In: Towards the Mathematics of Quantum Field Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-04564-1_24
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DOI: https://doi.org/10.1007/978-3-319-04564-1_24
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