PIRSA:25110103 Scientific Series Mathematical physics

DOI10.48660/25110103

Series: Mathematical Physics

(2025). Semiclassical Hodge Theory. Perimeter Institute. https://pirsa.org/25110103

Semiclassical Hodge Theory. Perimeter Institute, Nov. 21, 2025, https://pirsa.org/25110103 

@misc{ pirsa_PIRSA:25110103,
  doi = {10.48660/25110103},
  url = {https://pirsa.org/25110103},
  author = {},
  keywords = {Mathematical physics},
  language = {en},
  title = {Semiclassical Hodge Theory},
  publisher = {Perimeter Institute},
  year = {2025},
  month = {nov},
  note = {PIRSA:25110103 see, \url{https://pirsa.org}}
}

Abstract

A celebrated 1997 theorem of Kontsevich shows that every
Poisson manifold (classical phase space) can be "quantized" to produce a noncommutative algebra (the corresponding quantum observables). He
gives an explicit Feynman-style series expansion for the quantum
product, but as with many perturbative expansions, it is very difficult to calculate directly. Following a suggestion of Kontsevich, I will explain how Hodge theory can be used to construct natural "period
coordinates" on the moduli space of smooth Poisson varieties, in which the quantization can often be computed simply, explicitly and
nonperturbatively as the exponential map for a complex torus. This gives a conceptual explanation for the appearance of various classical transcendental functions in the relations defining well-known quantum algebras. This talk is based on ongoing joint work with Aidan Lindberg.

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