Hyunmoon Kim

I would like to thank the organizers of the SCS for creating this opportunity.

You can contact me at hktwo [at] snu [dot] ac [dot] kr

I will be a postdoc in the University of Toronto from August 2022.

Here are some responses to some questions I received during review. Thank you for the questions and helpful comments.

Q: What is the origin of the problem studied?

A: Dirac’s method to understand quantization (canonical quantization) has been very successful, but has also encountered some technical inconsistencies (Groenwald-van Hove). Geometric quantization (developed by Kostant, Souriau, Guillemin-Sternberg, Blattner, Robinson-Rawnsley, Snyaticki, Hess, etc.) is one way to implement this method rigorously, using the language of symplectic and differential geometry. It involves making additional choices (prequantum line bundle with connections, polarizations, etc) but to my knowledge it is not entirely resolved when and how all the additional choices give the same result. There are some results of Foth, Uribe, and Charles along these lines.

Q: What were your motivations to study the topic?

A: Because geometric quantization uses the language of symplectic geometry, there is the possibility that, a sufficiently elementary geometric description, can also become a justification (as had occurred historically with principal connections and gauge fields, and Einstein curvature and stress energy). This seemed to happen in the linear case in some works by M. Grossman and Daubechies, which was developed in a different direction from geometric quantization by Daubechies and Klauder.

I was trying to find an explicit map from a family of Heisenberg (or Segal-Shale-Weil) representations in Grossman-Daubechies work and a family of Heisenberg (or Segal-Shale-Weil) representations described by Satake. The parameter spaces for both families had a transitive action of the (real) symplectic group and the same stabilizer (its maximal compact subgroup), so such a map needed to exist. I discovered that not only these two families, but other families could also fit together, using the idea of transverse pairs of complex Lagrangian subspaces / polarizations, introduced earlier by Hess. (This construction is different from another one that can be extrapolated from a paper by A. Weil.) In hindsight, I observed that the partition of the complex Lagrangian Grassmannian seemed to work like an assembly blueprint for these different families. Here are some slides (https://drive.google.com/file/d/1Pa6FMyFK5bPtnGBFip6AC5wc6wuhzIHx/view?usp=sharing) and a paper (https://www.sciencedirect.com/science/article/pii/S0393044021001893) where I describe this part of the argument.

Q: To the solution of which problems can this decomposition be applied?

A: One of the things I would like to understand someday is how this can be related to Shiota's theorem in Abel-Jacobi theory.

Q: Can you say anything about the infinite dimensional case?

A: I do not know how the phenomenon happens in infinite dimensional symplectic vector spaces.

Q: Where can I find a detailed proof of the key theorem?

A: I have the writeup as a finished section of a currently unfinished document. I will be happy to provide the details by email.