Why is geometric quantization (esp. Berezin-Toeplitz quantization) interesting for a symplectic geometer/topologist?

Question

I know that many symplectic geometers are interested in quantization as well. From what I understood, quantization isn't expected to be used as a tool to answer symplectic questions (as in translating the problem to the quantum world, solve it there, and somehow go back), unlike the relation, for example between complex geometry and tropical geometry. Rather, it is supposed to be interesting and have appeal for reasons intrinsic to quantization.

As a person whose main interests are symplectic geometry, topology and algebraic geometry, why study quantization? What aspects or problems there may I find appealing?

For those who approach mathematics from the physics realm, I guess the answer is straightforward, as those ways of quantizing symplectic manifolds are attemps at formalization of some physical theories and beliefs.

I am looking for a more mathematically oriented answer, since I personally derive my interest and motivation from the intrinsic beauty of mathematics.

Thanks

Answer 1

The method of coadjoint orbits suggests that irreducible unitary representations of a Lie group $G$ are something like quantum mechanical systems. In fact finding all irreducible unitary representation of nilpotent lie groups correspounds to geometric quantization of coadjoint orbits. In fact quantiz ation problem comes down to a finite set of coadjoint orbits for each reductive group: the nilpotent orbits. The method was carried over to geometric quantization where you want to deform the pointwise multiplication of functions with the Poisson bracket to a noncommutative product with the Poisson bracket as a first order approximation. Ultimately this lead to star product quantization.

In symplectic geometry: quantization commutes with reduction :

states that the space of global sections of a line bundle satisfying the quantization condition on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of the line bundle.

In topology, Geometric quantization is related to K-theory: About the conjecture quantisation co mmutes with reduction for noncompact simple groups the only finite dimensional unitary representations are direct sums of the trivial one which by Landsman's idea we can replace the representation ring of a group by the K-theory of its $C^∗$-algebra, and the $K$-index by the analytic assembly map.