Sergeev, A. G. Twistor quantization of loop spaces and general Kähler manifolds. (English) Zbl 0907.58024 Ramírez de Arellano, E. (ed.) et al., Operator theory for complex and hypercomplex analysis. Proceedings of a conference, Mexico City, Mexico, December 12–17, 1994. Providence, RI: American Mathematical Society. Contemp. Math. 212, 221-228 (1998). Suppose we are given a Kähler phase space with Kähler form \(\omega\) such that the cohomology class of \(\omega/2\pi\) is integral. Then there is a Hermitian line bundle \(L\to M\) and a compatible connection on it whose curvature is \(\omega\). By some “physical reasons”, it is reasonable to take as quantization (Fock) space in the Kostant-Souriau prequantization scheme the space of holomorphic square integrable sections of \(L\). However, the Kostant-Souriau representation operators do not preserve, in general, this space because the symplectic diffeomorphisms of \(M\) generated by the elements of the algebra of observables on \(M\) (considered as a subalgebra of Hamiltonian vector fields) do not preserve the original complex structure of \(M\). It was proposed by the author and A. D. Popov [Aspects Math. E 25, 137-157 (1994; Zbl 0809.58017) and in: Quantization and infinite-dimensional systems (Bialowieza 1993), 43-51 (1994; MR 97i:81059)] to consider all complex structures obtained from the original one by the action of symplectic diffeomorphisms corresponding to the observables rather than to fix a complex structure on the symplectic manifold \((M, \omega)\), an idea inspired by twistor theory. The present paper contains a nice presentation of the author’s and A. Popov’s twistor reformulation of the geometric quantization scheme in the case when the phase space is the loop space of a compact simple Lie group or the group of translations of the Minkowski space \(\mathbb{R}^{d-1,1}\). The case of an arbitrary (finite- or infinite-dimensional) Kähler phase space is also discussed.For the entire collection see [Zbl 0881.00038]. Reviewer: J.Davidov (Sofia) MSC: 53D50 Geometric quantization 32L25 Twistor theory, double fibrations (complex-analytic aspects) 81S10 Geometry and quantization, symplectic methods 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 58D15 Manifolds of mappings Keywords:geometric quantization; twistor spaces; Kähler phase space; loop space Citations:Zbl 0809.58017 PDFBibTeX XMLCite \textit{A. G. Sergeev}, Contemp. Math. 212, 221--228 (1998; Zbl 0907.58024)