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On twistor quantization of loop spaces and Kähler manifolds. (English) Zbl 0895.58027

Dimiev, Stancho (ed.) et al., Topics in complex analysis, differential geometry and mathematical physics. Third international workshop on complex structures and vector fields, St. Konstantin, Bulgaria, August 23–29, 1996. Singapore: World Scientific. 148-157 (1997).
Let \((M^{2n},\omega)\) be a symplectic manifold such that \(\omega/2\pi\) represents an integral cohomology class in \(H^2(M,\mathbb{R})\). Then there exists a complex line bundle \(L\rightarrow M\) and a linear connection \(\nabla^L\) on \(L\) with curvature \(\omega\). Upon fixing a Hermitian structure \(\langle\cdot ,\cdot\rangle\) on \(L,\) one endows \({\mathcal L}^2(M,L;\omega),\) the Hilbert space of square-integrable sections of \(L\rightarrow M,\) with the inner product \((s_1,s_2)=\int_M\langle s_1,s_2\rangle\omega^n\). Then the Kostant-Souriau formula \[ f\mapsto r(f)=f-i\nabla_{X_f}^L, \] \(X_f\) being the Hamiltonian vector field associated with \(f,\) defines a representation of \(C^\infty (M)\) in \({\mathcal L}^2(M,L;\omega)\) which is reducible (for, roughly speaking, \({\mathcal L}^2(M,L;\omega)\) contains functions of both \(p\)’s and \(q\)’s, \((q,p)\) being symplectic coordinates on \(M)\). When \(M\) is Kähler, one gets around the problem by considering the subspace \({\mathcal L}^2_{{\mathcal O}}(M,L;\omega)\) of holomorphic elements of \({\mathcal L}^2(M,L;\omega)\). Since in general the Kostant-Souriau representation operators do not preserve \({\mathcal L}^2_{{\mathcal O}}(M,L;\omega)\) (for the symplectomorphisms generated by the elements of \(C^\infty (M)\) do not preserve the original complex structure of \(M),\) the author proposes, instead of fixing a complex structure on \(M,\) to consider the orbit of the original complex structure by the action of the symplectomorphisms generated by the elements of \(C^\infty(M)\). The elements of such an orbit can be identified with sections of a twistor bundle \(Z\rightarrow M\) which possesses a canonical almost complex structure. The author then reformulates the original quantization problem in terms of this twistor bundle endowed with its canonical almost complex structure (a problem that, so reformulated, does not depend on the choice of the complex structure on \(M),\) and, once the twistor problem is solved, he looks for conditions which allow that the twistor-solution can be pushed down from the twistor bundle \(Z\) to the base manifold \(M\).
For the entire collection see [Zbl 0872.00036].

MSC:

53D50 Geometric quantization
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58D15 Manifolds of mappings
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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