2010
DOI: 10.1016/j.jfa.2010.01.023
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Quantization of abelian varieties: Distributional sections and the transition from Kähler to real polarizations

Abstract: We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all non-negative polarizations. The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form corrected q… Show more

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Cited by 18 publications
(33 citation statements)
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“…Geometric quantization in this context (see e.g. [6]) has a different flavour to the discussion in this paper.…”
Section: Dissolved Vortices (Corresponding To the Critical Value τ = 4πdmentioning
confidence: 93%
See 1 more Smart Citation
“…Geometric quantization in this context (see e.g. [6]) has a different flavour to the discussion in this paper.…”
Section: Dissolved Vortices (Corresponding To the Critical Value τ = 4πdmentioning
confidence: 93%
“…The most natural polarisations to consider in the geometric quantization of a Kähler phase space are the complex polarisations compatible with the Kähler form. In our context, there are both (a) a natural family of such polarisations on S d Σ, namely, those determined by compatible complex structures J Σ induced from complex structures Σ on the surface Σ; and (b) a preferred polarisation in this family, namely, the one corresponding to the J j Σ determined by the particular complex structure j Σ featuring in one of the vortex equations (see (4) and (6) below), and which can therefore be regarded as part of the classical data. Once one takes heed of this preferred choice of complex structure J j Σ , the discussion of dependence of the quantization on complex polarisations is secondary, and perhaps even pointless.…”
Section: Introductionmentioning
confidence: 99%
“…In the second case, the need for half-forms is less evident. Nonetheless, it is commonly believed that the half-form correction is necessary also in the Kähler case and many arguments have been presented in its favor: the half-form correction renders the BKS pairing map unitary in the quantization of vector spaces with translation invariant complex structures [ADW91,KW06] and of Abelian varieties [BMN10] and allows for a transparent explanation of the vacuum energy shift in the Kähler quantization of symplectic toric varieties with toric Kähler structures [KMN10].…”
Section: Preliminariesmentioning
confidence: 99%
“…The linear observables, which generate translations and which, together with the constants, span the Heisenberg algebra, preserve the invariant polarizations and therefore have geometric-quantization induced actions on the Hilbert spaces H P of P-polarized quantum states, which integrate to irreducible representations of the Heisenberg group. According to the Stone-Von Neumann uniqueness theorem, all such representations are unitarily equivalent and then, from Schur's lemma, there is a unique-up-to-phase unitary operator U P,P ′ : H P −→ H P ′ , establishing equivalence of the quantizations for different polarizations within this class.When (M, ω) is a symplectic torus M = T 2n , the corresponding Stone-Von Neumann type theorem for the finite Heisenberg group also guarantees the equivalence between quantizations for translation invariant polarizations [1,23,2].In this paper, we will address this question in the case when M = T * K π −→ K is the cotangent bundle of a compact Lie group K equipped with the canonical symplectic form ω. One motivation for this is to address a question raised by Hall in [11,12] on finding an analogue, for the quantization of T * K, of the role played by Stone-Von Neumann theorem in establishing the unitary equivalence of quantizations for translation-invariant polarizations on a symplectic vector space.…”
mentioning
confidence: 99%