Journal of Symplectic Geometry

Volume 11 (2013)

Number 4

Degeneration of Kähler structures and half-form quantization of toric varieties

Pages: 603 – 643

DOI: http://dx.doi.org/10.4310/JSG.2013.v11.n4.a4

Authors

William D. Kirwin (Mathematics Institute, University of Cologne, Germany)

José M. Mourão (Center for Mathematical Analysis, Geometry and Dynamical Systems, and Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal)

João P. Nunes (Center for Mathematical Analysis, Geometry and Dynamical Systems, and Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal)

Abstract

We study the half-form Kähler quantization of a smooth symplectic toric manifold $(X,\omega)$, such that $[ \omega/ 2\pi]- c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}} )$ and is non-negative. We define the half-form corrected quantization of $(X,\omega)$ to be given by holomorphic sections of a certain Hermitian line bundle $L\to X$ with Chern class $[ \omega/ 2\pi]- c_{1}(X)/2$. These sections then correspond to integral points of a “corrected” polytope $P_{L}$ with integral vertices. For a suitably translated moment polytope $P_{X}$ for $(X,\omega)$, we have that $P_{L}\subset P_{X}$ is obtained from $P_{X}$ by a one-half inward-pointing normal shift along the boundary.

We use our results on the half-form corrected Kähler quantization to motivate a definition of half-form corrected quantization in the singular real toric polarization. Using families of complex structures studied in Baier-Florentino-Mourao-Nunes, which include the degeneration of Kähler polarizations to the vertical polarization, we show that, under this degeneration, the half-form corrected $L^{2}$-normalized monomial holomorphic sections converge to Dirac-delta-distributional sections supported on the fibers over the integral points of $P_{L}$, which correspond to corrected Bohr–Sommerfeld fibers. This result and the limit of the corrected connection, with curvature singularities along the boundary of $P_X$, justifies the direct definition we give for the corrected quantization in the singular real toric polarization. We show that the space of quantum states for this definition coincides with the space obtained via degeneration of the Kähler quantization.

We also show that the BKS pairing between Kähler polarizations is not unitary in general. On the other hand, the unitary connection induced by this pairing is flat.

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