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Register a new userGeometric quantization, complex structures and the coherent state transform
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It is shown that the heat operator in the Hall coherent state transform for a compact Lie group $K$ is related with a Hermitian connection associated to a natural one-parameter family of complex structures on $T^*K$. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of $T^*K$ for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin and Axelrod, Della Pietra and Witten.
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In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $Gtimes T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian flows of one $Gtimes G$-invariant function, $h$, and one $Gtimes T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $Gtimes G$-invariant Kahler structures gives new $Gtimes T$-invariant, but not $Gtimes G$-invariant, Kahler structures on $T^*G$. We study the Hilbert spaces ${mathcal H}_{tau,sigma}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kahler structures. On the other hand, by taking the vertical Schrodinger polarization as a starting point, the above $Gtimes T$-invariant Hamiltonian flows also generate families of mixed polarizations $mathcal{P}_{0,sigma}, sigma in {mathbb C}, {rm Im}(sigma) >0$. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kahler structure on the leaves of a foliation of $T^*G$. The geometric quantization of $T^*G$ with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [KMN1,KMN2].
We consider $G_2$ structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. The resulting system of partial differential equations can be regarded as an analogue of the Strominger system in $7$-dimensions. We initiate the study of the moduli space of solutions and show that it is finite dimensional using elliptic operator theory. We also relate the associated geometric structures to generalized geometry.
This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.
This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories are those conceived in a modified de Broglie-Bohm scheme and we note that identical classical and quantum trajectories for coherent states are obtained only in the present approach. The study is extended to Gazeau-Klauder and SUSY quantum mechanics-based coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller (PT) potential by solving for the trajectories numerically. For the coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the PT potential, though classical trajectories are not regained, a periodic motion results as t --> infty.
In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states and show that they satisfy some properties on an integral K$ddot{rm{a}}$hler manifold which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold and show that they satisfy similar properties. Finally we show a Berezin -type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$ where $U$ is an open set in ${mathbb C}P^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K$ddot{rm{a}}$hler manifold of real dimension $2n$. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${mathbb C}P^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.
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