arXiv subjects

José M. Mourão

Publications and source records attributed to José M. Mourão.

16 recordsLinked to original sources

arXiv2026

Fourier-Helgason transform as infinite geodesic time limit in geometric quantization

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ''horizontal'' polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

math.SG↗

arXiv2026

Quantum Hall States response to toroidal geometry deformation

In this paper, we apply techniques of geometric quantization to study the response of the integer and fractional quantum Hall effects to toroidal geometry deformation. The main method is that of using complex time Hamiltonian evolution to induce the geometry change and then the associated generalized coherent state transforms (gCST) to find the evolution of the Laughlin states. We consider two kinds of deformations. The first are flat toroidal deformations. Although Laughlin states for all flat toroidal geometries have been thoroughly studied before, we believe that our approach via the gCST is novel. It also serves as a testing ground to study the non-flat Kähler deformations. The Hamiltonians used in the flat deformations are quadratic in the generators of translations and therefore non periodic. The second kind of deformations involve nonflat Kähler toroidal deformations, generated by global, thus bi-periodic, Hamiltonians on the torus. The corresponding imaginary time flows are (elliptic curve modulus) $τ$-preserving Mabuchi geodesics in the space of Kähler metrics on the torus, hitting a curvature singularity in finite imaginary time. By restricting to $S^1$-invariant deformations we find explicit analytic expressions for the evolution of the toroidal geometry and of the Laughlin states all the way to the singularity.

math-ph↗

arXiv2025

A new look at unitarity in quantization commutes with reduction for toric manifolds

For a symplectic toric manifold we consider half-form quantization in mixed polarizations $\mathcal{P}_\infty$, associated to the action of a subtorus $T^p\subset T^n$. The real directions in these polarizations are generated by components of the $T^p$ moment map. Polarizations of this type can be obtained by starting at a toric Kähler polarization $\mathcal{P}_0$ and then following Mabuchi rays of toric Kähler polarizations generated by the norm square of the moment map of the torus subgroup. These geodesic rays are lifted to the quantum bundle via a generalized coherent state transform (gCST) and define equivariant isomorphisms between Hilbert spaces for the Kähler polarizations and the Hilbert space for the mixed polarization. The polarizations $\mathcal{P}_\infty$ give a new way of looking at the problem of unitarity in the quantization commutes with reduction with respect to the $T^p$-action, as follows. The prequantum operators for the components of the moment map of the $T^p$-action act diagonally with discrete spectrum corresponding to the integral points of the moment polytope. The Hilbert space for the quantization with respect to $\mathcal{P}_\infty$ then naturally decomposes as a direct sum of the Hilbert spaces for all its quantizable coisotropic reductions which, in fact, are the Kähler reductions of the initial Kähler polarization $\mathcal{P}_0$. This will be shown to imply that, for the polarization $\mathcal{P}_\infty$, quantization commutes unitarily with reduction. The problem of unitarity in quantization commutes with reduction for $\mathcal{P}_0$ is then equivalent to the question of whether quantization in the polarization $\mathcal{P}_0$ is unitarily equivalent with quantization in the polarization $\mathcal{P}_\infty$. In fact, this does not hold in general in the toric case.

math.SG↗

arXiv2024

Mabuchi rays, test configurations and quantization for toric manifolds

We consider Mabuchi rays of toric Kähler structures on symplectic toric manifolds which are associated to toric test configurations and that are generated by convex functions on themoment polytope, $P$, whose second derivative has support given by a compact subset $K<P$. Associated to the test configuration there is a polyhedral decomposition of $P$ whose components are approximated by the components of $P \setminus K$. Along such Mabuchi rays, the toric complex structure remains unchanged on the inverse image under the moment map of $(P \setminus \check {K})$, where $\check {K}$ denotes the interior of $K$. At infinite geodesic time, the Kähler polarizations along the ray converge to interesting new toric mixed polarizations. The quantization in these limit polarizations is given by restrictions of the monomial holomorphic sections of the Kähler quantization, for monomials corresponding to integral points in $P \setminus \check {K}$, and by sections on the fibers of the moment map over the integral points contained in $\check {K}$, which, along the directions parallel to $K$ are holomorphic and which along the directions transverse to $K$ are distributional. These quantizations correspond to quantizations of the central fiber of the test family, in the symplectic picture. We present the case of $S2$ in detail and then generalize to higher dimensional symplectic toric manifolds. Metrically, at infinite Mabuchi geodesic time, the sphere decomposes into two discs and a collection of cylinders, separated by infinitely long lines. Correspondingly, the quantization in the limit polarization decomposes into a direct sum of the contributions from the quantizations of each of these components.

math.DG↗

arXiv2024

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type $T^*(U/K)\cong U_\mathbb{C}/K_\mathbb{C}$, along Mabuchi rays of $U$-invariant Kähler structures. At infinite geodesic time, the Kähler polarizations converge to a mixed polarization $\mathcal{P}_\infty$. We show how a generalized coherent state transform relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional $\mathcal{P}_\infty$-polarized sections. Unlike in the case of $T^*U$, the gCST mapping from the Hilbert space of vertically polarized sections are not asymptotically unitary due to the appearance of representation dependent factors associated to the isotypical decomposition for the $U$-action. In agreement with the general program outlined in [Bai+23], we also describe how the quantization in the limit polarization $\mathcal{P}_\infty$ is given by the direct sum of the quantizations for all the symplectic reductions relative to the invariant torus action associated to the Hamiltonian action of $U$.

math.SG↗

arXiv2023

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.

math.SG↗

arXiv2021

Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics

We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hamiltonian $S^1$-symmetry. For geodesics associated with the square of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of $S^1$--orbits, corresponding to Bohr-Sommerfeld orbits of geometric quantization. The lifting of the Mabuchi geodesics to the bundle of quantum states, to which the Laughlin states belong, is achieved via generalized coherent state transforms, which correspond to the KZ parallel transport of Chern-Simons theory.

math-ph↗

arXiv2019

Partial coherent state transforms, $G \times T$-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups

In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $G\times T$-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 $G\times G$-invariant function, $h$, and one $G\times T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $G\times G$-invariant Kähler structures gives new $G\times T$-invariant, but not $G\times G$-invariant, Kähler structures on $T^*G$. We study the Hilbert spaces ${\mathcal H}_{τ,σ}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kähler structures. On the other hand, by taking the vertical Schrödinger polarization as a starting point, the above $G\times T$-invariant Hamiltonian flows also generate families of mixed polarizations $\mathcal{P}_{0,σ}, σ\in {\mathbb C}, {\rm Im}(σ) >0$. 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 Kähler 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].

math.DG↗

arXiv2018

Picard group and quantization of toric orbifolds

In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a combinatorial description of the orbifold Picard group for complete toric orbifolds, with the aim of detailing the relation between complex and symplectic aspects in the orbifold setting. In particular this permits to illustrate the breakdown of identification of (orbifold) line bundles by their Chern class (or moment polytope up to translations in $\mathfrak{t}^\ast$), and non-constancy of $h^0$ on representatives of the same Chern class. As an application, we discuss symplectic reduction with respect to restrictions of the action to sub-tori, and the associated Bohr--Sommerfeld conditions in mixed polarizations.

math.DG↗

arXiv2015

Quantization in singular real polarizations: Kähler regularization, Maslov correction and pairings

We study the Maslov correction to semiclassical states by using a Kähler regularized BKS pairing map from the energy representation to the Schrödinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of Kähler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of Kähler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration space are correctly reproduced.

math-ph↗

arXiv2014

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

Let $P$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $X_{P}$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $t = \sqrt{-1} s$. We relate the quantization of $X_{P}$ in two different toric Kähler polarizations by taking the time-$\sqrt{-1} s$ Hamiltonian "flow" of strongly convex functions on the moment polytope $P$. By taking $s$ to infinity, we obtain the quantization of $X_{P}$ in the (singular) real toric polarization. Recall that $X_{P}$ has an open dense subset which is biholomorphic to $({\mathbb{C}}^{*})^{n}$. The quantization of $X_{P}$ in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu--Guillemin symplectic potential $g$, at time $t=\sqrt{-1}$, to an appropriate finite-dimensional subspace of quantum states in the quantization of $T^{*}{\mathbb{T}}^{n}$ in the vertical polarization. By taking other imaginary times, $t= k \sqrt{-1}, k\in {\mathbb{R}}$, we describe toric Kähler metrics with cone singularities along the toric divisors in $X_{P}$. For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of $X_{P}$. We show that the pointwise and $L^2$-norms of quantum states are asymptotically vanishing on negative-definite regions.

math.DG↗

arXiv2012

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

We study the half-form Kaehler quantization of a smooth symplectic toric manifold $(X,ω)$, such that $[ω/2π]-c_{1}(X)/2 \in H^{2}(X,{\mathbb{Z}})$ and is nonnegative. We define the half-form corrected quantization of $(X,ω)$ to be given by holomorphic sections of a certain hermitian line bundle $L\rightarrow X$ with Chern class $[ω/ 2π]-c_{1}(X)/2$. 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 $(X,ω)$, 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 Kaehler 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:arXiv/0806.0606], which include the degeneration of Kaehler 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 Kaehler polarizations is not unitary in general. On the other hand, the unitary connection induced by this pairing is flat.

math.DG↗

arXiv2012

Coherent state transforms and the Mackey-Stone-Von Neumann theorem

Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The Kähler polarizations in the family are generated by (complex) time-$τ$ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of $T^{\ast}K$ is then studied by considering covariant pairs of representations of $K$ defined by geometric prequantization and of representations of $C^0(K)$ defined via Heisenberg time-$(-τ)$ evolution followed by time-$(+τ)$ geometric-quantization-induced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time-$(+τ)$ geometric-quantization-induced evolution with the time-$(-τ)$ evolution associated with the momentum space [W. D. Kirwin and S. Wu, Momentum space for compact Lie groups and the Peter-Weyl theorem, to appear] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.

math.DG↗

arXiv2012

Complex time evolution in geometric quantization and generalized coherent state transforms

For the cotangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between $L^{2}(K)$ and a certain weighted $L^{2}$-space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time $-τ$) within $L^{2}(K)$, followed by a polarization--changing geometric quantization evolution (for complex time $+τ$). In this case, our construction yields the usual generalized Segal--Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.

math.DG↗

arXiv2011

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

math.DG↗

arXiv2010

Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations

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 nonnegative polarizations. The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of $Sp(2g,\R)$. Moreover, these maps are shown to be unitary.

math.SG↗