arXiv subjects

João P. Nunes

Publications and source records attributed to João P. Nunes.

23 records · Page 2Linked to original sources

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↗

arXiv2011

Quantization of some moduli spaces of parabolic vector bundles on CP^1

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).

math.AG↗

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↗