Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.
problem Adapting classical theorems to prequantum systems.
method Establishing analogs of the Darboux, Moser, and Weinstein theorems.
result Prequantum systems with vanishing first cohomology are equivalent up to symplectomorphism and gauge transformation.
Surveying higher prequantum geometry and its applications.
problem Prequantizing local field theory locally and invariantly.
method Abstract cohesive homotopy theory.
result Existence of a solution in higher differential geometry.
Smooth structures found on Hilbert spaces varying with parameters.
problem Finding natural smooth structures on Hilbert spaces varying with parameters.
method Geometric quantization with half-form correction, producing a field of Hilbert spaces.
result A field of Hilbert spaces can have natural inequivalent smooth Hilbert bundle structures.
Constructs differential characters on nonlinear Graßmannians.
problem No specific problem stated; focuses on mathematical construction.
method Using a nonlinear version of the tautological bundle, a transgression map is constructed from M to nonlinear Graßmannians of submanifolds of fixed type. result Obtains prequantum circle bundles and central Lie group extensions.
Geometric quantization extended to arbitrary connected spaces using path integration.
problem Constructing a Prequantum Groupoid for arbitrary connected parasymplectic spaces.
method Define a Total Group of Periods and a Prequantum Groupoid with connected isotropy.
result The Prequantum Groupoid Tω is isomorphic to the group of symmetries of the Dynamical System. The paper quantizes the Hitchin system using geometric quantization.
problem Quantizing the Hitchin system using geometric methods.
method Geometric quantization of the Hitchin moduli space by modifying the Quillen metric.
result The modified Kahler form is integral and descends as a prequantum line bundle.
The paper studies geometric representations of submanifolds using complex-valued functions.
problem Exploring the geometry of codimension-2 submanifolds.
method Implicitly representing submanifolds by complex-valued functions and showing a prequantum bundle structure.
result The space of implicit representations admits a prequantum bundle structure over the space of submanifolds.
The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized ("extended", "multi-tiered") to a map on t…
New methods solve complex equations, proving some have solutions.
problem Solving complex equations in infinite dimensions.
method Infinite-dimensional prequantum line bundles and moment maps.
result Proves some perturbed equations have solutions for small parameters.
In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the L2 metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle…
We show that the prequantum line bundle on the moduli space of flat SU(2) connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of fl…
The Bergman metrics converge to symplectic forms at a rate of 1/p^2.
problem Understanding convergence rates of Bergman metrics on symplectic manifolds.
method Embedding symplectic manifolds in projective spaces and analyzing the convergence of induced Fubini-Study forms.
result The convergence rate of Bergman metrics to symplectic forms is 1/p^2.
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli sp…
Constructs a sheaf of Bargmann-Fock modules on Kähler manifolds.
problem Deformation quantization on Kähler manifolds.
method Extends Fedosov's method to Kähler manifolds with compatible Fedosov abelian connections.
result Explicit construction of sheaf of flat sections as a module over deformation quantization algebras.
Constructs 2-Hilbert space for line bundle gerbes.
problem Higher prequantisation and line bundle gerbes.
method Constructs a 2-Hilbert space category from morphism categories of line bundle gerbes.
result Shows the constructed 2-Hilbert space is semisimple and abelian.
The Madelung transform connects quantum mechanics and hydrodynamics.
problem Quantum mechanics and hydrodynamics equivalence for generic wave functions.
method Poisson geometry and coadjoint orbits of semidirect extensions of diffeomorphism groups.
result The Madelung transform provides a natural infinite-dimensional version of convexity results.
In this paper we construct a family of complex structures on a complex flag manifold that converge to the real polarization coming from the Gelfand-Cetlin integrable system, in the sense that holomorphic sections of a prequantum line bundle converge to delta-function sections supported on the Bohr-Sommerfeld fibers. Ou…
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of ∂ˉ-Laplacians on prequantum line bundles. The paper proves spectral convergence for a specific type of geometric quantization.
problem Spectral convergence of ∂-Laplacians on toric symplectic manifolds. method Study of a family of compatible complex structures converging to the large complex structure limit.
result Spectral convergence of ∂-Laplacians acting on Lk. Geometrically constructs representations for quantization on Kähler manifolds.
problem Quantization of Kähler manifolds using Berezin-Toeplitz method.
method Using peak sections to localize Hilbert spaces around points in the large volume limit.
result Geometric construction of representations for Berezin-Toeplitz quantization.
We formalize higher dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for s…
Study SO(3)-knot states for torus complements, linking to simplicial volume.
problem Capturing simplicial volume of knot complements via knot states.
method Holomorphic sections, geometric quantization, Witten-Chern-Simons theory.
result Large r asymptotics of L2-norm of SO(3)-knot states relate to simplicial volume. We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler struct…
The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures ΩΨ0 on the moduli space, parametrised by Ψ0, a section o…
Quantizes functions on Kähler manifolds without formal deformation.
problem Deforming smooth functions on Kähler manifolds to non-formal quantization.
method Using Fedosov connections and prequantum line bundles.
result Quantizable functions form a sheaf of twisted differential operators.
Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…
Geometric quantization adapted to polysymplectic manifolds.
problem Quantization of polysymplectic manifolds.
method Adapted geometric quantization framework to polysymplectic setting.
result Polysymplectic Guillemin-Sternberg conjecture is shown to be false with a complex polarization.
Quantum propagation studied for Berezin-Toeplitz operators.
problem Asymptotic behavior of quantum propagators and spectral projectors.
method Geometric analysis of Hamiltonian flows and Maslov indices.
result Introduction of quantum states associated with Lagrangian submanifolds.
We associate a homotopy Poisson-n algebra to any higher symplectic structure, which generalizes the common symplectic Poisson algebra of smooth functions. This provides robust n-plectic prequantum data for most approaches to quantization. UPDATE: It has been brought to my attention that the exterior product does not cl…
New contact structures defined on differentiable stacks.
problem Defining contact structures on differentiable stacks.
method Introducing 0-shifted and +1-shifted contact structures. result Shifted contact structures provide new insights into geometry.
Develops 2-Hilbert spaces for bundle gerbes in geometric quantization.
problem Higher geometric quantization of bundle gerbes.
method Introduces 2-Hilbert spaces, rig-categories, and duals for bundle gerbes.
result Constructs a 2-Hilbert space of sections for bundle gerbes.
Geometric quantization on hyperKähler manifolds via brane quantization.
problem Quantization of hyperKähler manifolds with Sp(1) symmetry.
method Intertwining Sp(1) action on cohomology of hyperKähler manifold.
result Establishes geometric quantization and proposes new mathematical definition.
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 m…
Extends geometric quantization to singular spaces.
problem Handling singular symplectic spaces in geometric quantization.
method Developed stratified pseudobundles to replace auxiliary information.
result Provided results for singular quotients of toric manifolds and cotangent bundles.
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.
problem Understanding the convergence of frame bundle metrics and their relation to Gromov-Hausdorff convergence.
method Analyzes the one-parameter family of Riemannian metrics on the frame bundle of a prequantum line bundle, considering the convergence to metric measure spaces with S1-actions. result The frame bundle metrics converge to metric measure spaces with S1-actions, which depend on the choice of base point in Bohr-Sommerfeld fibers. The paper proposes a noncommutative deformation of toric varieties.
problem Deforming complex structures into generalized complex structures.
method Using holomorphic Poisson tensors and generalized complex branes.
result Noncommutative deformation of homogeneous coordinate rings.
The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.
The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors has a well known description: it is given on the moduli space of fields by the fiber integration of the cup product square of classes in degree-(2k+2) differential cohomology. We first refine this statement from …
The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group G. The first chapter is intended to recall some facts about Lie groups. The mos…
An f-structure on a manifold M is an endomorphism field $φ\inΓ(M,\End(TM))$ such that φ3+φ=0. Any f-structure φ determines an almost CR structure $E_{1,0}\subset T_\C M$ given by the +i-eigenbundle of φ. Using a compatible metric g and connection ∇ on M, we construct an odd first-order differe…
The paper studies geometric quantization and theta functions for Lagrangian fibrations.
problem Geometric quantization of Lagrangian fibrations without compactness assumptions.
method Adiabatic limit and theta functions for Spin^c Dirac operators.
result Orthogonal systems of sections converging to delta-functions or zero under adiabatic limits.
This paper connects symplectic and Kähler manifolds via brane quantization.
problem Quantizing Kähler manifolds using brane techniques.
method Using physical proposals and geometric quantization, the authors relate A-model morphism spaces to quantizations of symplectic and Kähler manifolds.
result Chan-Leung-Li's work provides a mathematical realization of the action of A-branes on B-branes, linking deformation quantizations of symplectic and Kähler manifolds.
New method shows unitarity in quantization for toric manifolds.
problem Unitarity in quantization commutes with reduction for toric manifolds.
method Generalized coherent state transform (gCST) and geodesic rays of toric Kähler polarizations.
result Quantization commutes unitarily with reduction for the new mixed polarization.