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.
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…
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 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.
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.
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.
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.
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…
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.
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.
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. 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.
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.
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.
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. 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. 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.
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 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…
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.
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.
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.
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…
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.
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. 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.
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.
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 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…
We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …
Logarithmic Picard algebroids solve meromorphic line bundle prequantization.
problem Classifying meromorphic line bundles on smooth projective varieties.
method Introducing logarithmic Picard algebroids and using their cohomology.
result Logarithmic Picard algebroids classify meromorphic line bundles.
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
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 extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. The paper estimates the dimension of cohomology for semipositive line bundles on various manifolds.
problem Estimating the dimension of cohomology for semipositive line bundles over different types of manifolds.
method Asymptotic estimates for harmonic (0,q)-forms with high tensor powers of semipositive line bundles. result Asymptotic estimates for the dimension of cohomology of semipositive line bundles on various manifolds.
Study differential invariants for line bundle operators.
problem Conditions for equivalence of differential operators in line bundles.
method Use differential invariants and groups of automorphisms.
result Find conditions for equivalence of differential operators.
Let M be an irreducible smooth complex projective variety equipped with an action of a compact Lie group G, and let (L,h) be a G-equivariant holomorphic Hermitian line bundle on M. Given a compact connected Riemann surface X, we construct a G-equivariant holomorphic Hermitian line bundle $(L\,,…
Paper proves ε-regularity for line bundle mean curvature flow.
problem Proving regularity for a specific type of geometric flow.
method Develops a scale-invariant monotone quantity and defines self-shrinkers.
result Establishes ε-regularity theorem for line bundle mean curvature flow.
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…