Deformation Quantization of Endomorphism Bundles

Geometric Quantization of Vector Bundles

I repeat my definition for quantization of a vector bundle. For the case of Toeplitz and geometric quantization of a compact Kaehler Manifold, I give a construction for quantizing any smooth vector bundle which depends functorially on a choice of connection on the bundle.

Quantization of Equivariant Vector Bundles

The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (Coadjoint orbits are symplectic spaces with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle.

Parametrised strict deformation quantization of C*-bundles and Hilbert C*-modules

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrised strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrised stric…

Dynamical Yang-Baxter equation and quantum vector bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, {\em etc}. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and their Levi subalgebra. Using this twist, we obtain an equivariant star product quantizati…

The action of outer automorphisms on bundles of chiral blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik-Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applicat…

Deformation Quantization of Principal Fibre Bundles and Classical Gauge Theories

In this dissertation the notion of deformation quantization of principal fibre bundles is established and investigated in order to find a geometric formulation of classical gauge theories on noncommutative space-times. As a generalization, the notion of deformation quantization of surjective submersions is also discussed. It is shown that deformation quantizations of surjective submersions and principal fibre bundles always exist and are unique up to equivalence. These statements concerning complex-valued functions are moreover formulated and proved for sections of arbitrary vector bundles ove…

Noncommutative principal torus bundles via parametrised strict deformation quantization

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.

Nonassociative strict deformation quantization of C*-algebras and nonassociative torus bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with $H$-flux. We also show that the Octonions fit nicely into our theory.

A new construction of strict deformation quantization for Lagrangian fiber bundles

We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operators and principal symbols". We analyze the corresponding formal deformation quantization. We also investigate into relations between our construction and Berezin-Toeplitz deformation quantization.

Gauge Theory for Spectral Triples and the Unbounded Kasparov Product

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we …

Deformation quantization of principal bundles

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next we twist deform a subgroup of the group of authomorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations we obtain noncommutative principal bundles with noncommutative fiber and bas…

Deformation Quantization of Surjective Submersions and Principal Fibre Bundles

In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operator…

Quantization of Forms on Cotangent Bundle

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on $T^*M$ is made into a space of (full) symbols of operators acting on forms on $M$. This gives rise to the composition of symbols, which is a deformation of the (``super'')commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenböck like identities appear here.) Formulas for the traces corresponding to natural gradings of $Ω(T^*M)$ are esta…

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove that the classical moment map could be quantized if there exists an equivariant connection. Along the way we touch upon the general question of quantization with values in a bundle of algebras. We consider Fedosov's construction of deformation quantization in general. In the App…

Quantization of spectral curves and DQ-modules

Given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we construct a DQ-module supported by the spectral curve associated to this bundle. Then, we relate quantum curves arising in various situations (quantization of spectral curves of Higgs Bundles, quantization of the $A$-polynomial...) and DQ-modules and show that DQ-modules provide a suitable framework to study the quantization of spectral curves.

Fedosov connections on jet bundles and deformation quantization

We review our construction of star-products on Poisson manifolds and discuss some examples. In particular, we work out the relation with Fedosov's original construction in the symplectic case.

Quantization of spectral curves for meromorphic Higgs bundles through topological recursion

A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank $2$ meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees $D$-module on the base. The topological recursion then gives an all-order asymptotic expansion of its solution, thus determining a state vector corresponding to the spectral curve as a meromorphic Lagrangian. We establish a generalization of the topological recursion for a singular s…

Quantizations of Hitchin and Beauville-Mukai integrable systems

Spectral transformation is known to set up a birational morphism between the Hitchin and Beauville-Mukai integrable systems. The corresponding phase spaces are: (a) the cotangent bundle of the moduli space of bundles over a curve C, and (b) a symmetric power of the cotangent surface T^*(C). We conjecture that this morphism can be quantized, and we check this conjecture in the case where C is a rational curve with marked points and rank 2 bundles. We discuss the relation of the resulting isomorphism of quantized algebras with Sklyanin's separation of variables.

On the Deformation Quantization of super-Poisson Brackets

We show that for every vector bundle E over any given symplectic manifold M there exists an explicitly given super-Poisson bracket on the space of sections of the dual Grassmann bundle associated to E built out of symplectic structure of M, a fibre metric on E, and a connection in E compatible with the given fibre metric. Moreover, we construct a deformation quantization for this space of sections by means of a Fedosov type procedure.

Wick Quantization of Cotangent Bundles over Riemannian Manifolds

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal Kähler structure. The formality means that the metric is lifted from the Riemannian manifold $Q$ to its phase space $T^\ast Q$ in the form of formal power series in momenta with the coefficients being tensor fields on the base. The corresponding Kähler two-form on the total space of $T^\ast Q$ coincides with the canonical symplectic form, while the canonical projection of the Kähle…

通过量子化对射影空间余切丛的超Kähler结构的简单描述

A Simple Description of the Hyperkähler Structure of the Cotangent Bundle of Projective Space via Quantization

本文利用量子化将射影空间余切丛嵌入有界算子空间，证明该嵌入保持超Kähler与Hilbert–Schmidt几何，并显式刻画其结构。

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper $^*$-algebras $\mathcal A$ over rings $\ring C = \ring{R}(i)$, where $\ring R$ is an ordered ring and $i^2 = -1$, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) $\mathcal A$-valued inner product. For $A=C^\infty(M)$, M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star-product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that…

Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

Let $\XR$ be a (generalized) flag manifold of a non-compact real semisimple Lie group $\GR$, where $\XR$ and $\GR$ have complexifications X and G. We investigate the problem of constructing a graded star product on $Pol(T^*\XR)$ which corresponds to a $\GR$-equivariant quantization of symbols into smooth differential operators acting on half-densities on $\XR$. We show that any solution is algebraic in that it restricts to a G-equivariant graded star product star on the algebraic part R of $Pol(T^*\XR)$. We construct, when R is generated by the momentum functions $μ^x$ for G, a preferred choic…

The differential geometry of Fedosov's quantization

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally,…

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal M}$ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ${{\mathcal T}^*_ρ}{\ma…

On global deformation quantization in the algebraic case

We give a proof of Yekutieli's global algebraic deformation quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.

The Verlinde formula for Higgs bundles

We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We cons…

Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact K"ahler manifolds obtained via Berezin-Toeplitz operators. After choosing a holomorphic quantum line bundle the Berezin-Toeplitz operator associated to a differentiable function on the manifold is the operator defined by multiplying global holomorphic sections of the line bundle with this function and projecting the differentiable section back to the subspace of holomorphic sections. The results were obtained in (respectively based on) joint work wi…