arXiv subjects

Markus J. Pflaum

Publications and source records attributed to Markus J. Pflaum.

33 records · Page 2Linked to original sources

arXiv2011

Connes-Chern character for manifolds with boundary and eta cochains

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.

math.OA↗

arXiv2011

Liber Mathematicae: A Web-Based Documentation and Collaboration Project for Mathematics

Traditionally, mathematical knowledge is published in printed media such as books or journals. With the advent of the Internet, a new method of publication became available. To date, however, most online mathematical publications do not employ the full capabilities of the medium. We describe a project to modernize online mathematics presentation and build a community-focused environment in which the lines between "author" and "reader" are blurred, enhancing collaboration and improving publication quality.

math.HO↗

arXiv2008

On the existence of star products on quotient spaces of linear Hamiltonian torus actions

We discuss BFV deformation quantization of singular symplectic quotient spaces in the special case of linear Hamiltonian torus actions. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.

math.QA↗

arXiv2006

A homological approach to singular reduction in deformation quantization

We use the method of homological quantum reduction to construct a deformation quantization on singular symplectic quotients in the situation, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others one where the singularity type is worse than an orbifold singularity.

math-ph↗

arXiv2005

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann--Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.

math.KT↗

arXiv2005

Homology of formal deformations of proper etale Lie groupoids

In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper etale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

math.KT↗

arXiv2004

Phase Space Reduction of Star Products on Cotangent Bundles

In this paper we construct star products on Marsden-Weinstein reduced spaces in case both the original phase space and the reduced phase space are (symplectomorphic to) cotangent bundles. Under the assumption that the original cotangent bundle $T^*Q$ carries a symplectique structure of form $ω_{B_0}=ω_0 + π^*B_0$ with $B_0$ a closed two-form on $Q$, is equipped by the cotangent lift of a proper and free Lie group action on $Q$ and by an invariant star product that admits a $G$-equivariant quantum momentum map, we show that the reduced phase space inherits from $T^*Q$ a star product. Moreover, we provide a concrete description of the resulting star product in terms of the initial star product on $T^*Q$ and prove that our reduction scheme is independent of the characteristic class of the initial star product. Unlike other existing reduction schemes we are thus able to reduce not only strongly invariant star products. Furthermore in this article, we establish a relation between the characteristic class of the original star product and the characteristic class of the reduced star product and provide a classification up to $G$-equivalence of those star products on $(T^*Q,ω_{B_0})$, which are invariant with respect to a lifted Lie group action. Finally, we investigate the question under which circumstances `quantization commutes with reduction' and show that in our examples non-trivial restrictions arise.

math.SG↗

arXiv2002

On the deformation quantization of symplectic orbispaces

In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces which originally had been introduced by Chen. We explain the notion of a vector orbibundle and characterize the good sections of a reduced vector orbibundle as the smooth stratified sections. In the second part of the article we elaborate on the quantizability of a symplectic orbispace. By adapting Fedosov's method to the orbispace setting we show that every symplectic orbispace has a deformation quantization. As a byproduct we obtain that every symplectic orbifold possesses a star product.

math-ph↗

arXiv1999

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This allows to construct various trace functionals in a systematic way. Furthermore we study the higher-dimensional eta-invariants on algebras with parameter space $\R^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\R^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

math.FA↗

arXiv1998

On representations of star product algebras over cotangent spaces on Hermitian line bundles

For every formal power series $B=B_0 + λB_1 + O(λ^2)$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $κ\in [0,1]$ we construct a concrete star product $\star^B_κ$ on the cotangent bundle $π: T^*Q\to Q$. The star product $\star^B_κ$ is associated to the formal symplectic form on $T^*Q$ given by the sum of the canonical symplectic form $ω$ and the pull-back of $B$ to $T^*Q$. Deligne's characteristic class of $\star^B_κ$ is calculated and shown to coincide with the formal de Rham cohomology class of $π^*B$ divided by $\imλ$. Therefore, every star product on $T^*Q$ corresponding to the Poisson bracket induced by the symplectic form $ω+ π^*B_0$ is equivalent to some $\star^B_kappa$. It turns out that every $\star^B_kappa$ is strongly closed. In this paper we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on $Q$. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion.

math.QA↗

arXiv1997

The normal symbol on Riemannian manifolds

For an arbitrary Riemannian manifold $X$ and Hermitian vector bundles $E$ and $F$ over $X$ we define the notion of the normal symbol of a pseudodifferential operator $P$ from $E$ to $F$. The normal symbol of $P$ is a certain smooth function from the cotangent bundle $T^*X$ to the homomorphism bundle $Hom (E,F)$ and depends on the metric structures resp. the corresponding connections on $X$, $E$ and $F$. It is shown that by a natural integral formula the pseudodifferential operator $P$ can be recovered from its symbol. Thus, modulo smoothing operators resp. smoothing symbols, we receive a linear bijective correspondence between the space of symbols and the space of pseudodifferential operators on $X$. This correspondence comprises a natural transformation between appropriate functors. A formula for the asymptotic expansion of the product symbol of two pseudodifferential operators in terms of the symbols of its factors is given. Furthermore an expression for the symbol of the adjoint is derived. Finally the question of invertibility of pseudodifferential operators is considered. For that we use the normal symbol to establish a new and general notion of elliptic pseudodifferential operators on manifolds.

dg-ga↗

arXiv1996

Differential calculi on noncommutative bundles

We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a connection with respect to a differential calculus and consider questions of existence and uniqueness. At the end these constructions are applied to basic examples of noncommutative bundles over a coquasitriangular Hopf algebra.

q-alg↗

arXiv1996

Holomorphic deformation of Hopf algebras and applications to quantum groups

In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus giving the means to actually insert complex values for the deformation parameter. Moreover we establish a topological duality theory for locally convex Hopf algebras. Examples coming from the theory of quantum groups are reconsidered within our holomorphic deformation scheme and topological duality theory. It is shown that all the standard quantum groups comprise holomorphic deformations. Furthermore we show that quantizing the function algebra of a (Poisson) Lie group and quantizing its universal enveloping algebra are topologically dual procedures indeed. Thus holomorphic deformation theory seems to be the appropriate language in which to describe quantum groups as deformed Lie groups or Lie algebras.

q-alg↗

arXiv1996

A new concept of deformation quantization, I. Normal order quantization on cotangent bundles

In this work we give a deformation theoretical approach to the problem of quantization. First the notion of a deformation of a noncommutative ringed space over a commutative locally ringed space is introduced within a language coming from Algebraic Geometry and Complex Analysis. Then we define what a Dirac quantization of a commutative ringed space with a Poisson structure, the space of classical observables, is. Afterwards the normal order quantization of the Poisson space of classical polynomial observables on a cotangent bundle is constructed. By using a complete symbol calculus on manifolds we succeed in extending the normal order quantization of polynomial observables to a quantization of a Poisson space of symbols on a cotangent bundle. Furthermore we consider functorial properties of these quantizations. Altogether it is shown that a deformation theoretical approach to quantization is possible not only in a formal sense but also such that the deformation parameter $\hbar$ can attain any real value.

hep-th↗

arXiv1994

Quantum Groups on Fibre Bundles

It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the end $q$-deformed instanton models are introduced for every integral index.

hep-th↗