67 research outputs found
Laguerre semigroup and Dunkl operators
We construct a two-parameter family of actions \omega_{k,a} of the Lie
algebra sl(2,R) by differential-difference operators on R^N \setminus {0}.
Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from
the interpolation of the Weil representation of Mp(N,R) and the minimal unitary
representation of O(N+1,2) keeping smaller symmetries.
We prove that this action \omega_{k,a} lifts to a unitary representation of
the universal covering of SL(2,R), and can even be extended to a holomorphic
semigroup \Omega_{k,a}. In the k\equiv 0 case, our semigroup generalizes the
Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the
second author with G. Mano (a=1).
One boundary value of our semigroup \Omega_{k,a} provides us with
(k,a)-generalized Fourier transforms F_{k,a}, which includes the Dunkl
transform D_k (a=2) and a new unitary operator H_k (a=1), namely a Dunkl-Hankel
transform.
We establish the inversion formula, and a generalization of the Plancherel
theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty
inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and
F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining
operator.Comment: final version (some few typos, updated references
Noncommutative Geometry
Noncommutative Geometry applies ideas from geometry to mathematical structures determined by noncommuting variables. This meeting emphasized the connections of Noncommutative Geometry to number theory and ergodic theory
Arithmetic and Hyperbolic Structures in String Theory
This monograph is an updated and extended version of the author's PhD thesis.
It consists of an introductory text followed by two separate parts which are
loosely related but may be read independently of each other. In Part I we
analyze certain hyperbolic structures arising when studying gravity in the
vicinity of a spacelike singularity (the "BKL-limit"). In this limit, spatial
points decouple and the dynamics exhibits ultralocal behaviour which may be
described in terms of a (possibly chaotic) hyperbolic billiard. In all
supergravities arising as low-energy limits of string theory or M-theory, the
billiard dynamics takes place within the fundamental Weyl chambers of certain
hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden
infinite-dimensional symmetries of the theory. Part II of the thesis is devoted
to a study of how (U-)dualities in string theory provide powerful constraints
on perturbative and non-perturbative quantum corrections. These dualities are
described by certain arithmetic groups G(Z) which are conjectured to be
preserved in the effective action. The exact couplings are given by automorphic
forms on the double quotient G(Z)\G/K. We discuss in detail various methods of
constructing automorphic forms, with particular emphasis on non-holomorphic
Eisenstein series. We provide detailed examples for the physically relevant
cases of SL(2,Z) and SL(3,Z), for which we construct their respective
Eisenstein series and compute their (non-abelian) Fourier expansions. We also
show how these techniques can be applied to hypermultiplet moduli spaces in
type II Calabi-Yau compactifications, and we provide a detailed analysis for
the universal hypermultiplet.Comment: 346 pages, updated and extended version of the author's PhD thesi
C*-algebraic results in the search for quantum gauge fields
This thesis consists of two parts, both situated in operator theory, and both
motivated by the quest for rigorous quantizations of gauge theories. The first
part is based on [Skripka,vN - JST 2022], [van Suijlekom,vN - JNCG 2021], and
[van Suijlekom,vN - JHEP 2022], and concerns the spectral action of
noncommutative geometry and its perturbative expansions. We prove the existence
of a higher-order spectral shift function under the relative Schatten class
assumption, give a converging series expansion of the spectral action in terms
of Chern--Simons and Yang--Mills forms, and show one-loop renormalizability of
the spectral action in a generalized sense. The second part is based on
[Stienstra,vN 2020] and [vN - LMP 2022] and concerns a non-perturbative
approach to quantum gauge theory by means of Hamiltonian lattice gauge theory
and strict quantization. We construct C*-algebras of U(1)^n-gauge observables
on the lattice, show that they are conserved under the relevant time
evolutions, construct continuum limit C*-algebras, and show that the result
constitutes a strict deformation quantization.Comment: PhD thesis. 149 pages, 3 figure
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