6,795 research outputs found
Non-commutative Hilbert modular symbols
The main goal of this paper is to construct non-commutative Hilbert modular
symbols. However, we also construct commutative Hilbert modular symbols. Both
the commutative and the non-commutative Hilbert modular symbols are
generalizations of Manin's classical and non-commutative modular symbols. We
prove that many cases of (non-)commutative Hilbert modular symbols are periods
in the sense on Kontsevich-Zagier. Hecke operators act naturally on them.
Manin defines the non-commutative modilar symbol in terms of iterated path
integrals. In order to define non-commutative Hilbert modular symbols, we use a
generalization of iterated path integrals to higher dimensions, which we call
iterated integrals on membranes. Manin examines similarities between
non-commutative modular symbol and multiple zeta values both in terms of
infinite series and in terms of iterated path integrals. Here we examine
similarities in the formulas for non-commutative Hilbert modular symbol and
multiple Dedekind zeta values, recently defined by the author, both in terms of
infinite series and in terms of iterated integrals on membranes.Comment: 50 pages, 5 figures, substantial improvement of the article
arXiv:math/0611955 [math.NT], the portions compared to the previous version
are: Hecke operators, periods and some categorical construction
Quantum computation with Turaev-Viro codes
The Turaev-Viro invariant for a closed 3-manifold is defined as the
contraction of a certain tensor network. The tensors correspond to tetrahedra
in a triangulation of the manifold, with values determined by a fixed spherical
category. For a manifold with boundary, the tensor network has free indices
that can be associated to qudits, and its contraction gives the coefficients of
a quantum error-correcting code. The code has local stabilizers determined by
Levin and Wen. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. By studying braid
group representations acting on equivalence classes of colored ribbon graphs
embedded in a punctured sphere, we identify the anyons, and give a simple
recipe for mapping fusion basis states of the doubled category to ribbon
graphs. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
Combined with known universality results for anyonic systems, this provides a
large family of schemes for quantum computation based on local deformations of
stabilizer codes. These schemes may serve as a starting point for developing
fault-tolerance schemes using continuous stabilizer measurements and active
error-correction.Comment: 53 pages, LaTeX + 199 eps figure
Localization and the interface between quantum mechanics, quantum field theory and quantum gravity I (The two antagonistic localizations and their asymptotic compatibility)
It is shown that there are significant conceptual differences between QM and
QFT which make it difficult to view the latter as just a relativistic extension
of the principles of QM. At the root of this is a fundamental distiction
between Born-localization in QM (which in the relativistic context changes its
name to Newton-Wigner localization) and modular localization which is the
localization underlying QFT, after one separates it from its standard
presentation in terms of field coordinates. The first comes with a probability
notion and projection operators, whereas the latter describes causal
propagation in QFT and leads to thermal aspects of locally reduced finite
energy states. The Born-Newton-Wigner localization in QFT is only applicable
asymptotically and the covariant correlation between asymptotic in and out
localization projectors is the basis of the existence of an invariant
scattering matrix. In this first part of a two part essay the modular
localization (the intrinsic content of field localization) and its
philosophical consequences take the center stage. Important physical
consequences of vacuum polarization will be the main topic of part II. Both
parts together form a rather comprehensive presentation of known consequences
of the two antagonistic localization concepts, including the those of its
misunderstandings in string theory.Comment: 63 pages corrections, reformulations, references adde
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
Explicit methods for Hilbert modular forms
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of
Hilbert modular forms over a totally real field. We provide many explicit
examples as well as applications to modularity and Galois representations.Comment: 52 pages, 10 figures, many table
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