9,532 research outputs found
The Heisenberg group and conformal field theory
A mathematical construction of the conformal field theory (CFT) associated to
a compact torus, also called the "nonlinear Sigma-model" or "lattice-CFT", is
given. Underlying this approach to CFT is a unitary modular functor, the
construction of which follows from a "Quantization commutes with reduction"-
type of theorem for unitary quantizations of the moduli spaces of holomorphic
torus-bundles and actions of loop groups. This theorem in turn is a consequence
of general constructions in the category of affine symplectic manifolds and
their associated generalized Heisenberg groups.Comment: 45 pages, some parts have been rewritten. Version to appear in Quart.
J. Mat
Renormalization Group and Quantum Information
The renormalization group is a tool that allows one to obtain a reduced
description of systems with many degrees of freedom while preserving the
relevant features. In the case of quantum systems, in particular,
one-dimensional systems defined on a chain, an optimal formulation is given by
White's "density matrix renormalization group". This formulation can be shown
to rely on concepts of the developing theory of quantum information.
Furthermore, White's algorithm can be connected with a peculiar type of
quantization, namely, angular quantization. This type of quantization arose in
connection with quantum gravity problems, in particular, the Unruh effect in
the problem of black-hole entropy and Hawking radiation. This connection
highlights the importance of quantum system boundaries, regarding the
concentration of quantum states on them, and helps us to understand the optimal
nature of White's algorithm.Comment: 16 pages, 5 figures, accepted in Journal of Physics
Proof of the Kurlberg-Rudnick Rate Conjecture
In this paper we present a proof of the {\it Hecke quantum unique ergodicity
rate conjecture} for the Berry-Hannay model. A model of quantum mechanics on
the 2-dimensional torus. This conjecture was stated in Z. Rudnick's lectures at
MSRI, Berkeley 1999 and ECM, Barcelona 2000.Comment: In this version we add a proof that the character sheaf of the
Heisenberg-Weil representation is perverse, geometrically irreducible of pure
weight 0. Moreover, we supply invariant formulas for the character sheaf on
an appropriate open set, and we give also another alternative proof for the
rate conjecture that uses our invariant formula
A condensed matter interpretation of SM fermions and gauge fields
We present the bundle Aff(3) x C x /(R^3), with a geometric Dirac equation on
it, as a three-dimensional geometric interpretation of the SM fermions. Each C
x /(R^3) describes an electroweak doublet. The Dirac equation has a
doubler-free staggered spatial discretization on the lattice space Aff(3) x C
(Z^3). This space allows a simple physical interpretation as a phase space of a
lattice of cells in R^3. We find the SM SU(3)_c x SU(2)_L x U(1)_Y action on
Aff(3) x C x /(R^3) to be a maximal anomaly-free special gauge action
preserving E(3) symmetry and symplectic structure, which can be constructed
using two simple types of gauge-like lattice fields: Wilson gauge fields and
correction terms for lattice deformations. The lattice fermion fields we
propose to quantize as low energy states of a canonical quantum theory with
Z_2-degenerated vacuum state. We construct anticommuting fermion operators for
the resulting Z_2-valued (spin) field theory. A metric theory of gravity
compatible with this model is presented too.Comment: Minimal modifications in comparison with the published versio
Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties
We prove a new symplectic analogue of Kashiwara's Equivalence from D-module
theory. As a consequence, we establish a structure theory for module categories
over deformation quantizations that mirrors, at a higher categorical level, the
Bialynicki-Birula stratification of a variety with an action of the
multiplicative group. The resulting categorical cell decomposition provides an
algebro-geometric parallel to the structure of Fukaya categories of Weinstein
manifolds. From it, we derive concrete consequences for invariants such as
K-theory and Hochschild homology of module categories of interest in geometric
representation theory.Comment: Version 2. A number of minor edits and corrections. Comments welcom
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