235 research outputs found
On the fundamental group of the complement of a complex hyperplane arrangement
We construct two combinatorially equivalent line arrangements in the complex
projective plane such that the fundamental groups of their complements are not
isomorphic. The proof uses a new invariant of the fundamental group of the
complement to a line arrangement of a given combinatorial type with respect to
isomorphisms inducing the canonical isomorphism of the first homology groups.Comment: 12 pages, Latex2e with AMSLaTeX 1.2, no figures; this last version is
almost the same as published in Functional Analysis and its Applications 45:2
(2011), 137-14
Limits of Gaudin Systems: Classical and Quantum Cases
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new ''Gaudin'' algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ''Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
SU(3) Richardson-Gaudin models: three level systems
We present the exact solution of the Richardson-Gaudin models associated with
the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for
any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For
the rational case additional cubic integrals of motion are obtained, whose
number is added to that of the quadratic ones to match, as required from the
integrability condition, the number of quantum degrees of freedom of the model.
We discuss different SU(3) physical representations and elucidate the meaning
of the parameters entering in the formalism. By considering a bosonic mapping
limit of one of the SU(3) copies, we derive new integrable models for three
level systems interacting with two bosons. These models include a generalized
Tavis-Cummings model for three level atoms interacting with two modes of the
quantized electric field.Comment: Revised version. To appear in Jour. Phys. A: Math. and Theo
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
We give an explicit differential equation which is expected to determine the
instanton partition function in the presence of the full surface operator in
N=2 SU(N) gauge theory. The differential equation arises as a quantization of a
certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and
Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion,
appendix and references adde
Combinatorial Hopf algebras in quantum field theory I
This manuscript stands at the interface between combinatorial Hopf algebra
theory and renormalization theory. Its plan is as follows: Section 1 is the
introduction, and contains as well an elementary invitation to the subject. The
rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf
algebra theory and examples, in ascending level of complexity. Part II turns
around the all-important Faa di Bruno Hopf algebra. Section 7 contains a first,
direct approach to it. Section 8 gives applications of the Faa di Bruno algebra
to quantum field theory and Lagrange reversion. Section 9 rederives the related
Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf
algebras of Feynman graphs and, more generally, to incidence bialgebras. In
Section10 we describe the first. Then in Section11 we give a simple derivation
of (the properly combinatorial part of) Zimmermann's cancellation-free method,
in its original diagrammatic form. In Section 12 general incidence algebras are
introduced, and the Faa di Bruno bialgebras are described as incidence
bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us
to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next,
the general algebraic-combinatorial proof of the cancellation-free formula for
antipodes is ascertained; this is the heart of the paper. The structure results
for commutative Hopf algebras are found in Sections 14 and 15. An outlook
section very briefly reviews the coalgebraic aspects of quantization and the
Rota-Baxter map in renormalization.Comment: 94 pages, LaTeX figures, precisions made, typos corrected, more
references adde
JUNO Conceptual Design Report
The Jiangmen Underground Neutrino Observatory (JUNO) is proposed to determine
the neutrino mass hierarchy using an underground liquid scintillator detector.
It is located 53 km away from both Yangjiang and Taishan Nuclear Power Plants
in Guangdong, China. The experimental hall, spanning more than 50 meters, is
under a granite mountain of over 700 m overburden. Within six years of running,
the detection of reactor antineutrinos can resolve the neutrino mass hierarchy
at a confidence level of 3-4, and determine neutrino oscillation
parameters , , and to
an accuracy of better than 1%. The JUNO detector can be also used to study
terrestrial and extra-terrestrial neutrinos and new physics beyond the Standard
Model. The central detector contains 20,000 tons liquid scintillator with an
acrylic sphere of 35 m in diameter. 17,000 508-mm diameter PMTs with high
quantum efficiency provide 75% optical coverage. The current choice of
the liquid scintillator is: linear alkyl benzene (LAB) as the solvent, plus PPO
as the scintillation fluor and a wavelength-shifter (Bis-MSB). The number of
detected photoelectrons per MeV is larger than 1,100 and the energy resolution
is expected to be 3% at 1 MeV. The calibration system is designed to deploy
multiple sources to cover the entire energy range of reactor antineutrinos, and
to achieve a full-volume position coverage inside the detector. The veto system
is used for muon detection, muon induced background study and reduction. It
consists of a Water Cherenkov detector and a Top Tracker system. The readout
system, the detector control system and the offline system insure efficient and
stable data acquisition and processing.Comment: 328 pages, 211 figure
Transverse-momentum-dependent Multiplicities of Charged Hadrons in Muon-Deuteron Deep Inelastic Scattering
A semi-inclusive measurement of charged hadron multiplicities in deep
inelastic muon scattering off an isoscalar target was performed using data
collected by the COMPASS Collaboration at CERN. The following kinematic domain
is covered by the data: photon virtuality (GeV/), invariant
mass of the hadronic system GeV/, Bjorken scaling variable in the
range , fraction of the virtual photon energy carried by the
hadron in the range , square of the hadron transverse momentum
with respect to the virtual photon direction in the range 0.02 (GeV/ (GeV/). The multiplicities are presented as a
function of in three-dimensional bins of , , and
compared to previous semi-inclusive measurements. We explore the
small- region, i.e. (GeV/), where
hadron transverse momenta are expected to arise from non-perturbative effects,
and also the domain of larger , where contributions from
higher-order perturbative QCD are expected to dominate. The multiplicities are
fitted using a single-exponential function at small to study
the dependence of the average transverse momentum on , and . The power-law behaviour of the
multiplicities at large is investigated using various
functional forms. The fits describe the data reasonably well over the full
measured range.Comment: 28 pages, 20 figure
Leading-order determination of the gluon polarisation from semi-inclusive deep inelastic scattering data
Using a novel analysis technique, the gluon polarisation in the nucleon is
re-evaluated using the longitudinal double-spin asymmetry measured in the cross
section of semi-inclusive single-hadron muoproduction with photon virtuality
. The data were obtained by the COMPASS experiment at
CERN using a 160 GeV/ polarised muon beam impinging on a polarised LiD
target. By analysing the full range in hadron transverse momentum ,
the different -dependences of the underlying processes are separated
using a neural-network approach. In the absence of pQCD calculations at
next-to-leading order in the selected kinematic domain, the gluon polarisation
is evaluated at leading order in pQCD at a hard scale of . It is determined in three intervals
of the nucleon momentum fraction carried by gluons, , covering the
range ~ and does not exhibit a significant
dependence on . The average over the three intervals, at
, suggests that the gluon polarisation
is positive in the measured range.Comment: 14 pages, 6 figure
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