Quantum Hall Effect in Higher Dimensions

Following recent work on the quantum Hall effect on $S^4$, we solve the Landau problem on the complex projective spaces ${\bf C}P^k$ and discuss quantum Hall states for such spaces. Unlike the case of $S^4$, a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of ${\bf C}P^2$ in some detail considering both Abelian and nonabelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of ${\bf C}P^3$ is related to the case of $S^4$.Comment: 15 pages, LaTe

Noncommutative gravity: fuzzy sphere and others

Gravity on noncommutative analogues of compact spaces can give a finite mode truncation of ordinary commutative gravity. We obtain the actions for gravity on the noncommutative two-sphere and on the noncommutative ${\bf CP}^2$ in terms of finite dimensional $(N\times N)$-matrices. The commutative large $N$ limit is also discussed.Comment: LaTeX, 13 pages, section on CP^2 added + minor change

The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory

We consider different large ${\cal N}$ limits of the one-dimensional Chern-Simons action i\int dt~ \Tr (\del_0 +A_0) where $A_0$ is an ${\cal N}\times{\cal N}$ antihermitian matrix. The Hilbert space on which $A_0$ acts as a linear transformation is taken as the quantization of a $2k$-dimensional phase space ${\cal M}$ with different gauge field backgrounds. For slowly varying fields, the large ${\cal N}$ limit of the one-dimensional CS action is equal to the $(2k+1)$-dimensional CS theory on ${\cal M}\times {\bf R}$. Different large ${\cal N}$ limits are parametrized by the gauge fields and the dimension $2k$. The result is related to the bulk action for quantum Hall droplets in higher dimensions. Since the isometries of ${\cal M}$ are gauged, this has implications for gravity on fuzzy spaces. This is also briefly discussed.Comment: 37 pages, references and a comment adde

The effective action for edge states in higher dimensional quantum Hall systems

We show that the effective action for the edge excitations of a quantum Hall droplet of fermions in higher dimensions is generically given by a chiral bosonic action. We explicitly analyze the quantum Hall effect on complex projective spaces ${\bf CP}^k$ with a U(1) background magnetic field. The edge excitations are described by abelian bosonic fields on $S^{2k-1}$ with only one spatial direction along the boundary of the droplet relevant for the dynamics. Our analysis also leads to an action for edge excitations for the case of the Zhang-Hu four dimensional quantum Hall effect defined on $S^4$ with an SU(2) background magnetic field, using the fact that ${\bf CP}^3$ is an $S^2$ bundle over $S^4$.Comment: 22 pages, LaTeX, discussion on nature of edge states for S^4 added to section 5, final version to appear in Nucl. Phys.

On the Electromagnetic Interactions of Anyons

Using the appropriate representation of the Poincare group and a definition of minimal coupling, we discuss some aspects of the electromagnetic interactions of charged anyons. In a nonrelativistic expansion, we derive a Schrodinger-type equation for the anyon wave function which includes spin-magnetic field and spin-orbit couplings. In particular, the gyromagnetic ratio for charged anyons is shown to be 2; this last result is essentially a reflection of the fact that the spin is parallel to the momentum in (2+1) dimensions.Comment: 13 pages, RU-92-17-B,CU-TP-584, CERN-TH6768/9

Matrix Models

Matrix models and their connections to String Theory and noncommutative geometry are discussed. Various types of matrix models are reviewed. Most of interest are IKKT and BFSS models. They are introduced as 0+0 and 1+0 dimensional reduction of Yang--Mills model respectively. They are obtained via the deformations of string/membrane worldsheet/worldvolume. Classical solutions leading to noncommutative gauge models are considered.Comment: Lectures given at the Winter School on Modern Trends in Supersymmetric Mechanics, March 2005 Frascati; 38p

Non-Hermitian quantum mechanics in non-commutative space

We study non Hermitian quantum systems in noncommutative space as well as a \cal{PT}-symmetric deformation of this space. Specifically, a \mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not \mathcal{PT}-symmetric. A complex interacting anisotropic oscillator system has also been discussed.Comment: 5 pages, revised versio

Histopathological growth patterns as biomarker for adjuvant systemic chemotherapy in patients with resected colorectal liver metastases

Adjuvant systemic chemotherapy (CTx) is widely administered in patients with colorectal liver metastases (CRLM). Histopathological growth patterns (HGPs) are an independent prognostic factor for survival after complete resection. This study evaluates whether HGPs can predict the efectiveness of adjuvant CTx in patients with resected CRLM. Two main types of HGPs can be distinguished; the desmoplastic type and the non-desmoplastic type. Uni- and multivariable analyses for overall survival (OS) and disease-free survival (DFS) were performed, in both patients treated with and without preoperative chemotherapy. A total of 1236 patients from two tertiary centers (Memorial Sloan Kettering Cancer Center, New York, USA; Erasmus MC Cancer Institute, Rotterdam, The Netherlands) were included (period 2000–2016). A total of 656 patients (53.1%) patients received preoperative chemotherapy. Adjuvant CTx was only associated with a superior OS in non-desmoplastic patients that had not been pretreated (adjusted hazard ratio (HR) 0.52, 95% confdence interval (CI) 0.37–0.73, p<0.001), and not in desmoplastic patients (adjusted HR 1.78, 95% CI 0.75–4.21, p=0.19). In pretreated patients no signifcant efect of adjuvant CTx was observed, neither in the desmoplastic group (adjusted HR 0.83, 95% CI 0.49–1.42, p=0.50) nor in the non-desmoplastic group (adjusted HR 0.96, 95% CI 0.71–1.29, p=0.79). Similar results were found for DFS, with a superior DFS in non-desmoplastic patients treated with adjuvant CTx (HR 0.71, 95% CI 0.55–0.93, p<0.001) that were not pretreated. Adjuvant CTx seems to improve OS and DFS after resection of non-desmoplastic CRLM. However, this efect was only observed in patients that were not treated with chemotherap