Bf CP Research Articles

Towards Lefschetz Thimbles in Sigma Models, I

We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper.

Variation in Milk Yield and Composition of West African Dwarf Does fed Cassava Peel based Diets Supplemented with African Yambean Concentrate in the Humid Zone of Nigeria

This study investigated the variation in milk yield and composition of West African Dwarf goats (does) fed cassava peel meal based- diets supplemented with varying levels of African yambean seed meal (AYBM). Four experimental diets designated as T1, T2, T3 and T4 with 0, 10, 20 and 30% levels of AYBM respectively were used in the feeding trial. A total of twelve does with average weight of 19.6±1.12kg were used in this study. The does which were in their second to third parities were randomly divided into 4 groups of 3 animals each. Each group was assigned to one of the experimental diets in a Completely Randomized Design (CRD) experiment. Each animal was housed separately in cement floored pens measuring 1.85m x 1.70m (3.15m2). Dry hay material was used as bedding. During the first 4 months of pregnancy, the does were zero-grazed with forage consisting mainly Penisetum purpureum, Pueraria phaseoloides and Centrosema pubesens. Daily dry matter provision for each animal was based on 3% body weight. In the last trimester of pregnancy, each in-doe received 0.5kg of a concentrate diet in the morning (0800hr) and 1.0kg Penisetum purpureum in the afternoon (1400hr). This nutritional regime continued through parturition and into the 10th week of lactation for each doe. Water was provided ad libitum throughout the study duration. After the 10th day post –partum, the respective does were hand – milked daily and associated milk parameters such as yield and composition (lactose, total solids, butter fat, crude protein, solid–non fats, ash and gross energy) determined accordingly. Correlation coefficients were computed and simple linear regression was used in ascertaining the relationship between the parameters/constituents in goat milk. Results showed significant (P<0.05) differences in weekly variations in milk yield (1.88 – 3.32 kg) and constituents (4.05 – 4.52% lactose, 0.87 – 1.06% ash and 3.89 – 3.53 MJ/kg energy). The Milk yield generally rose with time post- partum and reached peak yield (3.32kg) in week 3 before declining. There were however significant differences (P<0.05) between the peak yield of week 4 and yield obtained in the 7th (2.43kg), 8th (2.43kg), 9th (2.55kg) and 10th (1.88kg) week, respectively. Milk yield was negatively and significantly correlated with total solids (r = -0.59; P<0.05), crude protein (r = 0.71; P<0.01), solid –non fats (r = - 0.87; P<0.01) and milk energy (r = -0.79; P<0.01), but poorly and positively correlated with butter fat (r ~ 0.01; P>0.05), and lactose (r = 0.42; P<0.05). Significantly high (P<0.05) positive correlation existed between BF and TS (r = – 0.55) and CP (r - 0.54), energy and TS (r = - 0.57) while a significantly high (P<0.01) but negative correlation existed between CP and SNF (r = -0.75). Lactose was negatively, poorly and non-significantly correlated with TS (P>0.05; r = -0.22) in this study. The study concludes that dairy goats fed 10% AYBM maintained milk high yield and optimum composition.

A recurrent intertrigo: benign familial chronic pemphigus

Benign familial chronic pemphigus (BF CP) is an autosomal dominant disorder characterized by recurrent vesicles and plaques mostly over the intertriginous areas. The defect is in the AT P2C1 gene which leads to suprabasilar acantholysis. Pregnancy, physical trauma, excessive sweating, skin infections and exposure to ultraviolet radiation are the important trigger factors. Self-medications or over the counter treatment usually makes a delay in the proper diagnosis.

On the geometry of four-qubit invariants

The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four-qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six lines and four planes in complex projective space ${\bf CP}^3$. For the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables. Moreover, their magnitudes are entanglement monotones that fit nicely into the geometric set of $n$-qubit ones related to Grassmannians of $l$-planes found recently. We also show that in terms of these invariants the hyperdeterminant of order 24 in the four-qubit amplitudes takes a more instructive form than the previously published expressions available in the literature. Finally in order to understand two, three and four-qubit entanglement in geometric terms we propose a unified setting based on ${\bf CP}^3$ furnished with a fixed quadric.

Rotating black hole, twistor-string and spinning particle

We discuss basic features of the model of spinning particle based on the Kerr solution. It contains a very nontrivial {\it real} stringy structure consisting of the Kerr circular string and an axial stringy system. We consider also the complex and twistorial structures of the Kerr geometry and show that there is a {\it complex} twistor-string built of the complex N=2 chiral string with a twistorial $(x,\theta)$ structure. By imbedding into the real Minkowski $\bf M^4$, the N=2 supersymmetry is partially broken and string acquires the open ends. Orientifolding this string, we identify the chiral and antichiral structures. Target space of this string is equivalent to the Witten's `diagonal' of the $\bf CP^3\times CP^{*3}.$

Construction of fuzzyS4

We construct a fuzzy $S^4$, utilizing the fact that ${\bf CP}^3$ is an $S^2$ bundle over $S^4$. We find that a fuzzy $S^4$ can be described by a block-diagonal form whose embedding square matrix represents a fuzzy ${\bf CP}^3$. We discuss some pending issues on fuzzy $S^4$, i.e., precise matrix-function correspondence, associativity of the algebra, and etc. Similarly, we also obtain a fuzzy $S^8$, using the fact that ${\bf CP}^7$ is a ${\bf CP}^3$ bundle over $S^8$.

COMPLETE HYPERBOLIC NEIGHBORHOODS IN ALMOST-COMPLEX SURFACES

We prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is ${\bf R^4}$ with an arbitrary almost-complex structure $J$ of class $C^{1.\alpha}$. Let $C$ be a non-singular $J$-complex curve passing through the origin. Our result cah be stated as follows: There exists a basis $\{U_j\}$ of neighborhoods of zero in ${\bf R^4}$, such that $(U_j,J)$ are complete hyperbolic in the sence of Kobayashi, moreover $(U_j\setminus C,J)$ are complete hyperbolic as well. The fact that this result remains true for any almost-complex structure is somewhat suprising. Really, given any germ of a non-singular real surface $C\ni 0$ in ${\bf R^4}$, one can easily construct an almost-complex structure $J$ in a neighborhood of zero, such that $C$ becomes a $J$-complex curve. Typical corollary is the following: Let ${\cal M}_{\omega, 5l}$ be the Banach manifold consisting of pairs $(J,\{D_j\}_{j=1}^5)$, where $J$ is any almost-complex structure on ${\bf CP^2}$ tamed by the Fubini-Studi form $\omega$ and $\{D_j\}_{j=1}^5$ the union of five $J$-complex lines in ${\bf CP^2}$ in general position. The set ${\cal H}_{\omega, 5l}$ consisting of $(J, \{D_j\}_{j=1}^5)$ with $Y=({\bf CP^2}\setminus \bigcup_{j=1}^5 D_j,J)$ hyperbolically imbedded into $({\bf CP^2}, J)$ is an open nonempty subset of ${\cal M}{\omega, 5l}$.

Generalization of Calabi-Yau/Landau-Ginzburg correspondence

We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg correspondence to a more general class of manifolds. Specifically we consider the Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf CP}^{N-1}$ for various values of k and N. When k<N, the 1-loop beta function of the sigma model on $M_N^k$ is negative and we expect the theory to have a mass gap. However, the quantum cohomology relation $\sigma^{N-1}={const.}\sigma^{k-1}$ suggests that in addition to the massive vacua there exists a remaining massless sector in the theory if k>2. We assume that this massless sector is described by a Landau-Ginzburg (LG) theory of central charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. We compute the topological invariants (elliptic genera) using LG theory and massive vacua and compare them with the geometrical data. We find that the results agree if and only if k=even and N=even. These are the cases when the hypersurfaces have a spin structure. Thus we find an evidence for the geometry/LG correspondence in the case of spin manifolds.

Exceptional Seiberg-Witten geometry with massive fundamental matters

We propose Seiberg-Witten geometry for N=2 gauge theory with gauge group $E_6$ with massive $N_f$ fundamental hypermultiplets. The relevant manifold is described as a fibration of the ALE space of $E_6$ type. It is observed that the fibering data over the base ${\bf CP}^1$ has an intricate dependence on hypermultiplet bare masses.

Topological orbifold models and quantum cohomology rings

We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.