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Front Matter The Fate of the Melcart Cleanor, Son of Lysis The Last of a Veteran Scipio A Great Scheme The Mission The Last of the Greeks A Corinthian Assembly At Thermopylae A Pinchbeck Alexander The Two Hasdrubals Scipio Sets to Work In the Roman Camp The Megara The Prisoners Baal Hammon Move and Countermove Help From the Hills The Battle of the Isthmus Treachery Polybius A Pleasure Trip Diplomacy In Sore Need A Refuge in the Storm Storming of the Upper City A Precious Book The End of Carthage At Delos Corinth Mummius The Slave-Dealer To Italy Misenum The World Well Lost Beyond the Sunset
Lords of the World - Alfred J. Church
At Misenum
Cornelia, the "Mother of the Gracchi", was at this time not far from fifty years of age, but retained by favour of nature, often so capricious in what she gives or takes away, much of the beauty of youth. Left a widow with a numerous family—she had borne twelve children to her husband, but all had not survived—she had found a royal suitor in Ptolemy, king of Egypt. This suit it had probably not caused her any effort to decline. A daughter of the great Cornelian house would have disdained in any case an alliance with so doubtful a race as the Ptolemies, and this particular Ptolemy, whose bloated appearance had earned him the name of Physcon, was a degenerate scion of it. But Cornelia had had serious troubles. Of her twelve children two only were now alive, Tiberius, now a lad of seventeen, and Caius, a child of five. Both, indeed, gave the fairest promise; the elder, though he had but lately assumed the manly gown, had exhausted such education as Roman teachers could then supply, and was already an accomplished rhetorician; the younger was a boy of singular beauty and intelligence. But Cornelia, a remarkably clear-sighted woman, had already begun to view with alarm the rapid development of Tiberius's character. The young man's political tendencies were strongly marked, and they seemed likely to bring him into dangerous collision with the aristocratic traditions of his mother's house. As for Caius, he was self-willed and imperious to an extraordinary degree. Still, no mother could have been prouder of her children, as none certainly could have been more devoted to them.
At this particular time, however, when Cleanor paid his first visit to the villa at Misenum, all was brightness and gaiety. Theoxena and her daughter had learned by this time to feel themselves thoroughly at home in Cornelia's hospitable house. The elder woman had suffered so much in the past that the best happiness which could be hoped for her was peace; but Daphne had blossomed out into a most attractive personality. There was a peculiar radiance about her beauty, which had all the greater charm because the girl's own disposition and the gracious example of her hostess, a very pearl among women, tempered it with a certain air of virginal reserve. Cleanor she met at first with her old sisterly frankness, but there was an ardour in the young man's glance, and a thrill in his voice—though he vainly attempted to subdue them into the greeting of a respectful affection—which seemed to alarm her. As for Cleanor, after the first day spent in her company, he could doubt no longer as to the real nature of his feelings. Daphne would be thenceforward the one woman in the world for him.
The holiday, which was prolonged to the beginning of the new year, passed only too quickly. The days were spent either in hare-hunting—larger game was not to be found in a region already thickly populated—or in excursions on the water, which were favoured by weather that, though it was the depth of winter, was remarkably calm and warm. Possibly the most delightful expedition of the season was the ascent of Vesuvius, then clothed almost to the summit with lovely woods, and giving no sign of the hidden forces which, two centuries later, were to spread desolation over the fairest region of Italy.
The evenings were begun by a meal, simply yet elegantly served, at which the whole party assembled, even the little Caius being allowed to be present for at least a time. The meal over there was no lack of entertainment. Tiberius was an accomplished reciter, and could give one of Terence's comedies with an artistic variety of voice and emphasis. Cleanor charmed the company with a passage from Homer, from Pindar, or from one of the great Athenian dramatists. Sometimes, by special request, he would dance the Pyrrhic dance, a pastime which in sterner Roman society would have more than savoured of frivolity. And now and then Daphne was persuaded to sing to the lute an exquisite little lyric from Stesichorus.
The last day of the year, which was also to be the last of the most delightful of visits, Cleanor determined to make as long as possible. Rising as soon as the first streaks of dawn began to show themselves in the sky, he began to explore more thoroughly than he had before an opportunity of doing, the beautifully ordered gardens which surrounded the villa. Following a path of velvet sward, sheltered on either side by shrubberies of box-wood, he came to a spot which gave him a wide prospect over the lovely bay of Naples. He noticed, but in the most casual way, the figure of a gardener, who was busy, as it seemed, in trimming the surrounding shrubs, the whole spot, except on that side which fronted the sea, being protected from the wind by a dense growth of box and laurel, arbutus and bay.
He threw himself down on a rustic bench and gazed on the scene before him. He was looking westward, and the sea at his feet lay in shadow, a dark purple in colour. In the distance the sun was just touching with golden light the crags of Prochyta and of the more remote Inarimé. For a time the beauty of the scene wholly occupied him, for nature stirred the hearts of the men of those days even as it stirs ours, though they had only begun to give their feelings articulate expression.
Then his thoughts recurred to what was the dominant emotion of the time with him, his love for Daphne. How, he asked himself, how should he make it known? How should he approach her? To speak directly, at least in the first instance, was not the custom of his race, though doubtless love, there as elsewhere, made exceptions of his own to the severest rule. Through her mother? But Theoxena was, he knew, only too thoroughly devoted to him. To her his wish would be a command; she would make it a matter of filial obedience with her daughter, and he wanted the voluntary submission that was wholly free. Through Cornelia? But would she favour such an alliance? She was a noble of the nobles, filled with the keenest sympathy for the people, but profoundly conscious the social difference between her class and them and with her own class she would certainly rank the well-born Cleanor.
Well, he said to himself, after a pause of reflection, which did not seem to make the matter clearer, "these things will settle themselves. I love her, and I think she loves me, so that nothing will keep us apart." And he broke into the beautiful choric song of the Antigone—for it was his habit, as it is the habit of all true lovers of poetry, thus to interrupt his solitary musings—
"O love invincible!"
After this came a stave of Alcæus, and after this again a piece of melodious tenderness from Sappho.
As he turned to retrace his steps to the house, for he had risen early, and the keen morning air made him feel that he had fasted long, he was startled to hear his name called from behind him, not the name by which he was known to the world, but the pet family name, which he had not heard since the home of his childhood had vanished in fire and blood.
"Cle," said the voice, and its tones seemed to be strangely familiar. He turned; no one was within sight but the gardener. The man had dropped the shears, and stood with his hands stretched out in a supplicating gesture.
"What is it?" he cried; "what or whom do you want?" He took two or three steps forward, and as he approached there seemed to be something strangely familiar in the figure before him.
"Yes, it is—" and the speaker swayed to and fro for a moment, and then fell unconscious to the ground. The wide-brimmed hat, which had been drawn down low over the face, to conceal, as it seemed, the features, was displaced by the fall, and revealed the graceful contour of the forehead, and the shapely head covered with short curls of sunny gold.
"Great Zeus!" cried Cleanor, as he lifted the prostrate figure from the ground. "Great Zeus! if I am not mad or dreaming, this is Cleoné come to life again."
Close by a tiny spring trickled down from a rock. Cleanor held his cap beneath it till it was half full, and dashed the water in his sister's face. She drew two or three deep breaths, and then opened her eyes. Vacant at first, for she could not remember where she was or what had happened, they soon became radiant with happy light.
"Dearest brother," she murmured, "have I found you again? But come to my little hut—it is close by. There you shall hear my story, and we will consider what is to be done."
Briefly put, for in the actual telling it was interrupted, as may be supposes, with numberless exclamations and questions, Cleoné's story was this:—
"I remember nothing after I was struck down by a blow from a soldier's sword in the market-place of Chelys, till I found myself in the hold of a ship at sea."
"Then you were not killed?" cried Cleanor.
"It seems not," said the girl with a merry laugh, "for even were I an Eurydice there was no Orpheus to bring me back from the house of Hades."
"Ah!" said the young man, "now I begin to understand what old Judas meant. He said, you must know, that they bribed the soldiers not to kill the prisoners, but to stun them."
"Well, as I was saying, I found myself in the hold of a ship which was evidently making very bad weather. I was lying with my head close to the deck, and I could hear two men talking just over me. There was such a roaring of the wind, and such a creaking of timbers, that I lost a good deal of what they said. Still I could make out something. Someone—I supposed it was the captain—was cursing his ill-luck. 'Here,' he said, 'is a bit of cursed spite—as good a speculation as ever I made in my life all comes to nothing. There are fifty as likely young fellows as I have had the handling of since I went into the business five-and-twenty years ago down there, and what is going to become of them? They are worth two hundred thousand sesterces if they are worth one, and now the whole lot is going to the bottom.' 'What is the odds?' growled the other, whom I took to be the steersman. 'What is the odds if you are going too?' 'I tell you what,' said the other again after a pause, 'you should give the fellows a chance. Open the hatches, and let them get to land if they can.' 'What is the good?' answered the captain sulkily; 'they may drown for all I care.' 'Nay, but you talk like a fool. If they live, they are still yours, and you may get hold of them, or, at least, of some of them again.' 'True,' said the owner, 'that is so. They shall have a chance.' A minute or two afterwards the hatches were opened, and the fellow cried, 'Up with you as quick as you can? The ship hasn't many minutes to float, and if you don't want to go to the bottom with her, now is your time.' About two score out of the fifty clambered upon deck. Some had never recovered from the blow which had stunned them—it can't be an easy thing to give just the right sort of stroke—and some, I take it, were so far gone with sea-sickness that they did not care to move. As for me, I felt a little dazed; sea-sickness never troubles me, as you know. We got up on deck only just in time, the ship was already close upon the rocks. The next minute she struck. What happened to the crew and to my companions is more than I can say; all I know is that I have never seen one of them since, except, indeed, sole dead bodies that I found on the shore next morning. I had a desperate struggle to get to land, and, indeed, I never should have done it, though, as you know, I am no bad swimmer, but that an extra big wave threw me up almost high and dry, and I had just strength enough to crawl away out of reach of the sea. The rest of the night—it was about the middle of the third watch, as near as I could guess, when this happened—I passed in a thicket in a bed of dry leaves, where I slept as soundly as ever I did in my life. The next day I rigged myself out with clothes that I took from the dead men on the shore—it was no robbery, I thought, poor fellows! I found some money, too, in their pockets. Following a road which led in-land, I came to a village where there was a tavern. Here I got some bread and a draught of sour wine. I thought it safest, I should tell you, to pretend to be deaf and dumb, and made them to understand by signs that I wanted something to eat and drink. I paid for what I had, but was careful to let the people know that I had very little, for I made up the few coppers that were wanted from one place and another. Then I got them to understand that I wanted to work for my living. First I made as if I were digging, then as if I were sawing wood. They happened to want someone, for it was a busy time of the year, and they saw that they could get the work done very cheaply, for they gave me no pay besides my food and lodging in an outhouse, which, happily, I had to myself. Here I stopped for about a month. Then I overheard some people talking of a great lady who lived in the neighbourhood. She was a widow, they said, and managed everything—house and garden and farm—all by herself. That, I thought to myself, is the place for me. Perhaps some day I shall be able to tell her my story. However, the day has never come. I got employment just in the same way as I did at the tavern, and I have the little hut to myself, where I look after some fowls and pigeons. But, somehow, I could never summon up courage to speak.
However, I always went on hoping and hoping, and now, dearest Cleanor, that you are come, all will be right."
"Yes," said the young man, "and the first thing, my dear Cleoné, will be to get you some proper clothes."
The girl blushed.
"By Castor!" she said, "I had almost forgotten that I was dressed as a man. But how will you manage it?"
"Easily enough," replied her brother. "The lady Cornelia has an excellent housekeeper with whom I am in high favour; I don't doubt that she will let me have everything I want. But I must go; the sooner we manage this the better."
Poor Cleoné, woman-like, felt the courage which had never failed before desert her when she had to part even for half an hour with her long-lost brother. She clung to him, and wept piteously. "Don't leave me," she sobbed.
The young man, to whom this sort of thing was quite a anew experience, looked at her with astonishment. "What, Cleoné, is the meaning of this after all you have gone through?"
"Yes," she said, smiling through her tears, "I am a fool. And besides," she went on, looking at her dirty and ragged garments, "I do want some decent clothes."
The good Pollia, who acted as wardrobe-keeper, mother-of-the-maids, and housekeeper in general to Cornelia, was not a little astonished when Cleanor asked her to supply him with the various articles of a young lady's toilet, not so numerous in those days, it should be mentioned, as they are now. He was a great favourite, however, and she asked no questions, probably thinking that some joke was being meditated. She searched accordingly among the treasures in her charge, and had no difficulty in finding all that was wanted.
Fashions did not change in those days as they change under the vagaries of modern taste. Women were careful, indeed perhaps more careful than they are now, to suit their dress to their age. But what the mother had worn at twenty, the daughter, reaching the same years, might wear without even the suspicion of oddity, and the garments might be handed down, if they were of the quality that was suited to so long a life, to yet another generation.
Cleanor was soon making his way with an armful of suitable apparel to the gardener's hut. Cleoné, who seemed to be bent on making up as quickly as possible for her enforced separation from all feminine vanities, received the precious burden with a shriek of delight. When she emerged, half an hour afterwards, from her hut it would have passed all human skill to recognize in the brilliant young beauty who held Cleanor's hand the shabby deaf-mute who for many months past had plied his solitary task in Cornelia's gardens.
All these confidences and preparations had taken time, and the house party had just assembled for the midday meal when the pair walked into the dining-room. Never since Misenum got its name had the place seen a more startling sight. At first it seemed as if Cleanor had found his double, for brother and sister were curiously alike. But the time that had passed since they were so tragically parted had changed them not a little. The young man had grown in height, and his frame, knit by the continual activities of an adventurous life, had developed the ampler proportions that became his sex. The girl was his very image, but now on a somewhat smaller scale. A fairer couple had never been seen in Italy.
"Cleanor has turned into Apollo," cried the little Caius, "and he has brought Diana with him."
As for the rest of the company, they gazed with an astonishment that was almost stupefaction on the scene. Cornelia was the first to recover herself. She advanced to greet the new-comer. "You are welcome," she said, "for your brother's sake—for Cleanor must surely be your brother—and, I am sure, for your own." Then Theoxena threw herself at the girl's feet and clasped her knees. "It Cleoné," she cried. "The gods have nothing more to give me." Little Cephalus kissed her hand, and Daphne, somewhat shy at first of the splendid stranger, was not long behind with an affectionate greeting.
"Not a word," said Cornelia, "till you have eaten and drunk. For the present," she said, smiling at the little Caius, "they will have to be content without ambrosia and nectar."
The meal ended, Cornelia heard the whole story. Her mind, always eminently practical, discerned at once the first thing that had to be done.
"We must assure without delay," she said, "this young lady's civil status. At present it would be very perplexing to say who or what she is."
A message was immediately despatched to the nearest town with a letter requiring the immediate presence of the resident notary. He arrived before sunset, and by a formal act of emancipation Cleoné, slave of Cornelia, was made free.
"Pardon me, my daughter," she said, "if I speak of you as my slave. And indeed my title is a very weak one; no one, however, is likely to make out a better. Meanwhile, as far as I can secure your freedom, you are free."


Today we'd like to introduce you to Vilma Machin.
Everything started in Cuba, my native country, where my love for ballet originated. Cubans dance with passion and without hesitation. We are people who understand that dancing begins in the soul. My mom remembers that I would always drop whatever I was doing and run into the room with the TV if any classical ballet was on one of the channels. I would sit in front of the screen, entranced by the dancers until the final note of the ballet. I also loved to dress up and imitate the many dance movements I had viewed on television. My mom could not help but notice how much I enjoyed classical music. My parents can also recall how I would constantly ask them to take me to the ballet school [National Cuban Ballet School]. If you ask anyone on the street, they will know about the National Cuban Ballet Company and its dancers. It is likely that this individual would even be able to tell you when and where the company will be performing next, Cubans appreciate ballet to that extent. Dancers in Cuba are admired, well-known and highly respected. Thousands of children audition each year in the hope that they will be accepted into the National Cuban Ballet's prestigious school. In Cuba, one has to pass a number of tests to be selected for the school. An applicant must have the ability and facilities to become a dancer (i.e. specific body conditions such as natural back, legs and feet flexibility, turn-out from the hips and pleasing body proportions). Prospective students must also pass a musicality exam to ensure that the student has a good ear and can count the music. One also has to undergo a physical exam to look for any bone abnormalities or any other factors that could limit the dancer's training. Finally, the child must complete a psychological test to ensure that the child is stable and can face the pressure and demanding nature of the intense training at such a young age. Acceptance into the National Ballet School is a privilege and a dream for many Cuban children. The school is free, so future pupils are only selected if they meet all the criteria. All the instructors at the school are Cuban and focus specifically on teaching the Cuban method of ballet. The government supports the arts in Cuba, and ballet is often given special attention.
A dancer's career is very hard and like many other dancers, I have faced many challenges and disappointments. Soon after presenting myself at the National Ballet School entrance exam at the age of nine, I suffered a horrible injury to my hamstring. I was not a very flexible dancer at the time and the exam was only a week away so my teacher was focusing on physically preparing my class. He needed to push me a little harder that day to see if I was going to be ready for the upcoming audition. As a result, I tore my hamstring and I was not able to pass the required exam to join the school. I was devastated, but I never gave up on my dream. I had to stay in bed for two months without being able to walk and then underwent a very slow recovery process that lasted for a year. During that year, I worked with a gymnastic teacher, who helped me gain confidence and improve my flexibility. My dream of becoming a professional ballerina was still alive, pushing me to work harder than ever. I remember my mom took me to the school the following year, knowing it was highly unlikely that I would be accepted because in Cuba you can only start dancing at the age of nine, not younger or older. I was ten years old at the time but I wanted to try again. We went and luckily the school had just started a new program that year. This program would accept students who were ten years old. These unique students would then be placed in a special group were during their second year of training they would be tasked with completing two years of ballet training curriculum over one year. My first year of ballet was very challenging, turn-out from my hips did not come naturally and it was a big limitation for me. I also had not achieved the level of technique required to pass. My ballet instructor spoke with my parents, telling them the simple truth. In order to keep my spot at the school, my technique would have to significantly improve and I would have to overcome my physical limitations. However, I was determined to work hard enough to hold onto my place at the school so my teacher offered to give me private lessons during the two months of vacation we have in Cuba. These added classes caused financial strain because my family was not wealthy and it was a real struggle for my parents to be able to afford private lessons for me seven days a week for three hours each day. My teacher believed in me so much that she did not request any specific payment from my parents and allowed them to pay what they could. My family and I will forever be grateful for her generosity, and how she enabled me to succeed in my career. Pursuing ballet was a big sacrifice not only for me but also for my family as I saw them struggle economically every day to ensure that I could achieve my dreams. However, no matter how hard the road was they never stopped encouraging me and I am here today because of their sacrifice and unconditional support. I remember my teacher and I worked for many hours every day for two straight months and when I returned to school for my second year of ballet, everybody was amazed by my improvement. That year was very difficult, but I pushed myself to accomplish two years of training in one.
I had academic classes in the morning from 7:30 am until 12:00 pm in the afternoon. Then I would complete my level two ballet classes from 1:00-5:00 pm. Afterwards from 5:00-9:00 pm I would complete my level three ballet classes. At the end of that day I would journey home, often walking because transportation was unavailable. I would arrive home exhausted, eat dinner (Cuba's economy was struggling at this time and often my family had barely enough food to eat), take a shower and finish my school work and finally go to bed. In the morning, my routine would begin again, repeating throughout the week from Monday through Saturday. Despite governmental support, the school has also felt the negative effects of the economic downturn in Cuba. I can remember one year when my school lacked the funds to purchase pointe shoes so my level lost a year of pointe instruction. A similar situation occurred with our dance uniforms. I was given a single leotard my first year at the school and I had the same leotard for five years because the school could not afford to give me anything else. I continued to take private ballet classes during my breaks for four years and all our hard work and collaborating paid off. In my fourth-year ballet exam I scored ninety-five out of a hundred possible points; receiving the best score in my class. My improvement continued, but many more challenges arose along the way. Not a single part of my training was easy, yet although I faced many challenges and hardships my determination and desire to achieve my dream helped me overcome them. With each new experience, I grew and came to the realization that anything worth fighting for requires sacrifice.
Each year at the National Cuban Ballet School, students must take academic and dance exams to determine if they will remain at the school. Dancers undergo a lot of pressure to pass and continue with their training each year. The training there is very demanding and there are a number of requirements that each dancer needs to follow in order to retain her spot at the school. Usually each level has around twenty students in each class. Over the next five years many students were eliminated for various reasons ranging from discipline issues, frequent absences from classes, lack of improvement in technical and artistic skills throughout the year (thus they do not pass their yearly exam), and academic issues (bad grades and lack of effort).
After eight years of intense training and a whirlwind of emotions, I finally completed my training. I began to prepare for the final exam that would decide if I was qualified to join the National Cuban Ballet Company under the direction of Alicia Alonso. Everyone in Cuba knows exactly who Alicia Alonso is, and her impact on ballet. Alicia Alonso is an icon of Cuban ballet on the island and internationally as well. She has been presented with numerous international awards, including: The Anna Pavlova Award from the University of Dance, Paris in 1966, the UNESCO Pablo Picasso Medal for her extraordinary contribution to dance in 1999 and an honorary doctorate from the University of Costa Rica in 2017 to name a few. However, a new physical challenge was causing me anxiety, my height. My teachers were concerned that my height was going to limit my chance to join the company, I was 5'9" and at that time the standard height for female dancers in the company was 5'6''. I was heartbroken, undergoing those eight years of rigorous training, sweating, stretching and strengthening every muscle, only to have my dream crushed by a factor outside of my control. My classmates, friends and family all asked what I would do if I didn't make it into the company. I honestly could never answer their questions. I could simply not see myself doing anything else. Since the beginning of my ballet journey I had wanted to become a professional dancer and I had decided I was not going to stop until I became one. My family and I had sacrificed too much along the way for me to fail now. So, I pulled myself together; I knew this would be my final opportunity to fulfill my dream. I decided I would give 300% in the upcoming exam and I would have to pass it. I entered the experience with extreme determination and confidence that I would simply have to be selected. After two exams and a performance for members of the National Cuban Ballet, I learned after a week of suspense that I had been selected along with one other girl to become a member of the company. I can say without a doubt that day was one of the happiest and most rewarding ones of my life. I had not only accomplished my dream but I had been able to overcome all my physical limitations that others had believed would stop me. After eight years of grueling training I graduated at the top of my class as a dancer/teacher and then joined the Cuban National Ballet Company for six years.
My first year in the company was very difficult because there were no other dancers of my height in the corps of ballet, so I was barely able to perform that year. Slowly however, the company started to increase the standard height of the female dancers that it selected for the company so everything started to fall into place. After six years in the company, I wanted a change. I dreamed of coming to the United States to search for more variety in my career and also to receive a higher salary so that I would be able to financially support my family in Cuba. On one of my many tours with the company, we traveled to Mexico and I decided to stay there at the performance, intending to cross the border. I was twenty-three years old at the time and I finally did it, crossing the border on my own and arriving in the U.S. Crossing alone as a woman was difficult but my faith and determination helped me to overcome the obstacles I faced along the journey. Once I reached the United States, the number of challenges I faced only grew. I faced many of the problems numerous immigrants battle against as they enter a foreign country. I did not have any family or support system here and with no money of my own I could not afford to live by myself. I was starting my life from zero and so I lived in many friends' houses for short periods of time and I faced what appeared an infinity of problems down the road. It was tough to audition for U.S. ballet companies because I had arrived after the traditional audition season (at the end of March). I also had no financial stability, so I could not afford to travel to different companies and audition on my own time. Thus, I was forced to wait a year before I could audition for a company and hopeful gain a job. Meanwhile, I struggled to find a place where I could train for almost a year until the audition season started again. It was difficult to even begin to find any sort of balance in my life. After what had seemed endless waiting, I was considering the possibility that I would have to give up on my dream of dancing so that I could find a job to support myself. However, I was given a new opportunity when I met Guillermo Leyva, my husband today.
Guillermo Leyva was also a Cuban principal dancer, and he owned a professional ballet company in Chicago. I was leaving Miami looking for work and he happened to be looking for new dancers for his upcoming company production. Another friend of mine, also a Cuban dancer, mentioned this opportunity to me and I contacted Guillermo immediately. After viewing a video of my dancing, he hired me for the production and I was able to travel to Chicago. Once I arrived I fell in love with the city and I knew right then that this was the place I wanted to live. After completing my time as a guest dancer with Guillermo's company, I decided to stay in Chicago. It was hard to start from zero again in a different city, and I was settling in just as the recent economic downturn was occurring in the U.S. Job opportunities were shrinking and I was lucky to even find a job as a waitress at a restaurant. Temporarily I worked many different jobs in restaurants, stores and other places. I simultaneously searched for positions as a ballet instructor in the suburbs which I soon found. I taught at the Chicago Academy of the Arts, the School of Dance West, and the Academy of Dance Art. I also worked at State Street Dance Studio and Faubourg School of Ballet for five years too. Fortunately, in Cuba, to be a dancer you must also take classes to become certified as a teacher as well. This extra training helped me to quickly succeed as a teacher here in Chicago.
After several years of teaching in the Chicagoland area with my husband Guillermo Leyva, we decided it was time for us to start our own ballet school. We both shared the same vision for our school and dreamed of giving children the opportunity to have the best training available in the suburbs. We felt the experience we had gained through from Cuban training, dancing professionally and teaching around the world was a blessing. Here in the U.S. I have danced as a soloist for a season with the Cuban Classical Ballet of Miami, and as a Principle dancer with the Alma Dance Company. My list of performance repertoire includes: Le Vivandiere, Bayadere, Swan Lake, Coppélia, Dying Swan, Majisimo, Classical Symphony, Frankenstein, Giselle, Don Quixote, Swan Lake, Nutcracker, Les Sylphides, Ballo de la Regina (Balanchine), and choreographic pieces staged by Alicia Alonso such as Romeo and Juliet and The Magic Flute to name a few. My husband, the Co-Director and Co-Owner of Alma has also had a very successful dance career. Guillermo was a principal dancer in Cuba for eleven years, a principal dancer at the Monterey Ballet for three years, a Principal dancer with the Joffrey Ballet of Chicago, Lexington Ballet, Lyric Opera of Chicago, Illinois Ballet, Cincinnati Ballet, Ballet de Santiago, Chile and the Royal Ballet de Wallonie in Belgium. He has performed roles in Swan Lake, Don Quixote, Giselle, Sleeping Beauty, Majisimo, and Nutcracker, to give a brief selection. It was this great combination of experiences that gave us all the tools we needed to make our school special and unique from others in the area. We underwent intense training and had robust dance careers. Our experiences in Cuba and internationally have prepared us for wonderful careers in ballet, and also for successful teacher positions around the world. In September of 2014 we made our dream a reality and Alma Dance School was born. Many people have asked us why we chose Alma as the name of the school, and we chose this name because it means soul in Spanish. To us, ballet is pure feeling, we dancers love our careers more than words can describe. Our love for ballet is authentic and goes far beyond any financial reward. For Guillermo and I, to dance is to feel it in one's soul.
As with sport or art, the time required to acquire a high level of proficiency is demanding and requires a high degree of self-sacrifice. Like an Olympic athlete, the family is intimately involved. Today's young student is torn between many activities and options beyond schoolwork. Creating an environment of support and dedication to one very difficult art form is a family affair! Parents rearrange schedules, shuttle students, alter vacation dates and wait, wait, wait for their child. It is a difficult journey which involves both student and parent in order to achieve success. However, this parental support is key to the success of the student. The path of the student is the path of the family, and it is filled with highs and lows.
To illustrate the sacrifice of time that must be made, I remember Fernando Alonso, the founder of the Cuban National School, once said, "It takes about five years with the periodicity of a daily lesson to develop the technical base of a dancer who starts between the ages of 8 and 10 years old. After that, training should be intensified, with a knowledge of theatre, and should continue until the student has managed to control his body and learned to project his personality in his movements".
Close cooperation between teachers, students and parents is key to prepare a dancer for a demanding and rigorous career in ballet. The student is trained both physically and mentally. Training is an important time for students to consolidate vital skills and learn the discipline and commitment necessary to achieve their dreams. From our career experience, Guillermo and I know the intrinsic value of serious and consistent training for a student who aspires to be a professional dancer. At Alma, our vision is focused on preparing students to achieve their goals and this belief sets the foundation for our daily training. Having a specific agenda for each class in terms of what step or artistic skill you will work on with your students is crucial. As a teacher, it is important that you know the areas in which your dancers need to practice and ensure that these aspects of their technique are addressed in their daily classes. I focus on creating exercises and combinations that will help the dancers correct and improve their dance weaknesses. At Alma, we strive to prepare our students for performances and this is the reason our dancers are able to dance technically challenging productions. The training received at Alma Dance School will impart dancers all the tools they need to pursue a career in ballet if they so wish.
Please tell us about Alma Dance School.
At Alma Dance School, we instruct our students in the Cuban Method of ballet. Below I have described this specific type of technique so readers can learn about the aspect of our training that makes our school so unique.
The trio of Fernando, Alberto and Alicia Alonso formed the foundations of the Cuban National Ballet School, which reflects the technique of the Cuban method of ballet. Fernando Alonso, was a master teacher who exceled at guiding dancers to a higher level of personal expression. This new level enhanced the emotional spectrum of the piece while advancing the technical vocabulary. Fernando studied and integrated the disciplines of psychology, philosophy, anatomy of the body and performance in his teaching. His goal was to draw the best from each of his students. Alberto Alonso, a master choreographer was capable of creating a stunning new dance language and new interpretations of classical pieces. Finally, Alicia Alonso, a world-famous dancer has become the epitome of Cuban ballet. When she danced, she was named a Prima Ballerina Assoluta, possessing the technical proficiency, spirit and dedication to bringing the art form of ballet to a new level.
The creation of the Cuban method of ballet dates back to the 1930s. Fernando Alonso's study of schools in France, Italy, Denmark, Russia and British formed the basis for developing his own methodology. The Cuban method has its origins in the Russian Vaganova method, which emphasizes dancing with the entire body. This results in harmonious movement between arms, legs and torso. As with other methods of ballet, in the Cuban method the torso is the foundation of all movement, the dancer's core is trained to be strong and well aligned. Movements are then achieved through control of the core, producing action which is clean and precise but not rigid. However, the Cuban method diverges from other styles of technique in that it has a romantic feel that also combines high Russian extensions and jumps with intricate Italian foot work, French arm artistry and British attention to detail. Of special importance is the emphasis in the work of the pas de deux, where the couple dances in tight synergy. In the Cuban method, communication between the couple is enhanced, so that the art of ballet is heightened through expression and drama. The unison and harmony between the couple is achieved through a true connection, one made through the eyes, the arms and the hands. The physical, emotional and spiritual aspects of Cuban ballet fuse to create a style of ballet which is like no other. The purpose of the creation of this Cuban method, also known as the "Alonso Method", was to create an ideal style of ballet which, unlike any other, pays attention to the form as a whole and not by its parts, turning the dancer into an esthetically perfect figure.
As professional dancers, ourselves, Guillermo and I trained in one method of ballet so that we could perfect our technique and become expressive artists. This is why we have carried on this tradition and continue to train our dancers' in a single style of ballet. In our past teaching experiences at other dance schools, we have often found that several different styles of ballet such as like Balanchine, Vaganova, and Cechetti are taught all together. We believe discrepancies in training between styles can create learning difficulties as a student strives to perfect his or her technique. Students become confused as they work to master the correct body positioning and to learn position names as each style has unique requirements. If dancers are constantly taught several different variations of a position, they can become confused and ultimately, their technique will suffer overall. Variations in style aside, the students' best interests are served by learning one pure style of ballet. Once they have mastered a single style the dancers can explore other styles or modify the style they trained in based on the wishes of the companies to which they are accepted. Otherwise, a student may become confused and his or her technique may be perceived as undisciplined and sloppy rather than a harmony of several training styles. Alma Dance School is distinctive because Guillermo and I only train our pre-professional division in the Cuban Method of ballet. Our shared vision for the school is to enable students to train so that they will have the skills and ability to reach the technical level required to become great dancers. The students are taught a powerful, lyrical and emotional form of ballet. Our school is known for the high-quality training we offer and how Guillermo and I dedicate our time to assist each dancer in individual progress. We strive to help each student fulfill their potential. Alma Dance School is also renowned for our annual high-quality productions. We have two professional productions each year, the Nutcracker and the Simply Classic Series. During these two shows we are able to showcase our dancers and also invite several international professional principal dancers to perform with our students. We are exceptionally proud of the technical quality that our dancers have achieved. Training is intense; however, the results are displayed every time the students take the stage. We are amazed by the amount of growth we see in each student from year to year. We are also delighted by the school's success over such a short period of time and with limited finances and resources. Guillermo and I have learned so much during our studio's journey and we are very excited for what the future holds in store for Alma Dance School.
I think that every struggle and every hard lesson learned throughout my life has been a blessing and has made me the person I am today. Maybe if my life would it be easier, I would not value it the same. I am thankful for each and every experience, the goods and bads, and I would not change any of them. All these experiences have made wiser, more generous and, most importantly, more appreciative of all the small things in life… and to always remember my roots and how far I have come.


text:
You told us: Most would demand a low price for Google's Apple One rival
Some readers wouldn't pay a cent, but a surprising number would drop $40 or more.
David Imel / Android Authority
Google has several services that could easily be lumped together as a single subscription, similar to Apple One. At present, Google users have to subscribe to some services individually. Apple users, however, can pay for a single subscription to get a number of services. The latter's bundle includes Apple TV Plus, Apple Music, Apple Arcade, and 50GB of iCloud storage at its base price. While it might work for Apple, would you pay for something similar if Google offered it?
In a recent poll, we asked you this question. Here's how you responded.
How much would you pay for a Google take on Apple One?
Apple One starts at $14.95, and that's unsurprisingly what most readers would be willing to pay for a similar service from Google. More than 60% of respondents see a price between $10 and $15 as the sweet spot for a service that includes YouTube Music, 100GB of Google Drive storage, Google Play Pass, and YouTube Premium. Only 16.5% of readers would be willing to drop between $1 and $4 more for that bundle.
Interestingly, the next most popular option was "$40 or more." We're aren't completely sure if these respondents read the context of our poll, but it's interesting to note that nearly 8% of readers would drop this amount on Google's cheapest Apple One competitor.
According to the comments, we may have left out an option. A few readers aren't willing to pay a cent for Google's bouquet. Peruse some more reader anecdotes below.
Here's what you told us
naduh: Where is the option for $0? Personally, I don't think Google's products are good enough to come out of pocket for. Plus why would I spend money on a product Google would kill off after a year or two.
Johnny Ravioli: $0.
Rick45TX: If one could trust that Google wouldn't snoop around in the associated storage and not track what one used for the services, I might consider it. But since it *is* Google, there's no way I'd pay.
Guy_Jorgen_Gold: The cost of all of these services combined is about 20$, so obviously less.
EasyCare: Will this ever happen? Many Google services come and go. Even if they are present, they are siloed and doesn't work well with each other.
sachouba: Why would I be willing to pay for any of that? The only service worth paying for is Google Drive. If all of the other services come for free with a Google Drive subscription, why not, but otherwise, I'll just stick to Spotify and YouTube Vanced.
That's it for this poll. Thanks for your votes and comments. If you have any further thoughts, be sure to drop them in the comments down below.
NewsSurveys

redpajama_set_name: RedPajamaCommonCrawl


text:
#ifndef APRINTER_NEWLIB_COMMON_H
#define APRINTER_NEWLIB_COMMON_H

#include <cstddef>

#ifndef APRINTER_NO_SBRK

#define APRINTER_HAS_GET_HEAP_USAGE 1

extern "C" std::size_t aprinter_get_heap_usage();

#endif

#endif

redpajama_set_name: RedPajamaGithub


text:
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In my forthcoming annual lecture I offer an explanation for our apparent inability to make progress on challenges most people would like to see addressed; like providing dignity and decent care for all old people, increasing social mobility or achieving a more sustainable model of economic growth. I argue that there are three basic sources of social power, hierarchical authority most often associated with the state, solidarity most often associated with community, and individualism most often associated with the market.
The problem is that in different ways, and together as a whole, these sources of power have dried up. I explore the problems facing hierarchy (incompetence and low public trust for example) and solidarity (as a consequence of diversity and social polarisation), and argue that while individualism is powerful, it is also narrow and in many ways self defeating.
However, the problem with individualism is not so much that it has, under its own volition, mutated into something malign. Like the other social forces, the dangers of individualism are the flip side of its strengths, the latter including creativity, ambition and drive. However – as we saw in the financial sector - individualism unchecked by wise and trusted authority or the binds of solidarity and social responsibility tends towards selfishness and irresponsibility.
The characteristics of a well functioning society, organisation or strategy is the balance of top down (doing what you're told), lateral (doing what others do), and bottom up (doing what you choose) ways of seeing and pursuing change. When one dimension fails not only is its power for constructive change lost, but it leads to the other dimensions being overloaded.
I came across an interesting case study of this phenomenon reading reviews of a new book causing much debate in America: The Twilight of the Elites by Chris Hayes. The book is an attempt to explain the multiple failings of authority in the US in the first decade of the 21st century (ranging from Enron and Worldcom to Katrina and Iraq, to Catholic priests and sports administrators).
I have read only the reviews (we will try to lure Mr Hayes over to the RSA some time), but I find it easy to believe the book's account of how meritocracy has failed, placing particular emphasis on how elites close ranks and self perpetuate. Hayes' solution is more controversial, although he is far from alone in advocating it. In essence he argues that meritocracy can only work within limits to overall levels of inequality. When society is highly unequal the rewards for staying at the top and the fear of dropping down make it impossible for all but the most socially responsible elite member not to use any means possible (including exploitation and gaming) to try to stay there and get their kids to join them.
The point is this: the individualistic ideal of meritocracy (remembering that social democrat Michael Young used the word disparagingly when he first defined it) only works alongside the solidaristic principle of social justice and, furthermore, that when solidarity fails to set limits to individualism the consequence is an incompetent and self-serving hierarchy.
Many American commentators have wondered aloud whether it would ever be possible to forge an alliance between the Tea Party and the Occupy movement, given that both share a profound hostility to the elite they believe are misruling their country. Given the degree to which both movements are driven by anger, the attribution of blame and a tendency to see conspiracies, I'm not sure this is either likely or to be desired. It is the system that is failing, something reflected in rising social pessimism, but the partisan nature of political campaigns means they only focus on one aspect of that failure. This is why when we listen to their message we often have the odd feeling of simultaneously agreeing with the critique while rejecting the overall analysis.
The problem is not a lack of leadership, nor a lack of social justice and responsibility, nor an inadequacy of personal responsibility and enterprise, but all these together acting on each other to create a deteriorating stasis. The answer is a counterintuitive combination of big thinking (whole system reform with implications for all of us) and moderation. The problem is functional (power to) rather than oppressive (power over).
Perhaps the system will correct itself (history provides plenty of grounds for both optimism and pessimism), but if we are waiting for a movement or a leader capable of articulating, let alone winning, an argument as profound but nuanced then looking at politics here, in Europe or – most depressingly – America, there are few reasons to be hopeful.
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An economy for all?
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The loss of an address can be a crippling blow for those who need support. It's time that was fixed.
How we 'pulled apart' and how we can 'come together' again
Anthony Painter
We shouldn't underestimate how far our societies have pulled apart. Yet there is hope for renewal, says Anthony Painter. The question is not whether we come together – but how.
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Iran Says It's Ready To Allow U.N. Nuclear Inspectors To Military Site
Published March 6, 2012 at 5:34 AM PST
Iran's semi-official Fars News Agency is reporting that the country is ready to allow United Nations nuclear inspectors into a military complex, where the West suspects Iran is undertaking secret nuclear work.
Inspectors for the International Atomic Energy Agency complained in unusually tough terms last month that their mission to Iran had failed because they had been denied entry into the military site at Parchin.
In a statement Iran's representative to the IAEA said the U.N. would be given access once the "modalities of cooperation" were settled.
Reuters adds that Western diplomats are rejecting the latest news, saying it just represents an effort by Iran to buy time.
But the announcement comes just before the five permanent members of the U.N. Security Council announced they had accepted an offer to return to the negotiating table with Iran. The United States and its allies are demanding that Iran cease any efforts to develop a nuclear weapon.
The AP reported that the group's point of contact, European Union foreign policy chief Catherine Ashton, responded positively to Tehran's overtures.
The AP adds:
"Ashton's says in a statement the EU hopes that Iran 'will now enter into a sustained process of constructive dialogue which will deliver real progress in resolving the international community's long-standing concerns on its nuclear program.'"
"The time and venue of the talks have not yet been determined."
Of course all of this news comes with the backdrop of a U.S. visit by Israeli Prime Minister Benjamin Netanyahu. President Obama met with Netanyahu yesterday and said that while the United States "will always have Israel's back," diplomacy is the best way forward with Iran.


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And the worst traffic in SA is…
Author: Compare Guru
Satnav specialist TomTom has released its fourth annual global traffic index for 2013. The report analyses traffic in major cities in SA such as Johannesburg, Pretoria, Durban and Cape Town.
JOHANNESBURG – Satnav specialist TomTom has released its fourth annual global traffic index for 2013. The report analyses traffic in major cities in SA such as Johannesburg, Pretoria, Durban and Cape Town. The results revealed that Cape Town is now the most congested city in South Africa, surpassing Johannesburg which was the most congested city in 2012. 10 DAYS PER YEAR IN TRAFFIC The report showed that traffic congestion on secondary roads is worse compared to main roads, and commuters are spending an unbelievable average of 10 working days per year stuck in traffic. It also showed South Africans spend an extra 13 minutes per hour stuck in traffic. TomTom reports that traffic "shortcuts" can increase travel time by up to 50%. Interestingly, the reports finds South Africa to be the only country in the world to experience its worst congestion on Monday mornings while the rest of the world experiences the worst congestion on Tuesdays. 'NO LONGER EFFECTIVE' TomTom South Africa manager Daan Henderickx said: "Traffic congestion is nothing new, and continues to be a global challenge. The traditional responses to congestion, such as building new roads or widening existing ones are no longer proving to be effective. "Real time traffic information can help drivers find the quickest shortcut on their journey, and assist governments to make smarter decisions to improve traffic flow for their cities." TomTom's Traffic Index compares travel times during non-congested hours with travel times in peak hours experienced by passenger vehicles. The report takes into account both local roads and highways and measures over 95 000kms of roadways in South Africa. South Africa has seen an increase in congestion in six major cities, with Cape Town and East London's congestion both increasing by 2%. SA had a congestion level of 21%, meaning travel time is increased by an average of 21% during heavy traffic. Worst traffic in SA: Global ranking included 1 Cape Town 27% (33rd highest congestion globally) 2 Johannesburg 25% (48) 3 East London 22% (65) 4 Pretoria 22% (73) 5 Durban 18% (98) 6 Bloemfontein 12% (134/138) The top ten most congested cities in 2013: 1 Moscow 74% 2 Istanbul 62% 3 Rio de Janeiro 55% 4 Mexico City 54% 5 São Paulo 46% 6 Palermo 39% 7 Warsaw 39% 8 Rome 37% 9 Los Angeles 36% 10 Dublin 35% Results summary: •South Africa has an overall congestion level of 21%. • South Africans spend on average an extra 13 minutes per hour stuck in traffic. • South Africans spend an average of 10 working days a year stuck in traffic. • Cape Town is the most congested city with a congestion level of 27%. • Bloemfontein is the least congested city in South Africa with a level of 12%. • Pretoria and East London share the same amount of congestion with a level of 22%. • Cape Town experiences the most congestion during morning peak periods with a congestion level of 71%. • Cape Town experiences the worst evening peak congestion period with a level of 58%. • All South African cities, except for East London, experience congestion during the morning peak periods compared to evening peak periods. • Bloemfontein experiences the least congestion with a level of 24% during morning peak periods and 22% during evening peak periods. • East London experiences the same amount of congestion during morning and evening peak periods at a level of 47%. • South African cities experiences more congestion on secondary roads than on highways. • Cape Town and Johannesburg share the same amount of congestion levels on their secondary roads with a level of 30%. • Joburgers on average will cut their delay time caused by congestion in half if they take the highway as opposed to a secondary road. • Congestion on secondary roads in Pretoria is three times the amount on its highways. • On average, Capetonians will save six minutes for every hour during peak periods if they take the highway. • South Africa is the only country in the world to experience their worst congestion on a Monday, the rest of the world experiences their worst day on a Wednesday. • The best morning peak days fall on a Friday for South Africans unless you live in Cape Town or Bloemfontein, then your best morning would be on a Thursday. • Johannesburg and East London both experience less congestion in both morning and evening peak times on a Friday. • On the 8 March 2013 Cape Town experienced 29mm of rain coincidentally this was the city's most congested day according to the report. This article was first published on Wheels24: http://www.wheels24.co.za/News/And-the-worst-traffic-in-SA-is-20140603
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text:
Is beverage alcohol an attractive investment for private equity?
Private equity involvement in beverage alcohol has been limited. Will this change going forward?
Despite CVC Capital Partners' £767m acquisition of Stock Spirits Group in August 2021, private equity involvement in beverage alcohol remains limited – with established spirits brand owners typically able to offer the benefits of enhanced routes to market and specialist experience in the sector.
Nonetheless, some industry observers believe that private equity players may possess the kind of capital, expertise and focus that is often needed to drive growth for neglected local and regional spirits brands, as well as encouraging nimble and speedy decision-making.
Bernstein analyst Trevor Stirling, for example, does not expect the CVC move for Stock – a spirits company with a strong focus on central and eastern Europe – to spark a wave of beverage alcohol M&A activity in beverage alcohol.
"A trade buyer can beat private equity because of the synergies they can bring to the table," he says. "An investment by a multinational spirits brand owner could bring additional distribution channels and routes-to-market, for example, which simply wouldn't be possible by a PE [private equity] firm.
"This is why PE, in general, is more involved in deals where there isn't an obvious trade buyer, or if the company is an early-stage business where distribution isn't so much an issue. PE can then nurture and grow the business until it's ready to be sold to a trade buyer."
For others, private equity can play an effective role in helping to develop local and regional brands that are sometimes neglected in investment terms by the bigger players. In these cases, private equity can help to drive growth, modernise operations and streamline decision-making.
For example, Terry Huffine, director of investment bank Raymond James' retail and consumer division, says, "beyond capital (which is important), these investors typically bring expertise in the form of executives from both within and outside the spirits sector, deep operational experience and considerable 'best practice' know-how from their other investments, experience and relationships."
Huffine contrasts the private equity approach with that of the larger strategic beverage alcohol companies. "Often, companies taking investment from strategics find … that bureaucracy and slow decision-making stifle growth, and that their brands take a back seat to higher-volume – and higher-priority – owned brands of the investor," he says. "Outside investment, on the other hand, can provide clarity of focus, speed and agility, and free companies from impositions inherent in having a strategic investor."
However, Stirling questions whether the central mission of private equity firms to secure a sometimes speedy return on their investment is compatible with an industry that often demands a long-term approach to category and brand development.
"When investing in a spirits company – or any company, for that matter – PE needs to have an exit strategy," he says. "They will be looking to accelerate and streamline business practices in order to grow the business for future exit.
"A trade buyer, however, will be looking to keep the brand for the longer term … The primary goal of PE investment will likely not be to reinvigorate a spirits category. PE invests where there is momentum already, and will look for businesses that can be made more efficient and support higher levels of debt."
This perceived gap between the respective approaches of strategic spirits brand owners and private equity firms can be partially bridged by investment from specialist private equity firms and incubator funds supported by larger beverage alcohol players.
Ventures such as consumer goods private equity firm L Catterton (backed by Moët Hennessy owner LVMH) and spirits incubator fund Distill Ventures (funded by Diageo) often retain a large degree of operational independence and decision-making – but benefit from the wealth of drinks experience offered by their major investor, who will be eyeing the incubator's brands with a view to future acquisition.
It remains to be seen if private equity starts to view the resilience of the beverage alcohol market, as well as the strong cash flows of many of the companies in this sector, as attractive investment opportunities. Vodka currently does not show the same growth trajectory as some of the more vibrant spirits categories. IWSR data shows that vodka, for example, declined by volume CAGR -0.7% (2016 -2020) and is expected to see a decline of -0.1% volume CAGR (2021-2025). Whisky, gin and Tequila, by contrast, have been growing over the past 5 years and are expected to continue on their growth trajectories, each with volume CAGRs of between +4% and +5% (2021-2025). In light of this, it will be interesting to see if CVC revitalises the vodka market over the next few years.
Vodka's premiumisation challenges and convergence with hard seltzers
Tequila volume overtakes bourbon and rum in the US – why?
5 key trends that will shape the global beverage alcohol market in 2021


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The design duo met at the Chelsea College of Art and Design, where Georgina's draped designs met with Keren's textile creations; the result was Marchesa. Incorporating vintage and Asian influences throughout the range of cocktail dresses, floor-sweeping gowns and evening separates, as well as shoes and jewelry, Marchesa is loved worldwide for its opulent embellishments, feather trims and exquisite draping.
Browse and shop the full range of Women's Marchesa Clothing, Handbags and Jewelry.


text:
\section{Introduction}
Recent progress in string theory \cite{maldacena, randall, sundrum} indicates
 scenarios
where our four dimensional space-time with standard model fields
corresponds to a D3-brane \cite{polchinski} embedded in a larger manifold.
Now, since D-branes correspond, in type II string theories, to the space
where the open string endpoints are attached, our space-time would be affected
by string boundary conditions (BC). One important consequence is the possible
noncommutativity of space-time coordinates at very small length scales
\cite{chu, schomerus, witten} since commuting coordinates are incompatible
 with open string
BC(s) in the presence of antisymmetric tensor backgrounds. This is one of the
main reasons of increasing interest in several aspects of noncommutative (NC)
quantum field theories \cite{witten, szabo}. Furthermore,
this illustrates the fact
that the string BC(s) may play a crucial role in the
phenomenology of four-dimensional physics.

\noindent Various methods have been applied to obtain this result
\cite{chu, chu1, ardalan, rb, agh, jing, our, sg}.
One of the most conventional methods to derive this noncommutativity
is to use Dirac'procedure \cite{dirac}-treat the mixed BC(s)
as primary constraints \cite{shirzad}. However, the interpretation of the
BC(s) as primary Dirac constraints lead to some ambiguities. One of them is,
in contrast to the standard Dirac's method, one obtains infinite secondary
constraint chains by consistency requirements \cite{shirzad, deghani}.
The other is that
the BC(s) are valid only at the boundaries, the Dirac function is introduced
so as to extend them to the neighborhood of the boundaries. In order to get
a finite result, one should regularise the Dirac function.
However, different regularisation procedures may lead to different
results \cite{loran}; a regularisation free proof is needed.

\noindent In a series of recent papers involving bosonic strings \cite{rb}
and superstrings \cite{agh},
it has also been shown explicitly
that noncommutativity can be obtained by modifying the canonical
bracket structure, so that it is compatible with the BC(s).
This is done in spirit to
the treatment of Hanson et.al \cite{hrt}, where modified Poisson brackets
(PBs) were obtained for the free Nambu-Goto string.
In \cite{jing}, PBs among the Fourier components were obtained using the
Faddeev-Jackiw symplectic formalism \cite{fj}, so that they are compatible
with these BCs. Using this the PBs
among the open string coordinates were computed revealing the NC structure
in the string end- points. It is important to note that all these analysis
were essentially confined to the classical level.
In a very recent paper \cite{our}, noncommutativity in an open bosonic string
moving in the presence of a
background Neveu-Schwarz two-form
field $B_{\mu \nu}$ is investigated in a conformal
field theory approach. The mode algebra is first obtained using the newly
proposed normal ordering \cite{godinho}, which satisfies both
equations of motion and BC(s). Using these
the commutator among the string coordinates is obtained. Interestingly,
this new normal ordering  yields the same algebra
between the modes as the one satisfying only  the equations of motion.
In this approach, we find that noncommutativity originates more
transparently and our results match with the existing results
in the literature.
In this paper, we shall extend the methodology of \cite{our} to
analyse an open super string propagating freely and one moving
in a constant antisymmetric background field.

\noindent The organisation of this paper is as follows.
In section 2, we review
the recent results involving new normal ordered products
(of fermionic operators) in \cite{br}.
In sec 3, we study the symplectic structure of the fermionic sector of
both free and interacting
super string using the method in \cite{our}
(borrowing relevant results from section 2).
We conclude in section 4. The computational details of some of the
key results in the paper are given in an appendix.

\section{New Normal ordering for fermionic string coordinates}
The action for a super string moving in the presence of
a constant background antisymmetric tensor field
${\cal B}_{\mu \nu}$ is given by:
\begin{eqnarray}
S &=&  \frac{- 1}{4 \pi \alpha^\prime} \int_{\Sigma} d\tau
 d{\s}\,\Big[ \, \partial_a X^\mu \partial^a X_\mu
\,+\, \epsilon^{ab} B_{\mu\nu} \partial_a X^\mu \partial_b X^\nu
\nonumber\\
 & & + i \psi_{\mu (-)} E^{\nu \mu} \partial_{+} \psi_{\nu (-)} +
i \psi_{\mu (+)} E^{\nu \mu} \partial_{-} \psi_{\nu (+)} \,\Big]
\label{1}
\end{eqnarray}
\noindent where, $\partial_{+} =  \partial_{\tau} + \partial_{\s},\; \;
\partial_{-} =  \partial_{\tau} - \partial_{\s} $
and $ E^{\mu\nu} \,=\, \eta^{\mu\nu} \, + {\cal B}^{\mu\nu}\,$.

\noindent Now since the bosonic and fermionic sectors decouple, we
can consider the fermionic sector seperately\footnote{The bosonic sector
was already discussed in \cite{our}.}.
The variation of the fermionic part of the action (\ref{1})
gives the classical equations of motion:
\begin{equation}
\partial_{+} \psi_{\nu (-)} \,=\, 0\quad,\quad
\partial_{-} \psi_{\nu (+)} \,=\, 0
\label{2}
\end{equation}
and a boundary term that yields the following BCs:
\begin{eqnarray}
 E_{\nu\mu}\,  \psi^\nu_{(+)} (0,\tau)\, &=&\,
  E_{\mu\nu} \, \psi^\nu_{(-)} (0,\tau)\, \nonumber\\
\label{4}
E_{\nu\mu}\,  \psi^\nu_{(+)} (\pi,\tau ) \, &=& \,\lambda
E_{\mu\nu} \, \psi^\nu_{(-)} (\pi, \tau )
\label{3}
\end{eqnarray}
\noindent at the endpoints $\s \,=\,0$ and $\s = \pi\,$ of the string,
where $\lambda = \pm 1 \,$ corresponds to Ramond and Neveu-Schwarz BC(s)
respectively.

It is convenient now to change to complex world-sheet coordinates and
therefore we first
make a Wick rotation by defining
$\s^2 = i\tau$. Then we introduce the complex world
sheet coordinates \cite{pol}:
$ z \,=\,  \s^1 +i \s^2 \, ; \,  \bar{z} \,=\, \s^1 - i \s^2$
and $\partial_z = \frac{1}{2}(\partial_1 - i\partial_2), \, \partial_{\bar{z}} =
\frac{1}{2}(\partial_1 + i\partial_2)$. In this notation the fermionic part of the
action (\ref{1}) reads:
\begin{eqnarray}
S_{F} =  \frac{-i}{4 \pi \alpha^\prime} \int_{\Sigma} dz
d \bar{z}[\psi_{\mu (-)} E^{\nu \mu} \partial_{\bar{z}} \psi_{\nu (-)} +
\psi_{\mu (+)} E^{\nu \mu} \partial_{z} \psi_{\nu (+)}]\,
\label{5}
\end{eqnarray}
while the classical equations of motion (\ref{2}) and the BCs (\ref{3})
take the form:
\begin{eqnarray}
\label{6}
\partial_{\bar{z}} \psi_{\nu (-)} \,=\, 0\,\,\,, \,\,
\partial_{z} \psi_{\nu (+)} \,=\, 0 \\
\left( E_{\nu\mu}\,  \psi^\nu_{(+)} (z,\bar{z})\, -\,
  E_{\mu\nu} \, \psi^\nu_{(-)} (z,\bar{z})\right)\vert_{z\,=\,
-\bar{z}\, , \, 2\pi - \bar{z}}\, =\, 0 \, .
\label{7}
\end{eqnarray}
We now study the properties of quantum operators
corresponding to the classical variables by considering the
expectation values\cite{pol}. Using the fact that the path integral of a
total functional derivative vanishes and considering the insertion of one
fermionic operator one finds:
\begin{equation}
\label{8}
\int [ d  \psi ] \left[  \frac{\delta}{\delta \psi^{\mu} _{(a) }(z,\bar{z})}
[e^{-S_{F}} \psi^{\nu}_{(b)} (z', \bar{z}')]
  \right] \,=\,0
\end{equation}
\noindent where, $a,b \,=\,+,-\,$.
Considering first the case of
$\psi^{\nu}_{(b)} (z', \bar{z}')\,$ inside the world-sheet
and not at the boundary,
this equation yields the following expectation values:
\begin{eqnarray}
\label{eqmov}
\langle \partial_{z} \psi^{\mu}_{(+)} (z,\bar{z}) \psi^{\nu}_{(+)}
(z',\bar{z}') \rangle &=& 2\,\pi \,i \, \alpha'\, \langle \eta^{\mu \nu}
\delta^2(z-z',\bar{z}-\bar{z}') \rangle \nonumber \\
\langle \partial_{\bar{z}} \psi^{\mu}_{(-)} (z,\bar{z})
\psi^{\nu}_{(-)}(z',\bar{z}') \rangle &=& 2\,\pi \,i \, \alpha '\, \langle
 \eta^{\mu\nu}  \delta^2(z-z',\bar{z}-\bar{z}') \rangle \nonumber \\
\langle \partial_{\bar{z}} \psi^{\mu}_{(-)} (z,\bar{z})
\psi^{\nu}_{(+)}(z',\bar{z}') \rangle &=&  \langle
 \partial_{z} \psi^{\mu}_{(+)} (z,\bar{z})
\psi^{\nu}_{(-)}(z',\bar{z}') \rangle =  0\,\,.
\end{eqnarray}
Using these results one finds the appropriate way to
define normal ordered products
that satisfy the equations of motion
for fermionic operators that are not at the world-sheet
boundary\cite{pol1, br}:
\begin{eqnarray}
\label{9}
: \psi^{\mu}_{(+)}(z,\bar{z})\,\,\, \psi^{\nu}_{(+)}(z',\bar{z}')  : &=&
  \psi^{\mu}_{(+)}(z,\bar{z})\,\, \,\psi^{\nu}_{(+)}(z',\bar{z}')
\,-\, \frac{i\, \alpha'\,}{{\bar z} - {\bar z}'}\,\eta^{\mu \nu}
\nonumber\\
: \psi^{\mu}_{(-)}(z,\bar{z})\,\,\, \psi^{\nu}_{(-)}(z',\bar{z}')  : &=&
  \psi^{\mu}_{(-)}(z,\bar{z})\,\,\, \psi^{\nu}_{(-)}(z',\bar{z}')
\,-\, \frac{i\, \alpha'\,}{ z -  z'}\,\eta^{\mu \nu}
\nonumber\\
: \psi^{\mu}_{(+)}(z,\bar{z})\,\,\, \psi^{\nu}_{(-)}(z',\bar{z}')  : &=& 0
\nonumber\\
: \psi^{\mu}_{(-)}(z,\bar{z})\,\,\, \psi^{\nu}_{(+)}(z',\bar{z}')  : &=& 0\,.
\end{eqnarray}
The above products satisfy the equations of motion (\ref{6})
at the quantum level,
but fails to satisfy the BC(s) (\ref{7}).

\noindent At this point it is more convenient to choose world sheet
coordinates, related to these $z$ coordinates by conformal
transformation, that simplify the representation of the boundary,
\begin{eqnarray}
\o\, = \,\mathrm{exp}\left(-iz\right)\, =\, e^{- i \s^1 + \s^2} \,\,\,;\,\,\,{\bar \o} = e^{i \s^1 + \s^2}.
\label{10}
\end{eqnarray}
Besides replacing
$\mathrm{exp}(-iz) \to \o$, we must transform the fields \cite{pol1},
\begin{eqnarray}
\label{13}
\psi^{\mu}_{\o^{\frac{1}{2}}}(\o) \,=\, \left(\partial_\o z\right)^{\frac{1}{2}}
\psi^{\mu}_{z^{\frac{1}{2}}}(z)
= \, i^{\frac{1}{2}} \o^{-\frac{1}{2}}\, \psi^{\mu}_{z^{\frac{1}{2}}}(z).
\end{eqnarray}
The subscripts are a reminder that these transform with half the weight
of a vector.
In this present coordinates the complete boundary corresponds
just to the region $\o = {\bar \o}$.
Further, the action (\ref{5})  along with equations of motion (\ref{6})
in terms of $\o, {\bar \o}$ has still the same form, while the form of BC(s)
(\ref{7}) change to the following:
\begin{eqnarray}
\left( E_{\nu\mu}\,  \psi^\nu_{(+)} (\o,\bar{\o})\, + i\,
  E_{\mu\nu} \, \psi^\nu_{(-)} (\o,\bar{\o})\right)\vert_{\o\,
=\,\bar{\o}}\, =\, 0 \, .
\label{28}
\end{eqnarray}

\noindent Let us now consider the case of an insertion of a fermionic
string coordinate $\psi^{\nu}_{(\pm )} (\o')\,$
located at the world-sheet boundary. Note that since $\o' =
\bar{\o}'$ at the boundary, the fermionic coordinate insertion at
the boundary depends only on $\o'$.
Working out equation
(\ref{8}), but now subject to constraint (\ref{28}) (with $\o$ replaced by
$\o'$ in (\ref{28})), we find\footnote{Note that the fields $\psi$'s with unprimed arguements are not
located at the boundary.} (see appendix for the computational details):
\begin{eqnarray}
\label{eqmov2}
\langle \partial_{\o} \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o') \rangle &=& \, 2\pi i\alpha^{\prime}\langle \eta^{\mu \nu}
\delta^2(\o-\o',\bar{\o}-\o') \rangle \nonumber \\
\langle \partial_{\o} \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o') \rangle &=& \, 2\pi i\alpha^{\prime}\langle \eta^{\mu \nu}
\delta^2(\o-\o',\bar{\o} - \o') \rangle \nonumber \\
\langle \partial_{\o} \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o') \rangle &=& \, -2\pi \alpha^{\prime}
\langle\Big[ ( \eta+{\cal B})^{^{-1}}\,
( \eta - {\cal B}) \Big]^{\nu \mu}
\delta^2(\o-\o',\bar{\o}-\o') \rangle \nonumber \\
\langle \partial_{\o} \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o') \rangle &=& \, 2\pi \alpha^{\prime}\langle
\Big[ ( \eta+{\cal B})^{^{-1}}\,
( \eta - {\cal B}) \Big]^{\nu \mu}
\delta^2(\o-\o',\bar{\o}-\o') \rangle \,\,.
\end{eqnarray}
So the appropriate normal ordering for fermionic string
coordinates at the boundary reads:
\begin{eqnarray}
:\psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o') : &=&  \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o')  \,-\, \frac{i\alpha^{\prime}}{\left(\bar{\o} - \o'
\right)} \eta^{\mu\nu}
\nonumber\\
: \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o') : &=& \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o')  \,-\, \frac{i\alpha^{\prime}}{\left(\o - \o'\right)}
\eta^{\mu\nu}
\nonumber\\
:  \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o')  : &=&   \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o') \,+\,\frac{ \alpha^{\prime}
\, \Big[ ( \eta + {\cal B})^{^{-1}}\,( \eta  -
{\cal B}) \Big]^{\nu \mu}}{\left(\bar{\o} - \o'\right)}
\nonumber\\
:  \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o')  : &=&  \psi^{\mu}_{(-)} (\o,\bar{\o}) \psi^{\nu}_{(+)}
(\o') \,-\,\frac{ \alpha^{\prime}
\, \Big[ ( \eta + {\cal B})^{^{-1}}\,(\eta  - {\cal B}) \Big]^{\nu
\mu} }{\left(\o - \o'\right)}
\label{11}
\end{eqnarray}
The above results of normal ordering of fermionic operators are new and
incorporates the effect of BC(s).

\noindent Now for the functional ${\cal{F}}[X]$ (representing
the combinations occuring in the left hand side of the above
equation), the new
normal ordering (in absence of the $\mathcal{B}$ field)
can be compactly written as:
\begin{eqnarray}
\label{compact}
:{\cal{F}}:= \exp\left(
\frac{i\alpha^{\prime}}{2}
\int d^2 \o^{\prime\prime}
d^2 \o^{\prime\prime\prime}\left[\frac{1}{(\omega^{\prime\prime}-
\omega^{\prime\prime\prime})}\frac{\delta}{\delta \psi^{\mu}_{(-)}
(\omega^{\prime\prime}, \bar{\omega}^{\prime\prime})}\,
\frac{\delta}{\delta\psi_{\mu(-)}
(\omega^{\prime\prime\prime}, \bar{\omega}^{\prime\prime\prime})}+\left(\o\leftrightarrow\bar\o,
-\leftrightarrow +\right)\right]\right)\cal{F}\nonumber\\
\end{eqnarray}
Note that the fields $\psi$'s with double prime and triple
prime arguements in (\ref{compact}) are not located at the boundary.

\noindent
We shall see now that normal ordered products are important to
compute the central charge which gives us the critical dimension.
\noindent The energy-momentum tensor
(in the absence of the $\mathcal{B}$ field) for the fermionic sector
for points inside the world-sheet (in the $z$-frame) is given by:
\begin{eqnarray}
\label{energy}
T^{zz}=-\frac{1}{2}\psi_{\mu(+)}\partial_{\bar{z}}\psi^{\mu}_{(+)}
\equiv\bar{T}\nonumber\\
T^{\bar{z}\bar{z}}=-\frac{1}{2}\psi_{\mu(-)}\partial_{z}\psi^{\mu}_{(-)}
\equiv T
\end{eqnarray}
while at the boundary, the BC(s) (\ref{7}) (with $\mathcal{B}=0$)
relating $\psi_{\nu(-)}$ to $\psi_{\nu(+)}$ lead t
:
\begin{eqnarray}
\label{energy1}
\bar{T}=-\frac{1}{2}\psi_{\mu(+)}\partial_{\bar{z}}\psi^{\mu}_{(+)}=-T
\end{eqnarray}
where we have used $\partial_{\bar{z}}=-\partial_{z}$ (since
$dz=-d\bar{z}$ at the boundary).
The central charge can now be computed from the most singular term
in the normal ordered product of energy-momentum tensor.
This involves two contractions of the fermionic coordinate operator
products and is proportional to \cite{br}:
\begin{eqnarray}
\label{energy2}
\int dz^{\prime}...dz^{\prime\prime\prime\prime}\frac{1}{2}
\left[\frac{i\alpha^{\prime}}{(z^{\prime}-z^{\prime\prime})}
\frac{\delta^{2}}{\delta\psi_{\mu(-)}(z^{\prime})
\delta\psi^{\mu}_{(-)}(z^{\prime\prime})}\right]
\left[\frac{i\alpha^{\prime}}{(z^{\prime\prime\prime}-
z^{\prime\prime\prime\prime})}
\frac{\delta^{2}}{\delta\psi_{\mu(-)}(z^{\prime\prime\prime})
\delta\psi^{\mu}_{(-)}(z^{\prime\prime\prime\prime})}\right] \nonumber \\
\times \left[T(z_{1})T(z_{2})\right]\nonumber \\
\sim \frac{D\alpha^{\prime 2}}{4(z_{1}- z_{2})^{4}}\quad \quad \quad
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad
\end{eqnarray}
where $\sim$ mean ``equal up to nonsingular terms"\footnote{The other
less singular terms are not given explicitly.}.
The above computation gives the well known result
$D/2$ as the central charge where $D$ is the dimension of space-time
\cite{pol1}, \cite{kaku}.
The results are also in conformity with \cite{br}.

\noindent We shall make use of the results discussed here in the next section
where we study both free and interacting open super strings.
\section{Mode expansions and Non(anti)Commutativity for super strings}
\subsection{Free open strings}
In this section, we consider the mode expansions of free
(${\cal B}_{\mu \nu} = 0$) open super strings.
We first expand $\psi^{\mu}_{(-)}(z)$ and
$\psi^{\mu}_{(+)}(\bar{z})$ in Fourier modes in ($z, \bar{z}$)
coordinates\cite{pol1}:
\begin{eqnarray}
\label{12}
\psi^{\mu}_{(-)}(z) = \frac{1}{\sqrt{2\pi}}\sum_{m \in {\bf{Z}}}
d^{\mu}_{m}\,\mathrm{exp}(imz)\ \ \ \ ;\ \ \ \
\psi^{\mu}_{(+)}(\bar{z}) = \frac{1}{\sqrt{2\pi}} \sum_{m \in {\bf{Z}}}
\tilde{d}^{\mu}_{m}\,\mathrm{exp}(-im\bar{z}).
\end{eqnarray}
Let us also write these as Laurent expansions in ($\o, \bar{\o}$) coordinates:
\begin{eqnarray}
\label{14}
\psi^{\mu}_{(-)}(\o) = \frac{i^{\frac{1}{2}}}{\sqrt{2\pi}}
\,\sum_{m \in {\bf{Z}}}
\frac{d^{\mu}_{m}}{\o^{m + \frac{1}{2}}}
\ \ \ \ ;\ \ \ \
\psi^{\mu}_{(+)}(\bar{\o}) = \frac{i^{-\frac{1}{2}}}{\sqrt{2\pi}}
\,\sum_{m \in {\bf{Z}}}
\frac{\tilde{d}^{\mu}_{m}}{\bar{\o}^{m + \frac{1}{2}}}.
\end{eqnarray}
Now the BC(s) (\ref{28}) in case of free open super strings
($\mathcal{B}_{\mu\nu} = 0$) requires
$d = \tilde{d}$ in the expansions (\ref{14}). The expressions (\ref{14})
can be equivalently written as:
\begin{eqnarray}
\label{15}
d^{\mu}_{m} &=& \frac{\sqrt{2\pi}}{\sqrt{i}}\oint \frac{d\o}{2\pi i}\,
\o^{m - \frac{1}{2}}\, \psi^{\mu}_{(-)}(\o)
 = \,-\sqrt{2\pi}\sqrt{i}\oint \frac{d\bar{\o}}{2\pi i}\,
\bar{\o}^{m - \frac{1}{2}}\, \psi^{\mu}_{(+)}(\bar{\o}).
\end{eqnarray}
The anticommutation relation between $d$'s can be worked out
from the contour arguement \cite{pol} and the operator product expansion (OPE)
(\ref{11}) (with $\mathcal{B_{\mu\nu}}=0$):
\begin{eqnarray}
\label{16}
\left\{d^{\mu}_{m}, d^{\nu}_{n}\right\} &=& \frac{1}{i}
\oint \frac{d\o_2}{2\pi i} \mathrm{Res}_{\o_1 \to \o_2}\left(
\o^{m - \frac{1}{2}}_1\,  \psi^{\mu}_{(-)}(\o_1)\,
\o^{n - \frac{1}{2}}_2\,  \psi^{\nu}_{(-)}(\o_2)\right) \nonumber \\
&=& 2\pi\alpha^{\prime}\,\eta^{\mu \nu}\, \delta_{m+n,0}
=\eta^{\mu \nu}\, \delta_{m+n,0}
\end{eqnarray}
where we have set $2\pi\alpha^{\prime}=1$.
The anti-commutation relations between $\psi^{\mu}_{(-)}(\o, \bar{\o})$ and
$\psi^{\nu}_{(+)}(\o^{\prime}, \bar{\o^{\prime}})$ are then obtained by using
(\ref{16}):
\begin{eqnarray}
\left\{\psi^{\mu}_{(-)}(\o, \bar{\o}), \psi^{\nu}_{(-)}(\o^{\prime},
\bar{\o^{\prime}})\right\} &=& \frac{i\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}}}\left(\o^{-m - \frac{1}{2}}\,
 \o^{\prime m - \frac{1}{2}} \right) \nonumber \\
\left\{\psi^{\mu}_{(+)}(\o, \bar{\o}), \psi^{\nu}_{(+)}(\o^{\prime},
\bar{\o^{\prime}})\right\} &=& -\frac{i\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}} }\left(\bar{\o}^{-m - \frac{1}{2}}\,
 \bar{\o}^{\prime m - \frac{1}{2}} \right)\nonumber \\
\left\{\psi^{\mu}_{(-)}(\o, \bar{\o}), \psi^{\nu}_{(+)}(\o^{\prime},
\bar{\o^{\prime}})\right\} &=& \frac{\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}}}\left(\o^{-m - \frac{1}{2}}\,
 \bar{\o}^{\prime m - \frac{1}{2}} \right).
\label{17}
\end{eqnarray}
To obtain the usual equal time ($\tau = \tau^{\prime}$)
anticommutation relation we first rewrite (\ref{17}) in ``$z$ frame''
using (\ref{10}, \ref{13}) and then in terms of $\s^1, \ \s^2$ to find:
\begin{eqnarray}
\left\{\psi^{\mu}_{(-)}(\s^1, \s^2), \psi^{\nu}_{(-)}(\s^{1 \prime},
\s^{\prime 2})\right\} &=& \frac{\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}}}\left[\exp\left({im(\s^1 + i \s^2
- \s^{\prime 1} - i \s^{\prime 2})} \right)\right]\nonumber \\
\left\{\psi^{\mu}_{(+)}(\s^1, \s^2), \psi^{\nu}_{(+)}(\s^{1 \prime},
\s^{\prime 2})\right\} &=& \frac{\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}}}\left[\exp\left({im(\s^1 - i \s^2
- \s^{\prime 1} + i \s^{\prime 2})} \right)\right]\nonumber \\
\left\{\psi^{\mu}_{(-)}(\s^1, \s^2), \psi^{\nu}_{(+)}(\s^{1 \prime},
\s^{\prime 2})\right\} &=& \frac{\eta^{\mu \nu}}{2 \pi}
 \sum_{m \in {\bf{Z}}}\left[\exp\left({im(\s^1 + i \s^2
+ \s^{\prime 1} - i \s^{\prime 2})} \right)\right].
\label{18}
\end{eqnarray}
Finally substituting $\tau = \tau^{\prime}$ (i.e. $\s^2 = \s^{2\,\prime}$)
and $\s^1 = \s$ we get
back the equal time anti-commutation relations:
\begin{eqnarray}
\left\{\psi^{\mu}_{(-)}(\s, \tau), \psi^{\nu}_{(-)}(\s^{\prime},
\tau)\right\} &=&\eta^{\mu \nu}
 \delta_{P}(\s - \s^\prime)\nonumber \\
\left\{\psi^{\mu}_{(+)}(\s, \tau), \psi^{\nu}_{(+)}(\s^{\prime},
\tau)\right\} &=& \eta^{\mu \nu}
 \delta_{P}(\s - \s^\prime)\nonumber \\
\left\{\psi^{\mu}_{(-)}(\s, \tau), \psi^{\nu}_{(+)}(\s^{\prime},
\tau)\right\} &=& \eta^{\mu \nu}
 \delta_{P}(\s + \s^\prime).
\label{19}
\end{eqnarray}
where, $\delta_{P}(\s - \s^\prime)$ is the so called periodic delta function
which is defined as:
\begin{eqnarray}
\delta_{P}(\s - \s^\prime) = \frac{1}{2 \pi}\sum_{m \in {\bf{Z}}}
\exp\left(im(\s - \s^{\prime})\right).
\label{20}
\end{eqnarray}
This structure of anticommutator is completely consistent with the
BCs (\ref{28}) for ${\mathcal{B}}_{\mu\nu} = 0$.
Note that not
only the usual Dirac delta function is replaced by the periodic delta
function but also the anticommutator among $\psi_{(-)}, \psi_{(+)}$ are
non-vanishing even in case of the free open fermionic string
\cite{agh, jing}.
\noindent The most important feature of the above analysis is that unlike
the bosonic case, the new normal ordering of the fermionic
operators (that incorporates the BC(s)) (\ref{11}) leads to the
nonanticommutative structures (\ref{19}) among the fermionic
string coordinates.

\subsection{Open superstring in the constant ${\mathcal{B}}$-field background}
We now analyse the open superstring moving in presence of a background
antisymmetric tensor field ${\cal{B}}_{\mu \nu}$.
To begin with, let us again consider the Laurent expansion of
$\psi_{(-)}^{\mu}(\o)$ and $\psi_{(+)}^{\mu}(\bar{\o})$
(\ref{14}). Now due to the BC(s) (\ref{28}) (with $\mathcal{B}_{\mu \nu} \neq 0$),
the modes $d$ and $\tilde{d}$ are no longer
independent but satisfy the following relation:
\begin{eqnarray}
E_{\mu \nu}\, d^{\nu}_{m}\, = \, E_{\nu \mu}\, \tilde{d}^{\nu}_{m}.
\label{21}
\end{eqnarray}
Hence there exists only one set of independent modes
$\alpha^{\mu}_{m}$, which can be thought of as the modes of free open
strings and is related to $d^{\mu}_{m}$ and $\tilde{d}^{\mu}_{m}$ by:
\begin{eqnarray}
\label{22}
d^{\mu}_{m} &=& \left(\delta^{\mu}_{\ \nu} - {\cal{B}}^{\mu}_{\ \nu}
\right)\alpha^{\nu}_{m} := \left[({1\!\mbox{l}} - {\cal{B}})\alpha
\right]^{\mu}_{m} \nonumber \\
\tilde{d}^{\mu}_{m} &=& \left(\delta^{\mu}_{\ \nu} + {\cal{B}}^{\mu}_{\ \nu}
\right)\alpha^{\nu}_{m} := \left[({1\!\mbox{l}} +
{\cal{B}})\alpha\right]^{\mu}_{m} .
\end{eqnarray}
Note that under world-sheet parity transformation
(i.e. $\sigma\leftrightarrow-\sigma$),
$d^{\mu}_{m} \leftrightarrow \tilde{d}^{\mu}_{m}$, since ${\cal{B}}_{\mu \nu}$
is a world-sheet pseudo-scalar (similar to bosonic part \cite{our}).
Substituting (\ref{22}) in (\ref{14}), we obtain  the
following Laurent expansions for $\psi^{\mu}_{-}$ and $\psi^{\mu}_{+}$:
\begin{eqnarray}
\label{23}
\psi^{\mu}_{(-)}(\o) &=& \frac{i^{\frac{1}{2}}}{\sqrt{2\pi}}
\sum_{m \in {\bf{Z}}}
\frac{\left[({1\!\mbox{l}} -
{\cal{B}}) \alpha\right]^{\mu}_{m}}{\o^{m + \frac{1}{2}}} \\
\psi^{\mu}_{(+)}(\bar{\o}) &=& \frac{i^{- \frac{1}{2}}}{\sqrt{2\pi}}
\sum_{m \in {\bf{Z}}}
\frac{\left[({1\!\mbox{l}} + {\cal{B}})
\alpha\right]^{\mu}_{m}}{\bar{\o}^{m + \frac{1}{2}}}\nonumber.
\end{eqnarray}
These are the appropriate mode expansions for the fermionic part of the
interacting superstring,
that satisfy both the equations of motion (\ref{6}) and the BC(s) (\ref{28}).

\noindent Now the expressions (\ref{23}) for interacting superstrings can
also be written as:
\begin{eqnarray}
\left[({1\!\mbox{l}} - {\cal{B}})\alpha\right]^{\mu}_{m} &=&
\frac{\sqrt{2\pi}}{i}\oint \frac{d\o}{2\pi i}\,
\o^{m - \frac{1}{2}}\, \psi^{\mu}_{(-)}(\o) \nonumber \\
\left[({1\!\mbox{l}} + {\cal{B}})\alpha\right]^{\mu}_{m} &=&
\,\frac{\sqrt{2\pi}}{i}\oint \frac{d\bar{\o}}{2\pi i}\,
\bar{\o}^{m - \frac{1}{2}}\, \psi^{\mu}_{(+)}(\bar{\o}) .
\label{24}
\end{eqnarray}
The anticommutation relation between $\alpha$'s can be obtained once again
from the contour arguement (using (\ref{24})) and the $\psi\, \psi$ OPE (\ref{11}):
\begin{eqnarray}
\label{25}
\left\{\alpha^{\mu}_{m}, \alpha^{\nu}_{n}\right\}
=  \left[\left({1\!\mbox{l}} - {\cal{B}}^2
\right)^{-1}\right]^{\mu \nu} \delta_{m, -n}
= \left(\mathcal{M}^{-1}\right)^{\mu \nu} \delta_{m, -n}
\end{eqnarray}
where, $\mathcal{M} = ({1\!\mbox{l}} - {\cal{B}}^2)$ ; $({\cal{B}}^2)^{\mu \nu}
= {\cal{B}}^{\mu}_{\ \rho}{\cal{B}}^{\rho \nu}$\footnote{Here we
should note that $\left({1\!\mbox{l}}\right)^{\mu \nu}
= \eta^{\mu \nu}$.}.
Now the anticommutator between the fermionic string coordinates
can be computed using (\ref{23}), (\ref{25}). The antibrackets between
$\left\{\psi^{\mu}_{(-)}, \psi^{\nu}_{(-)}\right\}$ and
$\left\{\psi^{\mu}_{(+)}, \psi^{\nu}_{(+)}\right\}$ are the same
as that of free case but the anticommutator between
$\psi^{\mu}_{(-)}$ and $\psi^{\mu}_{(+)}$ gets modified to the following form:
\begin{eqnarray}
\left\{\psi^{\mu}_{-}(\o, \bar{\o}), \psi^{\nu}_{+}(\o^{\prime},
\bar{\o^{\prime}})\right\} &=& \frac{1}{2 \pi} \,
 \sum_{m \in {\bf{Z}}} \left[\frac{\left({1\!\mbox{l}} - {\cal{B}}
\right)^{\mu}_{\ \rho}\, \left[\left({1\!\mbox{l}} - {\cal{B}}^2\right)^{-1}
\right]^{\rho \s}\,
\left({1\!\mbox{l}} - {\cal{B}}\right)_{\s}^{\ \nu}
}{\o^{m + \frac{1}{2}}\, \bar{\o}^{\prime -m + \frac{1}{2}}}\right].
\label{26}
\end{eqnarray}
Now proceeding as before, we can write the above anticommutation relation
in $(\tau, \s)$ coordinates to obtain the usual equal time
(i.e. $\tau = \tau^{\prime}$) anticommutation relation:
\begin{eqnarray}
\left\{\psi^{\mu}_{(-)}(\s, \tau), \psi^{\nu}_{(+)}(\s^{\prime},
\tau)\right\} &=& E^{\rho\, \mu}\,
\left[\left({1\!\mbox{l}} - {\cal{B}}^2\right)^{-1}
\right]_{\rho\, \s}\, E^{\nu \,\s}\,
 \delta_{P}(\s + \s^\prime).
\label{27}
\end{eqnarray}
The above result reduces to the the free case result in the
$\mathcal{B}_{\mu\nu}=0$ limit and also agrees with the existing results
in the literature \cite{agh}, \cite{jing}.

\section{Conclusions}
In this paper, we have used conformal field theoretic techniques
to compute the anticommutator among Fourier components of fermionic sector
of super strings. Using this the anticommutator between the
basic fermionic fields is obtained.
This is the extension of our earlier work on bosonic strings
\cite{our}. The method is also different from (\cite{jing}),
where the algebra among the Fourier components
have been computed using the Faddeev-Jackiw symplectic formalism.
The advantage of this approach (as mentioned in
\cite{our} also) is that the results one obtains takes
into account the quantum effects right from the beginning,
in contrary to the previous
investigations, which were made essentially at the classical level
\cite{chu, chu1, rb, jing, br1}. Interestingly,
the new normal ordering that takes into account the
effect of the BC(s) plays a crucial role in obtaining the
nonanticommutative symplectic structure among the fermionic string
coordinates (\ref{19}). This is in contrast to the analysis in case of the bosonic
strings where the new normal ordering has no bearing on the symplectic
structure.
\noindent Finally, we also computed the oscillator
algebra in presence of the $\mathcal{B}$
field which is a parity-odd field on the string world-sheet. As in the
bosonic case,
in presence of this $\mathcal{B}$ field,
the fourier modes appearing in the  Laurent series expansions
of the fermionic fields $\psi^{\mu}_{(-)}$ and $\psi^\mu_{(+)}$
of the closed string are
no longer equal when open string BCs (\ref{28}) are imposed. These rather get
related to the free oscillator
modes $d^{\mu}_{m}$.
Using these expressions of the modes (\ref{22}), we rewrite
the fermionic fields $\psi^{\mu}_{(-)} $ and $\psi^{\mu}_{(+)}$
entirely in terms
of the free oscillator modes $\alpha^{\mu}_{m}$ (\ref{22}). Then
a straight forward calculation, involving $\psi\psi$ OPE (\ref{11})
and contour arguement yields the NC anticommutator given in (\ref{27}),
thereby reproducing the results of \cite{agh}.
\section*{Appendix}
Here we would like to give some of the
computational details involved in deriving
(\ref{eqmov2}) from (\ref{8}) and (\ref{28}) (for convenience we treat the
free case, i.e. $\mathcal{B} = 0$).
Eq.(\ref{8}) with $z$ replaced by $\omega$ yields:
\begin{eqnarray}
\label{a1}
0&=&\int [d\psi]\left[  \frac{\delta}{\delta \psi^{\mu} _{(a)}
(\omega,\bar{\omega})}[e^{-S_{F}} \psi^{\nu}_{(b)} (\omega', \bar{\omega}')]
  \right] \nonumber\\
&=&\int[d\psi]e^{-S_{F}}
\left[-\frac{\delta S_{F}}{\delta\psi_{(a)}(\omega,\bar{\omega})}
\psi_{(b)}(\omega^{\prime},\bar{\omega}^{\prime})+
\frac{\delta\psi_{(b)}(\omega^{\prime},\bar{\omega}^{\prime})}
{\delta\psi_{(a)}(\omega,\bar{\omega})}\right]
\end{eqnarray}
Putting $a=+$, $b=-$; we obtain:
\begin{eqnarray}
\label{a2}
0&=&\int[d\psi]e^{-S_{F}}
\left[\frac{i}{2\pi\alpha^{\prime}}\partial_{\omega}\psi_{+}
(\omega,\bar{\omega})\psi_{-}(\omega^{\prime}, \bar{\omega}^{\prime})+
\frac{\delta\psi_{(-)}(\omega^{\prime},\bar{\omega}^{\prime})}
{\delta\psi_{(+)}(\omega,\bar{\omega})}\right]
\end{eqnarray}
where we have dropped the boundary term (arising from
the first term in (\ref{a1})) as it vanishes due to the
BC(s) (\ref{28}) (with $\mathcal{B}=0$).\\

\noindent Now we discuss two distinct cases seperately.

\noindent $\bullet$ Case 1: The insertion
$\psi_{(-)}(\omega^{\prime},\bar{\omega}^{\prime})$ is not located
at the boundary:

\noindent In this case
\begin{eqnarray}
\label{a3}
\frac{\delta\psi_{(-)}(\omega^{\prime},\bar{\omega}^{\prime})}
{\delta\psi_{(+)}(\omega,\bar{\omega})}=0
\end{eqnarray}
and therefore one finds:
\begin{eqnarray}
\label{a4}
\langle \partial_{\o} \psi^{\mu}_{(+)} (\o,\bar{\o}) \psi^{\nu}_{(-)}
(\o',\bar{\o}') \rangle &=& 0.
\end{eqnarray}

\noindent $\bullet$ Case 2: The insertion
$\psi_{(-)}(\omega^{\prime})$ is located at the boundary (since
$\o' = \bar{\o}'$ at the boundary, the insertion
$\psi_{(-)}(\omega^{\prime})$ depends only on the arguement $\o'$): \\
In this case the computation of the second term in (\ref{a2})
needs to be done more carefully. One finds
\begin{eqnarray}
\label{a5}
\frac{\delta\psi_{(-)}(\omega^{\prime},\bar{\omega}^{\prime})}
{\delta\psi_{(+)}(\omega,\bar{\omega})}\Big{\vert}_{\o' = \bar{\o}'} &=&
i\, \frac{\delta\psi_{(+)}(\omega^{\prime},\bar{\omega}^{\prime})}
{\delta\psi_{(+)}(\omega,\bar{\omega})}\Big{\vert}_{\o' = \bar{\o}'}
\nonumber \\
&=& i\, \delta^2 \left(\o-\o', \bar{\o}- \bar{\o}'\right)\Big{\vert}_{\o' = \bar{\o}'} \nonumber \\
&=&  i\, \delta^2 \left(\o-\o', \bar{\o}- \o'\right).
\end{eqnarray}
where we have used the BC (\ref{28}) (with $\o$ repaced by $\o^{\prime}$)
in the first line of (\ref{a5}).

\noindent Substituting (\ref{a5}) in (\ref{a2}) and equating the volume term to zero,
one finds the third of the equations in (\ref{eqmov2})
(with $\mathcal{B} = 0$).

\noindent
Similarly, for other choices of $a, b$
the rest of the equations in (\ref{eqmov2}) can be derived (with
$\mathcal{B} = 0$).

\section*{Acknowledgment}
The authors would like to thank the referee for very useful comments.

redpajama_set_name: RedPajamaArXiv


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