If I Only Had A Brane!

This thesis starts with a review of supersymmetric solutions of 11-dimensional supergravity; in particular flat M-branes and BPS configurations which can be constructed from them. The harmonic function rule is discussed and it is shown why this cannot be expected to apply to intersecting brane configurations where the intersection is localised. A new class of spacetimes is needed to cover these situations and the Fayyazuddin-Smith metric ansatz is introduced as the answer to this problem. The ansatz is then used to find supergravity solutions for M-branes wrapped on holomorphic curves. This method is discussed in detail and its various components explained; for instance, the way holomorphicity dictates the projection conditions on the Killing spinor and the construction of a spinor in terms of Fock space states. The method is illustrated via various examples. We then move on and discuss a mathematical concept known as calibrations. Calibrations are forms which can be used to pick out the minimal surfaces in a given background. Since BPS configurations of M-branes are minimal, it turns out that they must wrap cycles ’calibrated’ by such p-forms. For the case when the background contains no flux, calibrations have been classified. However, for more general cases where there is a non-trivial field strength, such a classification does not yet exist. This would be desirable for various reasons, one of which is the following. Given a particular calibrated form, there is a simple and very elegant method of writing down the supergravity solution for a brane wrapped on the corresponding calibrated cycle. So far, this method was applied only to Kahler calibrations as these were the only ones known to exist in backgrounds with non-trivial flux. We extend this method to a wider class which contains Kahler calibrations. A constraint is used to classify possible calibrations; this constraint incorporates the geometry of the space transverse to the submanifold which contains the supersymmetric cycle. A rule is given which can generate the required constraint for any given M-brane wrapped on a holomorphic cycle. Ways in which this constraint can be satisfied are also discussed. Acknowledgments My “Esteemed Advisor”, Ansar Fayyazuddin, took considerable time off from contemplating ’the logic of late capitalism’ to tend to his advisory duties. Regardless of the nature of the problem I pose him with, Ansar always manages to rise to the occasion. He has taught me much, but more than that I owe him gratitude for giving me back my confidence at a time when Grad School was beating it down. I feel privileged to know him both as a student and as a friend. It has been great fun having Fawad Hassan around. This thesis owes a lot to his endless supply of patience, chocolates and encouragement. He is always on hand when questions need to be answered and, together with Ioanna Pappa, is an excellent partner in crime when some crazy scheme is being hatched. The String Theory group here is rather like a library of friendly human encyclopedias. All queries can be addressed by walking down the corridor and knocking on the doors of Ulf Lindstrom, Bo Sundborg and earlier, Alberto Guijosa, and Subir Mukhopadhyay. Our all-girl group of students has to be a first – on this planet at least! I count on Cecilia Albertsson and Teresia Mansson for various things, from bailing me out of trouble with the printer, to a friendly chat in the middle of the day; when panic hits and reassurance is needed, Cecilia can be found at the desk next to mine. But before it was just us girls, Bjorn Brinne and Maxim Zabzine were around as well. I learnt a lot from both of them and still count on Bjorn for assorted help and advice. Were it not for Britta Schmidt, I would be buried somewhere under piles of paper-work, with my Swedish dictionary in hand. She is truly the Guardian Angel of the Physics Department and I don’t know what I would have done without her smiling face to turn to. Thanks are also due to the Theory Group at UT-Austin for their hospitality during my visit there. In particular, Jan Duffy for her welcoming warmth, Julie Blum for many useful discussions and Amer Iqbal for several fascinating conversations about Physics and everything else. I couldn’t go without expressing my heartfelt gratitude to Ishrat Phophi, who never let me feel the lack of family in Sweden and to Khalu Jan who got me started on my journey here. It’s amazing what friends can pull you through. Over the course of this PhD, many potential therapy sessions were averted by phone calls, emails and visits to Annie, Aisha, Afi, Sabir, Sidra, Erej, Hajira, Mahrukh, Aneeka, Roshani, Stella, Nicole, Saleha, Sherezade, Cristine and the HEPlons. Never has the world known kids more precious than Manni and Ali. They are the best cheer-upers on Earth with their undying faith in me and their unconditional love. Amir and Zaib have been adorable specially during my time in Austin. Despite their firm conviction that I really do nothing and this whole PhD thing is just an elaborate hoax they have been rather indulgent of it all. Shabeeb has suffered more than his fair share at the hands of my PhD – he has borne it well, with unwavering love and support. Maybe, now that the thesis is done, his suffering will be over as well .... no promises though! ;) Most of all, my love and my deepest gratitude is for Abbu and Ammi, who gave me the roots which steady me and wings with which to fly. Between them, they have created such a haven that no matter where I go “there’s no place like home”.