Field Theory Dynamics from Branes in String Theory D i s s e r t a t i o n zur Erlangung des akademischen Grades d o c t o r r e r u m n a t u r a l i u m Dr rer nat im Fach Physik eingereicht an der MathematischNaturwissenschaftlichen Fakultat I der HumboldtUniversitat zu Berlin von Andreas Karch MA geb oren am in Bamberg Prasident der HumboldtUniversitat zu Berlin Prof Dr Dr hc H Meyer Dekan der MathematischNaturwissenschaftlichen Fakultat I Prof Dr JP Rab e Gutachter Tag der m undlichen Pr ufung Contents Motivation Dbranes and nonp erturbative eects in string theory The breakdown of p erturbation theory A string theoretic description of Dbranes Dbranes and gauge theory Gauge theory on the worldvolume Compactications and Dbranes Engineering Gauge theories Geometric Engineering Branes as Prob es HananyWitten setups Dbranes and dualities String Dualities and Mtheory Matrix Theory Worldvolume theory of the NS brane Exploiting the SYM Dbrane corresp ondence Classical HananyWitten setups Branegineering Theres so much more one can do Solving the quantum theory Bending and quantum eects Lifting to Mtheory Applications of the brane construction Dualities in the brane picture Exact Sduality Seib erg duality and Mirror symmetry Sdual N pairs revisited Nontrivial RG xed p oints i App earance and applications Physics at nontrivial FP from branes d HananyWitten setups Extracting information ab out the FP theory Global Symmetries Deformations and Phase Transitions Correlators from branes A chiral nonchiral transition The equivalence of the various approaches The duality b etween orbifold and NS brane Hanany Witten versus branes at orbifolds The orbifold construction The classical branches Including D branes and orientifolds Applying Tduality Adiabatically expanding to d N Branes and geometry obtaining the Seib ergWitten curve The approaches A Tduality for b ent branes and branes ending on branes Unifying the dierent approaches Op en problems and directions for further research Brane Boxes Maldacena Conjecture Summary Bibliography Zusammenfassung Danksagung Selbstandigkeitserklarung Leb enslauf Publikationsliste ii Motivation and Introduction Motivation It is the main goal of physics to explain all ratios b etween measurable quantities The hop e is that in the end a very simple b eautiful and unique mathematical structure emerges the theory of everything Mo dern physics is based on two ma jor building blo cks general relativity and quantum physics The former expresses all phenomena of the macroscopic world and esp ecially gravity in terms of simple geometric concepts The latter is p owerful in explaining the microscopic world by replacing the notion of particles which o ccupy a certain p osition in space and carry a certain amount of momentum by the more abstract formalism of states in a Hilb ert space and observables as op erators acting on them Since the distinction b etween microscopic and macroscopic world seems to b e rather arbitrary these two building blo cks should b e unied in an underlying theory More than that as they stand the two concepts are even inconsistent Straight forward quantization of general relativity leads to innities in physical pro cesses that can not b e tolerated A larger theory unifying gravity and quantum theory is hence not only desired from an aestethic p oint of view but indeed required for consistency After many years of extensive search for this unifying theory a single candidate has emerged string theory String theory replaces the fundamental p oint like ob jects of particle physics by d strings thereby removing the innities encountered in quantizing general relativity General relativity reemerges as a low energy limit at large distances where it was tested exp erimentally However at small distances stringy physics takes over and even our concepts of space and time break down Similarly ordinary particle physics as describ ed by the standard mo del can reemerge in the limit where gravitational interactions b etween the particles can b e neglected The energy scale at which b oth gravity and quantum eects b ecome imp ortant is set by the Planck scale and is roughly GeV Since this scale is so huge it is imp ossible to just create the fundamental degrees of quantum gravity in an accelerator and then lo ok what they are Even though string theory has all the ingredients required by mo dern physics it is dicult to make contact with physics as we know it The ma jor obstacle is that string theory allows for a variety of dierent vacua each of which leads to dierent physics No precise predictions ab out the lowenergy physics like answers to the why questions left by the standard mo del can b e made without nding a pro cess that determines the right vacuum However the most imp ortant concepts app earing in string theory namely gauge theory gravity and sup ersymmetry are indeed known or b elieved to dominate the real world While the former two are the bases for all physics describ ed by the standard mo del and general relativity the latter is b elieved to b e of similar imp ortance for next generation collider physics Exp erimental verication of sup ersymmetry in the real world would b e even more supp ort that string theory is not just the only known consistent quantum theory of gravity but indeed the fundamental theory realized in our world Of sp ecial interest are also ob jects which prob e the regime of quantum gravity that is they are small enough for quantum eects to b e imp ortant and heavy enough to require gravity Examples of such ob jects are black holes close to their singularity or our universe in very early times close to big bang Treatment of these imp ortant issues as well as the vacuum selection problem requires nonp erturbative information ab out string theory So far the p erturbative expansion of string theory in terms of worldsurfaces was the only denition we had of string theory Only very recently to ols have emerged that allow us control certain asp ects of the nonp erturbative physics b ehind string theory raising the hop e that these fundamental issues can nally b e addressed Introduction During the so called second string revolution it has b ecome p ossible to gain control over asp ects of string theory that were not contained in its p erturbation expansion in terms of worldsurfaces The ma jor achievements were the discovery of Dbranes as one of the nonp erturbative ob jects in string theory and the realization of the role of duality symmetries in string theory relating two seemingly dierent theoretical descriptions to one and the same physical situation Dualities have b een of similar imp ortance in gauge theories Using dualities it has b een p ossible to solve the IR b ehaviour of certain quantum eld theories exactly and get a lot of nonp erturbative information even in situations with less restrictive symmetries However in all those setups sup ersymmetry is a vital ingredient and it is not clear yet how these metho ds can b e generalized to nonsup ersymmetric situations Nonp erturbative string theory and Sup er Yang Mills SYM gauge theories are indeed deeply related The dynamics of the Dbranes which are so imp ortant for our understand ing of nonp erturbative stringtheory is basically governed by SYM This connection can b e used in a twofold way dualities and other obscure asp ects of eld theory only dis covered recently like nontrivial xed p oints of the renormalization group with mutual nonlo cal ob jects b ecoming massless nd their natural place in string theory where they can b e easily visualized On the other hand some problems which seem intractable in string theory can b e mapp ed to questions in gauge theory which are under much b etter control Indeed it has b een prop osed that all nonp erturbative asp ects of string theory can b e enco ded in large N SYM theory It is the purp ose of this work to highlight some of the insights gained with the help of this deep connection b etween gauge theory and string theory In Chapter I will review the construction of Dbranes and explain how gauge theory determines their dynamics I will also comment on the imp ortant role Dbranes played in understanding string theory dualities since these dualities lead to the discovery of an dimensional theory called Mtheory that basically summarizes all the nonp erturbative insights we gained ab out string theory and is