Recent Developments in String Theory From Perturbative Dualities to MTheory Lectures given by D L ust at the Saalburg Summer School in September y y Michael Haack Boris Kors Dieter L ust MartinLutherUniversitatHalleWittenberg Institut f urPhysik Friedemann Bach Platz HalleSaale Germany y Humboldt Universitatzu Berlin Institut f urPhysik Invalidenstr Berlin Germany Abstract These lectures intend to give a p edagogical introduction into some of the developments in string theory during the last years They include p erturbative Tduality and non p ertur bative S and Udualities their unavoidable demand for Dbranes an example of enhanced gauge symmetry at xed p oints of the Tduality group a review of classical solitonic so lutions in general relativity gauge theories and tendimensional sup ergravity a discussion of their BPS nature Polchinskis observations that allow to view Dbranes as RR charged states in the non p erturbative string sp ectrum the application of all this to the computation of the black hole entropy and Hawking radiation and nally a brief survey of how everything ts together in Mtheory hep-th/9904033 6 Apr 1999 1 Email MichaelHeraPhysikUniHalleDe 2 Email Ko ersPhysikHUBerlinDe 3 Email LuestPhysikHUBerlinDe Contents Introduction Perturbative string theory TDuality Nonp erturbative dualities Sduality Uduality Stringstringduality Duality to elevendimensional sup ergravity and Mtheory Tduality Closed b osonic string theory Compactication on a circle D Compactication on a torus T Heterotic string theory Type I IA and I IB sup erstring theory Op en strings Non p erturbative phenomena Solitons in eld theory Black holes Magnetic monop oles BPS states Solitons in string theory Extended charges as sources of tensor elds Solutions of the sup ergravity eld equations pbranes Dbranes as pbranes Black holes in string theory Mtheory A Compactication on T and Tduality Introduction These notes are a summary and a substantial extension of the material that D L ustpresented in his lectures at the summer school at Saalburg in They are intended to give a basic overview over non p erturbative eects and duality symmetries in string theory including recent developments After a short review of the status of p erturbative string theory as it presented itself b efore the second string revolution in and a brief summary of the recent progress esp ecially concerning non p erturbative asp ects the main text falls into two pieces In chapter we will go into some details of Tduality Afterwards chapter and chapter will fo cus on some non p erturbative phenomena The text is however not meant as an introduction to string theory but rather relies on some basic knowledge see eg the lectures given at this school by O Lechtenfeld or The references we give are never intended to b e exhaustive but only to display the material that is essentially needed to justify our arguments and calculations Perturbative string theory Before string theory was only dened via its p erturbative expansion As the string moves in time it sweeps out a two dimensional worldsheet which is embedded via its co ordinates in a Minkowski target space M X M This worldsheet describ es after a Wick rotation in the time variable a Riemann surface p ossibly with b oundary Propagators or general Greens functions of scattering pro cesses can b e expanded in the dierent top ologies of Riemannian surfaces which corresp onds to an expansion in the string coupling constant g see g The reason for this is that all string diagrams can S + + + ... Figure String p erturbation expansion b e built out of the fundamental splitting resp ectively joining vertex g This vertex comes along with a factor g which is given by the vacuum exp ectation value VEV of a scalar eld S the so called dilaton hi g e S As there is no p otential for the dilaton in string p erturbation theory its VEV is an arbitrary g s Figure Fundamental string interaction vertex parameter which can b e freely chosen Only if it is small the ab ove expansion in Riemann surfaces makes sense Statements ab out the strong coupling regime on the other hand require some knowledge ab out non p erturbative characteristics of string theory such as duality relations combining weakly coupled string theories with strongly coupled ones First quantizing the string amounts to quantizing the embedding co ordinates regarded as elds of a two dimensional con formal eld theory living on the world sheet This is a two dimensional analog of p oint particle quantum mechanics A sensible second quantized string eld theory is very dicult to achieve and will not b e discussed here any further In the p erturbative regime there exist ve consistent Type Gauge group of sup ercharges N Heterotic E E Heterotic SO I includes op en strings SO I IA nonchiral I IB chiral Table The ve consistent sup erstring theories in d ten dimensional sup erstring theories see table Type I IA and I IB at rst sight do not contain any op en strings In fact they do however app ear if one introduces the non p erturbative ob jects called Dbranes which are hyperplanes on which op en strings can end Thus from the world sheet viewp oint a Dbrane manifests itself by cutting a hole into the surface and imp osing Dirichlet b oundary conditions These ob jects will b e studied in more detail b elow To get a string theory in lower dimensional spacetime such as in the phenomenologically most interesting case d one has to compactify the additional space dimensions There are several metho ds to construct fourdimensional string theories and in fact there are many dierent ways to get rid of the extra dimensions A priori each compactication gives rise to a dierent string vacuum with dierent particle content gauge group and couplings This huge vacuum degeneracy in four dimensions is known as the vacuum problem But despite of the large number of dierent known vacua it has not yet b een p ossible to nd a compactication yielding in its low energy approximation precisely the standard mo del of particle physics TDuality Tduality or target space duality denotes the equivalence of two string theories compactied on dierent background spaces Both theories can in fact b e considered as one and the same string theory as they contain exactly the same physics The equivalence transformation can thus b e considered as some kind of transformation of variables in which the theory is describ ed Nev ertheless we will always use the usual terminology sp eaking of dierent theories when we actually mean dierent equivalent formulations of the same physical theory Tduality is a p erturbative symmetry in the sense that the Tduality transformation maps the weak coupling region of one theory to the weak coupling regime of another theory Thus it can b e tested in p erturbation theory eg by comparing the p erturbative string sp ectra Examples of Tdualities are p 0 Tdual R p Het on S with radius R Het on S with radius D 0 R p 0 Tdual R p I IA on S with radius I IB on S with radius R D 0 R These are sp ecial cases of the so called mirror symmetry As we will see in chapter Tduality transformations for closed strings exchange the winding number around some circle with the corresp onding discrete momentum quantum number Thus it is clear that this symmetry relation has no counterpart in ordinary p oint particle eld theory as the ability of closed strings to wind around the compactied dimension is essential Nonp erturbative dualities At strong coupling the higher top ologies of the expansion g b ecome large and the series expansion do es not make sense anymore Non p erturbative eects dominantly contribute to the scattering pro cesses Their contributions b ehave like g g S S A e or A e The second exp onential with g g is the typical non p erturbative suppression factor in S YM gauge eld theoretic amplitudes involving solitons like magnetic monop oles or instanton eects Solitons also play a role in general relativity in the form of black holes In general solitons are non trivial solutions of the eld equations which have a nite action integral Their energy is lo calized in space and they have prop erties similar to p oint particles Clearly it is of some interest to ask what kind of solitonic ob jects app ear in string theory giving rise to the b ehavior of eq The answer is that the string solitons are extended pspatialdimensional at ob jects called pbranes ie p dimensional hypersurfaces in spacetime The sp ecial values p therefore give p oint particles strings and membranes resp ectively Such ob jects can indeed b e found as classical solutions of the eective low energy eld theories derived from the various sup erstring theories see section It was however Polchinskis achievement to realize that some of them namely those which do not arise in the