![]() ![]() We construct brane solutions of supergravity, discuss their BPS properties and identify those carrying R-R charge as D-branes. ![]() We will discuss the concept of dualities and the role of BPS states in establishing non-perturbative dualities. The five superstring theories which we have encountered so far will be argued to be different perturbative limits of one underlying theory, connected via a web of perturbative and non-perturbative dualities. ![]() The Dirac-Born-Infeld action, which governs the dynamics of the gauge field on a D-brane, will also be discussed. We also include a discussion of eleven-dimensional supergravity. We present the bosonic sectors of the ten-dimensional supergravity theories which are related to the ten-dimensional superstring theories. Often the leading order (in α′) effective actions are already uniquely determined by symmetries, such as gauge symmetries or supersymmetry. We also compute the four-point functions of open and closed string tachyons and discuss some of their properties. the interactions of graviton, two-form, dilaton and of gauge fields at leading order. We construct the vertex operators and compute various three-point functions which are needed to extract e.g. Such effective actions can be deduced from on-shell string scattering amplitudes which are computed as correlation functions of physical state vertex operators. To relate string theory to the usual description of particles and their interactions in terms of quantum field theories, it is important to have tools at hand to derive the effective point particle interactions for the massless excitation modes of the string. The derivations of some results which are used in the main text are also relegated to the appendix. In an appendix we fix our notation and review some concepts of Riemannian geometry. With them at hand we consider compactifications of the type II and heterotic superstring on Calabi-Yau manifolds and discuss the structure of their moduli spaces. We then introduce some of the mathematical tools which are required for an adequate treatment of Calabi-Yau compactifications. This leads to the so-called Buscher rules. We then derive a generalization of T-duality to manifolds with isometries. But we start with a brief discussion of the string equations of motion as the requirement of vanishing beta-functions of the non-linear sigma model for a string moving in a curved background. We describe this approach in detail for a class of backgrounds which preserve some amount of space-time supersymmetry in four-dimensions: compactification on Calabi-Yau manifolds. Moreover, we introduce two additional superstring theories in ten-dimensions, which are hybrid theories of a right-moving superstring and a left-moving bosonic string, whose additional sixteen dimensions are compactified on the weight-lattice of \(\mathrm\). The simplest such class are toroidal orbifolds. To break supersymmetry, however, one has to compactify on non-flat spaces. These feature a new symmetry, called T-duality. We first study the simplest examples, toroidal compactifications of the bosonic string and the type II superstring theories. One option to make contact with the four-dimensional world is to compactify the closed string theories on compact spaces. So far we discussed the 26-dimensional bosonic string and three kinds of 10-dimensional superstring theories, the type IIA/B theories and the type I theory. ![]()
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