Strings, Branes and Symmetries

Recent dramatic progress in the understanding of the non-perturbative structure of superstring theory shows that extended objects of various kinds, collectively referred to as <i>p</i>-branes, are an integral part of the theory. In this thesis, comprising an introductory text divided in two parts and seven appended research papers (Papers <i>I-VII</i>), we study various aspects of <i>p</i>-branes with relevance for superstring theory. <p /> The first part of the introductory text is a brief review of string theory focusing on the role of <i>p</i>-branes. In particular, we consider the so-called D-branes which currently are attracting a considerable amount of attention. The purpose of this part is mainly to put into context the results of Papers <i>IV</i>, <i>V</i> and <i>VI</i> concerning action functionals describing the low-energy dynamics of D-branes. More specifically, in Paper <i>IV</i> we study reformulations of the Dirac-Born-Infeld action involving an auxiliary world-volume metric. In Papers <i>V</i> and <i>VI</i> we then construct the supersymmetric and kappa-symmetric actions for D-branes propagating in curved type II supergravity backgrounds (for <i>p</i>=3 in Paper <i>V</i> and then for arbitrary <i>p</i> in Paper <i>VI</i>). The discussion of perturbative string theory given in this part of the introduction is also intended to provide some background to Paper <i>II</i> which contains an application of the Reggeon-sewing approach to the construction of string vertices. <p /> The second part covers a rather different subject, namely higher-dimensional loop algebras and their cohomology, with the aim of facilitating the reading of Papers <i>I</i>, <i>III</i> and <i>VII</i>. The relation to <i>p</i>-branes is to be found in Paper <i>I</i> where we introduce a certain higher-dimensional generalization of the loop algebra Map(<i>S</i>¹, <i>g</i>) and discuss its potential applicability as a symmetry algebra for <i>p</i>-branes. Papers <i>III</i> and <i>VII</i> are mathematically oriented outgrowths of this paper addressing the issue of realizing algebras of this kind, known in physics as current algebras, in terms of pseudodifferential operators (PSDOs). The main result of Paper <i>III</i> is a proof of the equivalence between certain Lie-algebra cocycles on the space of second-quantizable PSDOs. This result is of general validity in the sense that it is not restricted to any particular current algebra that one might wish to realize in this manner. Finally, in Paper <i>VII</i> we present a partially successful attempt at finding such a realization of the algebra introduced in Paper <i>I