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In 1986 Polyakov published his theory of rigid string. I investigate the instantons associated with the consequent new fine structure of strings in four dimensional Euclidean space-time. I reduce the self-dual equation of rigid string instantons to a simple form and show that (p,q) torus knots satisfy the equation, thus forming an interesting new class of solutions. I calculate by computer the world-sheet self-intersection number of the first few such closed knotted strings and derive a very simple formula for the self-intersection number of a torus knot. I consider an interpretation in terms of the first Chem number and discover the empirical formula Q = q - p for the inslanton number, Q, of torus knots and links. In 1987 Biran, Floratos and Savvidy pioneered an approach for constructing self-dual equations for membranes. I present some new solutions for self-dual membranes in three dimensions. In 1989 Grabowski and Tze pointed out a new class of exceptional immersions for which self-dual equations can be constructed and for which there are no known non-trivial solutions. By analogy with (p,q) torus knots, I describe an algorithm for generating a class of potential solutions of self-dual lumps in eight dimensions. I show how these come to within a single sign change of solving all the required constraints and come very close to solving all the 32 self-dual (4;8)-brane equations.

This thesis explores the geometry of string theory backgrounds and the nongeometric features of string theory that arise due to T-duality. For this reason, it is divided into two complementary parts. Part I deals with the superalgebras of symmetries of string theory and M-theory backgrounds, the so-called Killing superalgebras. It is shown that one can define a Lie superalgebra consisting of the infinitesimal field-preserving isometries and the supersymmetries of the background. We also explore the extension of a Killing superalgebra with brane charges. Part II deals with non-geometric backgrounds. In particular, we adopt the framework of the doubled geometry, also known as the doubled torus. We analyze the hamiltonian dynamics of the system and quantize a model T-fold. Finally we extended the doubled torus system to include worldsheet supersymmetry. Throughout part II, we focus on the equivalence, classical and quantum, of the doubled formalism with the conventional formulation.

This thesis is concerned with mathematical interpretations of some recent develop- ments in string theory. All theories are considered before quantisation. The rst half of the thesis investigates a large class of Lagrangians, L, that arise in the physics literature. Noether's famous theorem says that under certain conditions there is a bijective correspondence between the symmetries of L and the \conserved currents" or integrals of motion. The space of integrals of motion form a sheaf and has a bilinear bracket operation. We show that there is a canonical sheaf d1;0 J1( ) that contains a representation of the higher Dorfman bracket. This is the rst step to de ne a Courant algebroid structure on this sheaf. We discuss the existence of this structure proving that, for a re ned de nition, we have the necessary components. The pure spinor formalism of string theory involves the addition of the algebra of pure spinors to the data of the superstring. This algebra is a Koszul algebra and, for physicists, Koszul duality is string/gauge duality. Motivated by this, we investigate the intimate relationship between a commutative Koszul algebra A and its graded Lie superalgebra Koszul dual to A, U(g) = A!. Classically, this means we obtain the algebra of syzygies AS from the cohomology of a Lie subalgebra of g. We prove H (g 2;C) ' AS again and extend it to the notion of k-syzygies, which we de ne as H (g k;C). In particular, we show that H B er(A) ' H (g 3;C), where H Ber(A) is the Berkovits cohomology of A.