Abstract:
We consider Dirichlet p-branes in type II string theory on a space which has been toroidally compactified in d dimensions. We give an explicit construction of the field theory description of this system by putting a countably infinite number of copies of each brane on the noncompact covering space, and modding out the resulting gauge theory by Z^d. The resulting theory is a gauge theory with graded fields corresponding to strings winding around the torus an arbitrary number of times. In accordance with T-duality, this theory is equivalent to the gauge theory for the dual system of (d + p)-branes wrapped around the compact directions, where the winding number is exchanged for momentum in the compact direction.

Abstract:
These lectures provide an introduction to the subject of tachyon condensation in the open bosonic string. The problem of tachyon condensation is first described in the context of the low-energy Yang-Mills description of a system of multiple D-branes, and then using the language of string field theory. An introduction is given to Witten's cubic open bosonic string field theory. The Sen conjectures on tachyon condensation in open bosonic string field theory are introduced, and evidence confirming these conjectures is reviewed.

Abstract:
A general algorithm is presented which gives a closed-form expression for an arbitrary perturbative diagram of cubic string field theory at any loop order. For any diagram, the resulting expression is given by an integral of a function of several infinite matrices, each built from a finite number of blocks containing the Neumann coefficients of Witten's 3-string vertex. The closed-form expression for any diagram can be approximated by level truncation on oscillator level, giving a computation involving finite size matrices. Some simple tree and loop diagrams are worked out as examples of this approach.

Abstract:
Tachyon condensation in the open bosonic string is analyzed using a perturbative expansion of the tachyon potential around the unstable D25-brane vacuum. Using the leading terms in the tachyon potential, Pad\'e approximants can apparently give the energy of the stable vacuum to arbitrarily good accuracy. Level-truncation approximations up to level 10 for the coefficients in the tachyon potential are extrapolated to higher levels and used to find approximants for the full potential. At level 14 and above, the resulting approximants give an energy less than -1 in units of the D25-brane tension, in agreement with recent level-truncation results by Gaiotto and Rastelli. The extrapolated energy continues to decrease below -1 until reaching a minimum near level 26, after which the energy turns around and begins to approach -1 from below. Within the accuracy of this method, these results are completely consistent with an energy which approaches -1 as the level of truncation is taken to be arbitrarily large.

Abstract:
Open string field theory is considered as a tool for deriving the effective action for the massless or tachyonic fields living on D-branes. Some simple calculations are performed in open bosonic string field theory which validate this approach. The level truncation method is used to calculate successive approximations to the quartic terms \phi^4, (A^\mu A_\mu)^2 and [A_\mu, A_\nu]^2 for the zero momentum tachyon and gauge field on one or many bosonic D-branes. We find that the level truncation method converges for these terms within 2-4% when all massive fields up to level 20 are integrated out, although the convergence is slower than exponential. We discuss the possibility of extending this work to determine the structure of the nonabelian Born-Infeld theory describing the gauge field on a system of many parallel bosonic or supersymmetric D-branes. We also describe a brane configuration in which tachyon condensation arises in both the gauge theory and string field theory pictures. This provides a natural connection between recent work of Sen and Zwiebach on tachyon condensation in string field theory and unstable vacua in super Yang-Mills and Born-Infeld field theory.

Abstract:
This note focuses on the coupling of a type IIA D2-brane to a background B field. It is shown that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by bulk contributions, for a compact D2-brane wrapping a homotopically trivial cycle in space-time. In M-theory this cancellation is a straightforward consequence of momentum conservation. This result resolves a puzzle recently posed by Bachas, Douglas and Schweigert related to the quantization of R-R charges on stable spherical D2-branes on the group manifold SU(2).

Abstract:
These lecture notes give a pedagogical and (mostly) self-contained review of some basic aspects of the Matrix model of M-theory. The derivations of the model as a regularized supermembrane theory and as the discrete light-cone quantization of M-theory are presented. The construction of M-theory objects from matrices is described, and gravitational interactions between these objects are derived using Yang-Mills perturbation theory. Generalizations of the model to compact and curved space-times are discussed, and the current status of the theory is reviewed.

Abstract:
A Yang-Mills solution is constructed on T^6 which corresponds to a brane configuration composed purely of 0-branes and 6-branes. This configuration breaks all supersymmetries and has an energy greater than the sum of the energies of its components; nonetheless, the configuration is stable classically, at least to quadratic order. An analogous construction is also given for a system of 0-branes and 8-branes on T^8. These constructions may prove to be useful for describing 6-branes and 8-branes in M(atrix) theory.

Abstract:
The level truncation approach to string field theory is used to study the zero-momentum action for vector excitations on a bosonic D-brane which has been annihilated by tachyon condensation. It is shown that in the true vacuum the translation zero modes associated with transverse scalars on the D-brane are lifted by spontaneous generation of mass terms. Similarly, the U(1) gauge field on the brane develops a nonzero mass term.

Abstract:
A self-contained review is given of the matrix model of M-theory. The introductory part of the review is intended to be accessible to the general reader. M-theory is an eleven-dimensional quantum theory of gravity which is believed to underlie all superstring theories. This is the only candidate at present for a theory of fundamental physics which reconciles gravity and quantum field theory in a potentially realistic fashion. Evidence for the existence of M-theory is still only circumstantial---no complete background-independent formulation of the theory yet exists. Matrix theory was first developed as a regularized theory of a supersymmetric quantum membrane. More recently, the theory appeared in a different guise as the discrete light-cone quantization of M-theory in flat space. These two approaches to matrix theory are described in detail and compared. It is shown that matrix theory is a well-defined quantum theory which reduces to a supersymmetric theory of gravity at low energies. Although the fundamental degrees of freedom of matrix theory are essentially pointlike, it is shown that higher-dimensional fluctuating objects (branes) arise through the nonabelian structure of the matrix degrees of freedom. The problem of formulating matrix theory in a general space-time background is discussed, and the connections between matrix theory and other related models are reviewed.