Table of contents

We study a Wess-Zumino-Witten model with target spaceAdS₃ × (S³ × S³ × S¹)/₂. This allows us to construct space-time = 3 superconformal theories. By combining left-, and right-moving parts through a GSO and a ₂ projections, a new asymmetric (,bar−) = (3,1) model is obtained. It has an extra gauge (affine) SU(2) symmetry in the target space of the type IIA string. An associated configuration is realized as slantwise intersecting M5-M2 branes with a ₂-fixed plane in the M-theory viewpoint.

Recently, it was conjectured that parallel branes at conical singularities are related to string/M theory on AdS × X where X is an Einstein manifold. In this paper we consider coincident M2 branes near a conifold singularity when M theory is compactified on AdS₄ × Q^1,1,1, where Q^1,1,1 = (SU(2) × SU(2) × SU(2))/(U(1) × U(1)) is a seven dimensional Sasaki-Einstein manifold. We argue that M theory on AdS₄ × Q^1,1,1 is dual to a three dimensional superconformal field theory. We use the fact that the three dimensional self-mirror duality is preserved by exact marginal operators, as observed by Strassler.

Index theorems for the Dirac operator allow one to study spinors on manifolds with boundary and torsion. We analyse the modifications of the boundary Chern-Simons correction and APS η invariant in the presence of torsion. The bulk contribution must also be modified and is computed using a supersymmetric quantum mechanics representation. Here we find agreement with existing results which employed heat kernel and Pauli-Villars techniques. Nonetheless, this computation also provides a stringent check of the Feynman rules of de Boer et al. for the computation of quantum mechanical path integrals. Our results can be verified via a duality relation between manifolds admitting a Killing-Yano tensor and manifolds with torsion. As an explicit example, we compute the indices of Taub-NUT and its dual constructed using this method and find agreement for any finite radius to the boundary. We also suggest a resolution to the problematic appearance of the Nieh-Yan invariant multiplied by the regulator (mass)² in computations of the chiral gravitational anomaly coupled to torsion.

We use toric geometry to investigate the recently proposed relation between a set of D3 branes at a generalized conifold singularity and type IIA configurations of D4 branes stretched between a number of relatively rotated NS5 branes. In particular we investigate how various resolutions of the singularity corresponds to moving the NS branes and how Seiberg's duality is realized when two relatively rotated NS-branes are interchanged.

Wilson's approach to renormalization group is reanalyzed for supersymmetric Yang-Mills theory. The usual demonstration of exact renormalization group equation must be modified due to the presence of the so called Konishi anomaly under the rescaling of superfields. We carry out the explicit computation for N = 1 SUSY Yang-Mills theory with the simpler, gauge invariant regularization method, recently proposed by Arkani-Hamed and Murayama. The result is that the Wilsonian action S_M consists of two terms, i.e. the non anomalous term, which obeys Polchinski' s flow equation and Fujikawa-Konishi determinant contribution. This latter is responsible for Shifman-Vainshtein relation giving the exact β-function.

We evaluate the couplings of pions in the transitions of positive parity heavy mesons, (1⁺,2⁺) and (0⁺,1⁺), to negative parity ones (0⁻,1⁻) using a technique which is not limited to the soft-pion limit. This is made through a constituent quark-meson model (the CQM model) where the amplitudes are obtained by quark loops with mesons on the external lines. The results are in good agreement with experimental data.

We construct the effective chiral lagrangean for chiral perturbation theory in the mesonic even-intrinsic-parity sector at order p⁶. The lagrangean contains 112 in principle measurable + 3 contact terms for the general case of n light flavours, 90+4 for three and 53+4 for two flavours. The equivalence between equations of motion and field redefinitions to remove spurious terms in the lagrangeans is shown to all orders in the chiral expansion. We also discuss and implement other methods for reducing the number of terms to a minimal set.

We obtain = 1 SU(N)^k gauge theories with bifundamental matter and a quartic superpotential as the low energy theory on D3-branes at singular points. These theories generalize that on D3-branes at a conifold point, studied recently by Klebanov and Witten. For k = 3 the defining equation of the singular point is that of an isolated D₄ singularity. For k > 3 we obtain a family of multimodular singularities. The considered SU(N)^k theories flow in the infrared to a non-trivial fixed point. We analyze the AdS/CFT correspondence for our examples.

Sen has shown that tachyon condensation in Dbrane--anti-Dbrane configurations can lead to remarkable connections between string theories. A consequence of his results is that there is a minimal value of the radial coordinate r_c∝(α')^1/2(gN)^1/(9−p) such that N units of Dpbrane charge cannot be localized to values smaller than r_c. At this value ofr_c the curvature is of order 1/α' and the gradient of the Ramond-Ramond field strength is of order 1/(α')^3/2, and the vacuum, regarded as a D-brane-anti-D-brane configuration with a tachyon condensate, is rendered unstable, leading to a separation of the D-brane and the anti-D-brane. This value of r_c lies in the region intermediate between the near-horizon regime and the asymptotic regime for D-brane classical solutions for small g. This vacuum stability bound on the curvature can be interpreted as an uncertainty relation for D-brane charge.

We present a geometric formulation of (p,q)-strings in which the SL(2,)-doublet of the two-form gauge potentials is constructed as second order in the supersymmetric currents. The currents are constructed using a supergroup manifold corresponding to the (p,q)-string superalgebra, which contains fermionic generators in addition to the supercharges and transforms under the SL(2,). The properties of the superalgebra and the generalizations to higher p-branes are discussed.

Noncommutative torus compactification of Matrix model is shown to be a direct consequence of quantization of the open strings attached to a D-membrane with a non-vanishing background B field. We calculate the BPS spectrum of such a brane system using both string theory results and DBI action. The DBI action leads to a new transformation property of the compactification radii under the SL(2,)_N transformations.

This is the written version of a set of introductory lectures to string theory. The lectures were given at the Universidad Autónoma de Madrid in the semester 1997/98 and at the VI Escuela de Otoño de Física Teórica, held in Santiago de Compostela (10-23 september 1998).

A general parametrization of the amplitudes for the rare two-body decays B→πK is introduced, which makes maximal use of theoretical constraints arising from flavour symmetries of the strong interactions and the structure of the low-energy effective weak Hamiltonian. With the help of this parametrization, a model-independent analysis of the branching ratios and direct CP asymmetries in the various B→πK decay modes is performed, and the impact of hadronic uncertainties on bounds on the weak phase γ = arg(V_ub*) is investigated.

We study orbifolds of = 4 U(n) super-Yang-Mills theory given by discrete subgroups of SU(2) and SU(3). We have reached many interesting observations that have graph-theoretic interpretations. For the subgroups of SU(2), we have shown how the matter content agrees with current quiver theories and have offered a possible explanation. In the case of SU(3) we have constructed a catalogue of candidates for finite (chiral) = 1 theories, giving the gauge group and matter content. Finally, we conjecture a McKay-type correspondence for Gorenstein singularities in dimension 3 with modular invariants of WZW conformal models. This implies a connection between a class of finite = 1 supersymmetric gauge theories in four dimensions and the classification of affine SU(3) modular invariant partition functions in two dimensions.

We provide evidence for the validity of AdS/CFT correspondence in the Coulomb branch by comparing the Yang-Mills effective action with the potential between waves on two separated test 3-branes in the presence of a large number of other 3-branes. For constant gauge fields excited on the branes, this requires that the supergravity potential in a AdS₅ × S⁵ background is the same as that in flat space, despite the fact that both propagators and couplings of some relevant supergravity modes are different. We show that this is indeed true, due to a subtle cancellation. With time-dependent gauge fields on the test branes, the potential is sensitive to retardation effects of causal propagation in the bulk. We argue that this is reflected in higher derivative (acceleration) terms in the Yang-Mills effective action. We show that for two 3-branes separated in flat space the structure of lowest order acceleration terms is in agreement with supergravity expectations.

Low-energy, near-horizon scaling limits of black holes which lead to string theory on AdS₂ × S² are described. Unlike the higher-dimensional cases, in the simplest approach all finite-energy excitations of AdS₂ × S² are suppressed. Surviving zero-energy configurations are described. These can include tree-like structures in which the AdS₂ × S² throat branches as the horizon is approached, as well as disconnected AdS₂ × S² universes. In principle, the black hole entropy counts the quantum ground states on the moduli space of such configurations. In a nonsupersymmetric context AdS_D for general D can be unstable against instanton-mediated fragmentation into disconnected universes. Several examples are given.

We find the expectation value of the energy-momentum tensor in the CFT corresponding to a moving black hole in AdS. Boosting the black hole to the speed of light, keeping the total energy fixed, yields a gravitational shock wave in AdS. The analogous procedure on the field theory side leads to ``light cone'' states, i.e., states with energy-momentum tensor localized on the light cone. The correspondence between the gravitational shock wave and these light cone states provides a useful tool for testing causality. We show, in several examples, how the CFT reproduces the causal relations in AdS.

We present nontrivial examples of d = 3 gauge theories with sixteen and eight supercharges which are infrared dual at special points in the moduli space. This duality is distinct from mirror symmetry. To demonstrate duality we construct the gauge theories of interest using D2-branes and orientifolds and then consider their lift to M-theory. We also discuss the strong coupling limit of orientifold two-planes and orbifolds of orientifold six-planes.

We study D-branes on a three complex dimensional nonabelian orbifold ³/Γ with Γ a finite subgroup of SU(3). We present general formulae necessary to obtain quiver diagrams which represent the gauge group and the spectrum of the D-brane worldvolume theory for dihedral-like subgroups Δ(3n²) and Δ(6n²). It is found that the quiver diagrams have a similar structure to webs of branes.

We describe the type IIB supergravity background on AdS₃⊗S₃⊗T⁴ using the potentials of AdS_3|4⊗S₃⊗T⁴ and we use the supersolvable algebra associated to AdS_3|4 to compute the κ gauge fixed type IIB string action. From the explicit form of the action we can clearly see how passing from pure NSNS backgrounds to pure RR backgrounds the WZW term disappears.

We discuss string theories with small numbers of non-compact moduli and describe constructions of string theories whose low-energy limit is described by various pure supergravity theories. We also construct a D = 4,N = 4 compactification of type II string theory with 34 vector fields.

We discuss the concept of connected, reparameterization invariant matter correlators in quantum gravity. We analyze the effect of discretization in two solvable cases: branched polymers and two-dimensional simplicial gravity. In both cases the naively defined connected correlators for a fixed volume display an anomalous behavior, which could be interpreted as a long-range order. We suggest that this is in fact only a highly non-trivial finite-size effect and propose an improved definition of the connected correlator, which reduces the effect. Using this definition we illustrate the appearance of a long-range spin order in the Ising model on a two-dimensional random lattice in an external magnetic field H, whenH→0 and β = β_C.

We calculate, using noncommutative supersymmetric Yang-Mills gauge theory, the part of the spectrum of the toroidally compactified Matrix theory which corresponds to quantized electric fluxes on two and three tori.

We consider brane solutions where the tensor degrees of freedom are excited. Explicit solutions to the full non-linear supergravity equations of motion are given for the M5 and D3 branes, corresponding to finite selfdual tensor or Born-Infeld field strengths. The solutions are BPS-saturated and half-supersymmetric. The resulting metric space-times are analysed.

Predictions for the diffractive photoproduction of the Υ-family at HERA energies, within the framework of the analysis by Frankfurt, Koepf and Strikman, are presented. Two novel effects lead to a significant enhancement of the original calculation: the non-diagonal (or skewed) kinematics, calculated to leading-log(Q²) accuracy, and the large magnitude of the real part of the amplitude. The resultant cross sections are found to agree fairly well with recent preliminary data from ZEUS and H1. A strong correlation between the mass of the diffractively produced state and the energy dependence of the cross section is found. In particular, a considerably stronger rise in energy is predicted than that found in J/ψ-production.

In the context of brane solutions of supergravity, we discuss a general method to introduce collective modes of any spin by exploiting a particular way of breaking symmetries. The method is applied to the D3, M2 and M5 branes and we derive explicit expressions for how the zero-modes enter the target space fields, verify normalisability in the transverse directions and derive the corresponding field equations on the brane. In particular, the method provides a clear understanding of scalar, spinor, and rank r tensorial Goldstone modes, chiral as well as non-chiral, and how they arise from the gravity, Rarita-Schwinger, and rank r+1 Kalb-Ramond tensor gauge fields, respectively. Some additional observations concerning the chiral tensor modes on the M5 brane are discussed.