Gauge Theories, Applied Supersymmetry and Quantum Gravity II

The following sections are included:

• Preface

• Network Participation

• Contents

This is a concise forward to, rather than a review of, D-brane physics.

We give an elementary introduction to the theory of supermembranes.

Some notions in non-perturbative dynamics of supersymmetric gauge theories are being reviewed. This is done by touring through a few examples.

In these lectures we review some of the recent developments in string theory on an introductory and qualitative level. In particular we focus on S – T – U dualities of toroidally compactified ten-dimensional string theories and outline the connection to M-theory. Dualities among string vacua with less supersymmetries in six and four space-time dimensions is discussed and the concept of F-theory is briefly presented.

A brief review of superstring phenomenology is presented including effective lagrangians, model independent results, model building, supersymmetry breaking and some phenomenological implications of the strong/weak coupling duality symmetries.

We analyze how string theory dualities may be described in M theory. T dualities arise from scalar-vector dualities in the worldvolume of the membrane of M theory. “Electric-magnetic” dualities arise from a duality transformation in M theory compactified on a 3-torus, which takes the membrane into a fivebrane wrapped around the 3-torus.

We show that the known interactions between strings and D-branes in type II_A and type II_B arise in appropriate limits from “natural” interactions between M theory membranes and five-branes.

In this lecture moduli dependent charges for p-extended objects are analyzed for generic N-extended supergravities in dimensions 4 ≤ D < 10. Differential relations and sum rules among the charges are derived.

We show that in a topological field theory in which the boundary of the moduli space has codimension one, only a subset of the equivariant BRS closed operators have gauge-fixing independent correlators. The theory on the four-sphere and instanton number k = 1 is studied in detail.

We consider solutions to the string effective action corresponding to p-Branes, D-Branes and M-Branes and discuss some of their properties.

We show how BPS states of supersymmetric SU(2) Yang-Mills with matter-both massless and massive-are described as self-dual strings on a Riemann surface. This connection enables us to prove the stability and the strong coupling behavior of these states. The Riemann surface naturally arises from type-IIB Calabi-Yau compactifications whose three-branes wrapped around vanishing two-cycles correspond to one-cycles on this surface.

The nonlinear vector-tensor multiplet is coupled off-shell to an N = 2 vector multiplet such that its central charge transformations are realized locally. A gauged central charge is a necessary prerequisite for a coupling to supergravity and the strategy underlying our construction uses the potential for such a coupling as a guiding principle. The results for the action and transformation rules take a nonlinear form and necessarily include a Chern-Simons term. After a duality transformation the action is encoded in a homogeneous holomorphic function consistent with special geometry.

We briefly review the covariant formulation of the Green-Schwarz superstring by Berkovits, and describe how a detailed tree-level and one-loop analysis of this model leads, for the first time, to a derivation of the low-energy effective action of the heterotic superstring while keeping target-space supersymmetry manifest. The resulting low-energy theory is old-minimal supergravity coupled to tensor multiplet. The dilaton is part of the compensator multiplet.

We describe how to derive non perturbative effective potentials and how to identify the emergence of massless excitations in N = 2 and N = 1 SUSY gauge theories.

A method is presented by which a hidden N=2 superconformal symmetry can be exhibited in a string theory or indeed in a topological conformal field theory. More precisely, we present strong evidence, based on calculations with string theories, in favour of the conjecture that any topological conformal field theory can be obtained by twisting an N=2 superconformal field theory.

We discuss zero-brane solutions in three-dimensional Minkowski space-time annihilated by half of the supersymmetries. The other half of the supersymmetries which should generate the supersymmetric multiplet are ill-defined and lead to a non bose-fermi degenerate solitonic spectrum.

We gauge the non-Abelian isometries of a sigma model with boundaries. Forcing the field strength of the gauge fields to vanish renders the gauged model equivalent to the ungauged one provided that boundary conditions are taken into account properly. Integrating out the gauge fields gives the T-dual model. We observe that T-duality interchanges Neumann (or mixed) boundary conditions with Dirichlet boundary conditions.

Generic partial supersymmetry breaking of N = 2 supergravity with zero vacuum energy and with surviving unbroken arbitrary gauge groups is exhibited. Specific examples are given.

Boundary states for D-branes are constructed using the light cone gauge. The D-brane breaks half the spacetime supersymmetry giving rise to fermionic zero modes living on the brane. The nonlinear realization of the broken supersymmetry on the open string degrees of freedom is analysed and the influence of boundary terms coming from closed string vertex operators is discussed.

The superconformal properties of N = 4 Yang-Mills theory are most naturally studied using the formalism of harmonic superspace. Superconformal invariance is shown to imply that the Green's functions of analytic operators are invariant holomorphic sections of a line bundle on a product of certain harmonic superspaces and it is argued that the theory is soluble for a class of such operators.

We present the exact solution of a matrix model which interpolates between fiat and random lattices. The importance of the results is twofold: Firstly, we have developed a new large N technique capable of treating a class of matrix models previously thought to be unsolvable. Secondly, we are able to make a first precise statement about two-dimensional R² gravity.

The quantum consistency of sigma-models describing the dynamics of extended objects in a curved background requires the cancellation of their world-volume anomalies, which are conformal anomalies for the heterotic string and SO(1,5) Lorentz-anomalies for the heterotic five-brane, and of their ten dimensional target space anomalies. In determining these anomalies in a D = 10 Lorentz-covariant back-ground gauge we find that for the heterotic string the worldvolume anomalies cancel for 32 heterotic fermions while for the conjectured heterotic five-brane they cancel for only 16 heterotic fermions, this result being in contrast with the string/five-brane duality conjecture. For what concerns the target space anomalies we find that the five-brane eight-form Lorentz-anomaly polynomial differs by a factor of 1/2 from what is expected on the basis of duality. Possible implications of these results are discussed.

We report on recent progress in the use of string techniques for the computation of field theory amplitudes. We show how one-loop renormalization constants in Yang-Mills theory can be computed using the open spinning string, we review the calculation of two-loop scalar amplitudes with the bosonic string, and we briefly indicate how the technique can be applied to the two-loop vacuum bubbles of Yang-Mills theory.

In this talk we consider the vector and the higher derivative gravitational couplings in four-dimensional N = 2 strings which have both a heterotic and a dual type II description. These effective couplings serve to provide a successful test for the N = 2 string-string duality.

We review the evidence for the various dualities amongst the five D=10 superstring theories and for the existence of M-theory using the associated effective supergravity theories. We also summarize the combinatorial techniques developed for constructing BPS solutions in D = 11 supergravity theory and conjecture that all the BPS solutions of D < 11 supergravity theories can be derived from the BPS solutions of D = 11 supergravity that preserve 1/2 the supersymmetry. To demonstrate this, we derive the dyonic p-brane solutions from eleven dimensions.

We discuss the role of the CPT transformation in first quantized string theories, both on the world-sheet and in the space-time. We explicitly show that the space-time CPT theorem holds for all first-quantized (perturbative) string theories in a Minkowski background of even dimension D > 2.

We review recent work on the construction of Liouville observables and their three point functions in the operator approach. We demonstrate that there exists a Coulomb gas formulation of the theory without the need for analytic continuation procedures.

We review non-linear σ-models with (2,1) and (2,2) supersymmetry. We focus on off-shell closure of the supersymmetry algebra and give a complete list of (2,2) super-fields. We provide evidence to support the conjecture that all N = (2,2) non-linear σ-models can be described by these fields. This in its turn leads to interesting consequences about the geometry of the target manifolds. One immediate corollary of this conjecture is the existence of a potential for hyper-Kähler manifolds, different from the Kähler potential, which does not only allow for the computation of the metric, but of the three fundamental two-forms as well. Several examples are provided: WZW models on SU(2) × U(1) and SU(2) × SU(2) and four-dimensional special hyper-Kähler manifolds.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim σ-models corresponding to coset conformal field theories.

We find the complete solution to ten-dimensional supergravity coupled to a three-form field strength, given the “standard ansatz” for the fields, and show that in addition to the well-known elementary and solitonic (heterotic) string solutions, one of the possibilities is an (unstable) elementary type I string solution.

We discuss special extreme black holes in 4 and 5 dimensions which can be interpreted as compactifications of BPS configurations of intersecting p-branes in 10 or 11 dimensions.

We obtain a continuous Wick rotation for Dirac, Majorana and Weyl spinors ψ → exp (½ θ γ⁴ γ⁵)Ψ which interpolates between Minkowski and Euclidean field theories.

The essential elements in the construction of the couplings of vector multiplets to supergravity using the conformal approach are repeated. This approach leads automatically to the basic quantities on which the symplectic transformations, the basic tools for duality transformations, are defined. A recent theorem about the existence of a basis allowing for a prepotential is discussed.

The following sections are included:

• List of participants