International Journal of Modern Physics A

I describe our understanding of physics near the Planck length, in particular the great progress of the last four years in string theory. The lectures were presented at the 1998 SLAC Summer Institute.

The multiplets that occur in four-dimensional rigidly supersymmetric theories can be described either by chiral superfields in Minkowski superspace or analytic superfields in harmonic superspace. The superconformal Ward identities for Green's functions of gauge-invariant operators of these types are derived. It is shown that there are no chiral superconformal invariants. It is further shown that Green's functions of analytic operators in harmonic superspace are severely restricted by the superconformal Ward identities when internal analyticity is taken into account.

We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.

In most of the models of hadronic collisions, the number of exchanged color strings grows with energy and atomic numbers of the projectile and target. At high string densities interaction between them becomes important, which should melt them into the quark–gluon plasma state in the end. It is shown that under certain reasonable assumptions about the string interaction, a phase transition to the quark–gluon plasma indeed takes place in the system of many color strings. It may be of the first or second order, depending on the particular mechanism of the interaction. The critical string density is about unity in both cases. In the latter case the percolation of strings occurs above the critical density. The critical density may have already been reached in central Pb–Pb collisions at 158A GeV.

We present some approaches to the perturbative analysis of the classical and quantum gravity. First we introduce a graphical representation for a global SO(n) tensor (∂)^dh_αβ, which generally appears in the weak field expansion around the flat space: g_μν=δ_μν+h_μν. Making use of this representation, we explain (1) Generating function of graphs (Feynman diagram approach), (2) Adjacency matrix (Matrix approach), (3) Graphical classification in terms of "topology indices" (Topology approach), (4) The Young tableau (Symmetric group approach). We systematically construct the global SO(n) invariants. How to show the independence and completeness of those invariants is the main theme. We explain it taking simple examples of ∂∂h-, and (∂∂h)²-invariants in the text. The results are applied to the analysis of the independence of general invariants and (the leading order of) the Weyl anomalies of scalar-gravity theories in "diverse" dimensions (2,4,6,8,10 dimensions).

We examine a nonlocal interaction that results from expressing the QCD Hamiltonian entirely in terms of gauge-invariant quark and gluon fields. The interaction couples one quark color-charge density to another, much as electric charge densities are coupled to each other by the Coulomb interaction in QED. In QCD, this nonlocal interaction also couples quark color-charge densities to gluonic color. We show how the leading part of the interaction between quark color-charge densities vanishes when the participating quarks are in a color singlet configuration, and that, for singlet configurations, the residual interaction weakens as the size of a packet of quarks shrinks. Because of this effect, color-singlet packets of quarks should experience final state interactions that increase in strength as these packets expand in size. For the case of an SU(2) model of QCD based on the ansatz that the gauge-invariant gauge field is a hedgehog configuration, we show how the infinite series that represents the nonlocal interaction between quark color-charge densities can be evaluated nonperturbatively, without expanding it term-by-term. We discuss the implications of this model for QCD with SU(3) color and a gauge-invariant gauge field determined by QCD dynamics.

The manifestly Poincaré-invariant formulations for SO(1,10) and SO(2,9) superstring actions are proposed. The actions are invariant under a local fermionic κ-symmetry as well as under a number of global symmetries, which turn out to be on-shell realization of the known "new supersymmetry" S-algebra. Canonical quantization of the theory is performed and the relation of the quantum state spectrum with that of type IIA Green–Schwarz superstring is discussed. Besides, a mechanical model is constructed, which is a zero tension limit of the D=11 superstring and which incorporates all essential features of the latter. A corresponding action invariant under off-shell closed realization of the S-algebra is obtained.