On adelic formulas for the p-adic string

For the p-adic bosonic string, a simple regularization procedure is applied to the four-particle adelic formula. Arguments that have been made against the validity of the five-particle adelic formula are criticised.     

Spinor-vector duality in heterotic string vacua

Classification of the N=1 space–time supersymmetric fermionic Z₂×Z₂ heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t↔V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z₂ heterotic string models, which preserve N=2 space–time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S_t↔V duality map. We present a novel basis to generate the free fermionic models in which the ten-dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.     

Gravitational waves as string vacua I

Recent results on the role of gravitational waves in string theory are reviwed. Motivations for the research and an organic view of the topic are furnished. Short exposition of the elements of string theory in the Polyakov formalism is provided, with particular emphasis on the evaluation of the Weyl and super-Weyl anomalies. Relevant results on the plane wave solutions of the Einstein equations are listed and a way of solving a quantum field theory in such backgrounds is presented. The results obtained for theS-matrix of the field theory and (singular) expectation values of its energy momentum tensor will be linked to string physics in subsequent parts of this review.This review article is based on the PhD thesis of the author defended in 1989 at SISSA, Trieste. It is divided into three parts with independent abstracts and tables of contents.     

Gravitational waves as string vacua III

We describe perturbative construction of in and out scalar vertex operators in nonlinear-models corresponding to gravitational plane wave backgrounds. The calculation is performed to all loops in the weak field limit and up to three loops in the usual perturbation expansion. The closed expressions for the vertex operators are given. They exhibit peculiar singularities in the target space which are closely related to the focusing phenomena in such backgrounds described in the previous parts of this review. Then the appendix is provided in which finite planar shell shock waves are studied. The review ends up with conclusions and a short list of related results, published since it was written.This is the third part of the review based on the PhD thesis of the author defended in 1989 at SISSA, Trieste.     

Gravitational waves as string vacua II

Classical propagation of (super)strings through gravitational shock waves is analyzed. The exact classical solutions are used for quantization and for the identification of the exact quantumS-matrix describing string scattering by the wave. ThisS-matrix coincides with theS-matrix of the string-string scattering in theflat space-time for particular profile of the shock wave! This is interpreted as the generation of curved geometry from the flat space-time string theory. The quantum consistence of (super)string motion in gravitational plane wave backgrounds is then studied. It turns out that for the standard dimensionsD=26 (D=10) the vanishing of the Ricci tensor for the plane wave is sufficient condition for vanishing of the Weyl (superWeyl) anomaly. Thus, plane wave solutions of the Einstein equations are automatically the classical (super)string vacua. For particular plane waves the anomaly can be evaluated even nonperturbatively.This is the second part of the review based on the PhD thesis of the author defended in 1989 at SISSA, Trieste.     

Mirror symmetry and the web of Landau-Ginzburg string vacua

We present some mathematical aspects of Landau-Ginzburg string vacua in terms of toric geometry. The one-to-one correspondence between toric divisors and some of the (−1, 1) states in the Landau-Ginzburg model is presented for superpotentials of typical types. The Landau-Ginzburg interpretation of non-toric divisors is also presented. Using this interpretation, we propose a method to solve the so-called “twisted sector problem” by orbifold construction. Moreover, this construction shows that the moduli spaces of the original Landau-Ginzburg string vacua and their orbifolds are connected. By considering the mirror map of the Landau-Ginzburg models, we obtain the relation between Mori vectors and the twist operators of our orbifoldization. This consideration enables us to argue the embedding of the Seiberg-Witten curve in the defining equation of the Calabi-Yau manifolds on which the type II string gets compactified. Related topics concerning the Calabi-Yau fourfolds and the extremal transition are discussed.     

Quantization of degenerate vacua and symmetry breaking in string theory

We consider the implications of the existence of continuous families of classically degenerate vacua in constructing quantum string theory in compact toroidal spacetimes. We show that in the presence of unbroken supersymmetry when quantum corrections do not lift the degeneracy, symmetries of particular compactifications can be broken by the presence of the neighbouring vacua. We explicitly demonstrate this in case of the heterotic string where the gauge group SO(32) or E₈×E₈ breaks down to U(1)¹⁶ due to neighbouring configurations of arbitrary Wilson lines.     

The string theory landscape: a tale of two hydras

String theory is the leading theoretical approach to finding a consistent quantum theory that unifies gravity with the other three forces. However, the theory is only consistent in ten dimensions. In order for string theory to describe observable physics, the six extra dimensions must be sufficiently small to be unobservable. Originally it was hoped that there would only be one consistent way of doing this, but now it is realized that this problem may have as many as 10⁵⁰⁰ possible solutions. I describe where this number comes from and the approaches, both statistical and anthropic, taken in dealing with this superfluity of solutions.     

What we learn on the heterotic string vacua from anomaly-free supergravity

The recently constructed D = 10 anomaly-free supergravity (AFS) has been argued to contain the full effective theory of the heterotic string. The solutions of the effective theory must be solutions of AFS, while the converse is not necessarily true since string theory might specify the boundary conditions for the AFS torsion equation. We show that Calabi-Yau spaces are exact solutions of AFS, while compact group and coset manifolds are not. This is due to a positivity argument, which is the extension to anomaly-free supergravity of the “ten into four won't go” theorem of Freedman, Gibbons and West for the Chapline-Manton theory.     

Stability of string vacua: (I). A new picture of the renormalization group

Ordinary second-quantized field theories are rearranged to look like first-quantized strings expanded around an unstable classical vacuum. The β function of the field theory becomes the tadpole amplitude of the string, the anomalous dimensions become the string mass matrix. This map should be useful in both directions: it gives a new picture of renormalization, and it shows how to do spontaneous symmetry breaking for strings. Dynamical symmetry breaking, where string loop divergences cancel against those of the 2D field theory, is also conceivable.     

General N=1 supersymmetric flux vacua of massive type IIA string theory

We derive conditions for the existence of four-dimensional N=1 supersymmetric flux vacua of massive type IIA string theory with general supergravity fluxes turned on. For an SU(3) singlet Killing spinor, we show that such flux vacua exist when the internal geometry is nearly K?hler. The geometry is not warped, all the allowed fluxes are proportional to the mass parameter, and the dilaton is fixed by a ratio of (quantized) fluxes. The four-dimensional cosmological constant, while negative, becomes small in the vacuum with the weak string coupling.     

Adèlic formulas for string amplitudes in fields of algebraic numbers

On the basis of the analysis on adèle groups (Tate's formula) for any field of algebraic numbers, a regularization of infinite adèlic products of gamma and beta functions of local fields is proposed. The formulas obtained are applied to representations of the four-point crossing symmetric Veneziano and Virasoro-Shapiro amplitudes through regularized adèlic products of the corresponding string (open and closed, resp.) amplitudes.     

Charting the landscape of supercritical string theory

Special solutions of string theory in supercritical dimensions can interpolate in time between theories with different numbers of spacetime dimensions and different amounts of world sheet supersymmetry. These solutions connect supercritical string theories to the more familiar string duality web in ten dimensions and provide a precise link between supersymmetric and purely bosonic string theories. Dimension quenching and c duality appear to be natural concepts in string theory, giving rise to large networks of interconnected theories.