String Geometry and the Noncommutative Torus

We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive Dirac fermion fields. Received: 26 October 1998/ Accepted: 9 April 1999     

O(d,d) symmetry in quantum cosmology

We analyze the quantum cosmology of one-loop string effective models which exhibit an O(d, d) symmetry. It is shown that due to the large symmetry of these models the Wheeler-de Witt equation can completely be solved. As a result, we find a basis of solutions with well-defined transformation properties under O(d, d) and under scale factor duality in particular. The general results are explicitly applied to 2-dimensional target spaces while some aspects of higher dimensional cases are also discussed. Moreover, a semiclassical wave function for the 2-dimensional black hole is constructed as a superposition of our basis.     

The heterotic string is a soliton

It is shown that the Type IIA superstring compactified on K3 has a smooth string soliton with the same zero mode structure as the heterotic string compactified on a four-torus, thus providing new evidence for a conjectured exact duality between the two six-dimensional string theories. The chiral worldsheet bosons arise as zero modes of Ramond-Ramond fields of the IIA string theory and live on a signature (20,4) even, self-dual lattice. Stable, finite loops of soliton string provide the charged Ramond-Ramond states necessary for enhanced gauge symmetries at degeneration points of the K3 surface. It is also shown that Type IIB strings toroidally compactified to six dimensions have a multiplet of string solutions with Type II worldsheets.     

Rigidity and Defect Actions in Landau-Ginzburg Models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x ^d , where defects are described by matrix factorisations of x ^d − y ^d . The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N = 2{\mathcal N = 2} conformal field theories. We find that the two actions differ by phases.     

Supergravity and M-theory

Supergravity provides the effective field theories for string compactifications. The deformation of the maximal supergravities by non-abelian gauge interactions is only possible for a restricted class of charges. Generically these ‘gaugings’ involve a hierarchy of p-form fields which belong to specific representations of the duality group. The group-theoretical structure of this p-form hierarchy exhibits many interesting features. In the case of maximal supergravity the class of allowed deformations has intriguing connections with M/string theory. This study is based on a talk presented at Quantum gravity: challenges and perspectives, Heraeus Seminar, Bad Honnef, 14–16 April 2008.     

Vector Bundles and F Theory

To understand in detail duality between heterotic string and F theory compactifications, it is important to understand the construction of holomorphic G bundles over elliptic Calabi-Yau manifolds, for various groups G. In this paper, we develop techniques to describe these bundles, and make several detailed comparisons between the heterotic string and F theory. Received: 6 February 1997 / Accepted: 29 May 1997     

Non critical super strings on world sheets of constant curvature

We consider correlation functions in Neveu-Schwarz string theory coupled to two dimensional gravity. The actionfor the 2D gravity consists of the string induced Liouville action and the Jackiw-Teitelboim action describing pure 2D gravity. Then gravitational dressed dimensions of vertex operators are equal to their bare conformal dimensions. There are two possible interpretations of the model. Considering the 2D dilaton and the Liouville field as additional target space coordinates one gets ad+2-dimensional critical string. In thed-dimensional non critical string picture gravitational fields retain their original meaning and ford=4 one can get a mass spectrum via consistency requirements. In both cases a GSO projection is possible.     

The T-Domain and Extreme Matter Phases Inside Spherically Symmetric Black Holes

Black hole interiors (the T-domain) are studied here in great detail. Both the general and particular T-domain solutions are presented including non-singular ones. Infinitely many local T-domain solutions may be modeled with this scheme. The duality between the T and R domains is presented. It is demonstrated how generally well behaved R-domain solutions will give rise to exotic phases of matter when collapsed inside the event horizon. However, as seen by an external observer, the field is simply that of the Schwarzschild vacuum with well behaved mass term and no evidence of this behaviour may be observed. A singularity theorem is also presented which is independent of energy conditions.     

Cosmological evolution in compactified Horava-Witten theory induced by matter on the branes

The combined Einstein equations and scalar equation of motion in the Horava-Witten scenario of the strongly coupled heterotic string compactified on a Calabi-Yau manifold are solved in the presence of additional matter densities on the branes. We take into account the universal Calabi-Yau modulus with potentials in the 5-d bulk and on the 3-branes, and allow for an arbitrary coupling of the additional matter to and an arbitrary equation of state. No ad hoc stabilization of the five dimensional radius is assumed. The matter densities are assumed to be small compared to the potential for on the branes; in this approximation we find solutions in the bulk which are exact in y and t. Depending on the coupling of the matter to and its equation of state, various solutions for the metric on the branes and in the 5-d bulk are obtained: solutions corresponding to a ”rolling radius”, and solutions with a static 5-d radius, which reproduce the standard cosmological evolution. Received: 8 April 2002 / Published online: 26 July 2002     

Letter: O(d + 1, d + n + 1)-Invariant Formulation of Stationary Heterotic String Theory

We present a pair of symmetric formulations of the matter sector of the stationary effective action of heterotic string theory that arises after the toroidal compactification of d dimensions. The first formulation is written in terms of a pair of matrix potentials Z ₁ and Z ₂ which exhibits a clear symmetry between them and can be used to generate new families of solutions on the basis of either Z ₁ or Z ₂; the second one is an O(d + 1, d + n + 1)-invariant formulation which is written in terms of a matrix vector W endowed with an O(d + 1, d + n + 1)-invariant scalar product which linearizes the action of the O(d + 1, d + n + 1)symmetry group on the coset space O(d + 1, d + n + 1)/[O(d + 1) × O(d + n + 1)]; this fact opens as well a simple solution-generating technique which can be applied on the basis of known solutions.     

Large <Emphasis Type="Italic">N</Emphasis> Expansion of <Emphasis Type="Italic">q</Emphasis>-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras

We derive the q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU(N) Yang-Mills theory. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl duality of quantum groups. The appearance of Euler characters of configuration spaces of Riemann surfaces in the expansion persists. We discuss the geometrical meaning of these formulae.     

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.     

Geometrical string and dual spin systems

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual hamiltonian represents a new type of spin system with local gauge invariance. At each vertex ξ there are d (d − 1) /2 Ising spins ∧_{μ, η} = ∧_{η,μ N}. ≠ P = 1, … , d and one Ising spin Γ on every linkξ ξ + e,). For the frozen spin Γ  1 the dual hamiltonian factorizes into d (d − 1) /2 two-dimensional Ising ferromagnets and into antiferromagnets in the case Γ  −1. For fluctuating F it is a sort of spin-glass system with local gauge invariance. The generalization to p-membranes is given.     

(2,0) Heterotic superstring with Wess-Zumino coupling

By means of the identity operator and vertex operator technique, reparametrization invariance and BRST symmetry are proven for the heterotic string with Wess-Zumino term coupling the fiber bundle. The motion space of the string is assumed to be a direct productM _d G of a Minkowski spaceM _d of dimensiond with an intrinsic group manifoldG of dimensiond _G , and turns out to give the critical dimension.     

Plane waves in effective field theories of superstrings

Exact plane wave solutions of d = 10, N = 1 Einstein-Yang-Mills supergravity theory are presented and their possible modifications in superstring effective theories are examined. It is found that the solutions are not affected by any of the known heterotic string corrections. It is argued that plane waves satisfy the effective field equations of the heterotic string theory in all powers of the string tension parameter α′, possibly after including a totally antisymmetric torsion field and reinterpreting the constants. Similar remarks also apply for type I superstrings. Further properties of the solutions are discussed.     

Stability at a local maximum in higher dimensional anti-deSitter space and applications to supergravity

The stability of anti-deSitter background solutions of gravity/scalar systems with respect to small fluetuations of the scalar fields is analyzed. As in the four dimensional case, one finds stability even about a local maximum provided that the effective scalar field masses are not “too tachyonic,” and for sufficiently low mass there are two possible boundary conditions that allow a well-defined and conserved energy. Unlike the d = 4 case only one boundary condition is applicable to the known gauged supergravity theories in d = 5 and d = 7. For d = 7 the stability criterion is satisfied while for d = 5 stability is marginal.     

Self-dual Yang-Mills fields in d=7,8, octonions and ward equations

The Yang-Mills theories in d=7 and d=8 with the arbitrary gauge group G are considered. Generalized self-duality-type relations for gauge fields are reduced to systems of nonlinear differential equations on functions of one variable (Ward equations). Ward equations may be reduced to equations which follow from Yang-Baxter equations. This permits us to obtain new classes of explicit solutions of the Yang-Mills equations in d=7 and d=8.     

Global strings in extra dimensions: The full map of solutions,matter trapping,and the hierarchy problem

We consider (d ₀ + 2)-dimensional configurations with global strings in two extra dimensions and a flat metric in d ₀ dimensions, endowed with a warp factor e ^2γ depending on the distance l from the string center. All possible regular solutions of the field equations are classified by the behavior of the warp factor and the extradimensional circular radius r(l). Solutions with r → ∞ and r → const > 0 as l → ∞ are interpreted in terms of thick brane-world models. Solutions with r → 0 as l → l _c > 0, i.e., those with a second center, are interpreted as either multibrane systems (which are appropriate for large enough distances l _c between the centers) or as Kaluza-Klein-type configurations with extra dimensions invisible due to their smallness. In the case of the Mexican-hat symmetry-breaking potential, we build the full map of regular solutions on the (ɛ, Γ) parameter plane, where ɛ acts as an effective cosmological constant and Γ characterizes the gravitational field strength. The trapping properties of candidate brane worlds for test scalar fields are discussed. Good trapping properties for massive fields are found for models with increasing warp factors. Kaluza-Klein-type models are shown to have nontrivial warp factor behaviors, leading to matter particle mass spectra that seem promising from the standpoint of hierarchy problems. The text was submitted by the authors in English.