Narrow Results

On the way towards quantum gravity and the unification of interaction, several ideas have been rejected and avenues avoided because they were perceived as physically unviable. But in the literature there are works in which it was found the contrary, namely that those rejected topics make sense after all. Such topics, reviewed in this paper, are negative energies occurring in higher derivative theories and ultrahyperbolic spaces, ordering ambiguity of operators in curved spaces, the vast landscape of possible compactifications of extra dimensions in string theory, and quantization of a 3-brane in braneworld scenarios.

Motivated by string theory results, we study Liouville black hole solutions and their thermodynamics on noncommutative space. In particular, we present explicit solutions of black hole equations of motion, then we find their classical properties such as the ADM-mass, the horizon geometry and the scalar Ricci curvature. Thermodynamic properties of such noncommutative black hole solutions including the Hawking temperature and entropy function are also discussed for three different regions of the moduli space of the theory.

We use Berkovits' pure spinor quantization to compute various three-point tree correlation functions in position-space for the Type IIB superstring. We solve the constraint equations for the vertex operators and obtain explicit expressions for the graviton and axion components of the vertex operators. Using these operators we compute tree level correlation functions in flat space and discuss their extension to the AdS₅ × S⁵ background.

I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac–Moody algebras, I explain how to construct the nonlinear realisation based on the Kac–Moody algebra $E_{11}$ and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space–time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space–time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of $E_{11}$ we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the $E_{11}$ conjecture given many years ago.

Following arXiv:hep-th/0412336 we use the nonlinear realisation of the semi-direct product of $E_{11}$ and its vector representation to construct brane dynamics. The brane moves through a space-time which arises in the nonlinear realisation from the vector representation and it contains the usual embedding coordinates as well as the worldvolume fields. The resulting equations of motion are first order in derivatives and can be thought of as duality relations. Each brane carries the full $E_{11}$ symmetry and so the Cremmer–Julia duality symmetries. We apply this theory to find the dynamics of the IIA and IIB strings, the M2 and M5 branes, the IIB D3 brane as well as the one and two branes in seven dimensions.

We construct the $E$ theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of $E_{11}$ with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in $E$ theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of $E_{11}$. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

We consider the growth of the action for black hole space–time with a fundamental string. Our interest is to find the difference of the behavior between black holes with three different topologies in the scenario of complexity-action conjecture. These black holes have positive, negative and zero curvatures. We would like to calculate the action growth of these systems with a probe fundamental string according to the complexity-action conjecture. We find that for the case where the black holes have the toroidal horizon structure this probe string behaves very differently from the other two cases.

An upper bound for sorting permutations with an operation estimates the diameter of the corresponding Cayley graph and an exact upper bound equals the diameter. Computing tight upper bounds for various operations is of theoretical and practical (e.g., interconnection networks, genetics) interest. Akers and Krishnamurthy gave a Ω(n! n²) time method that examines n! permutations to compute an upper bound, f(Γ), to sort any permutation with a given transposition tree T, where Γ is the Cayley graph corresponding to T. We compute two intuitive upper bounds γ and δ′ each in O(n²) time for the same, by working solely with the transposition tree. Recently, Ganesan computed β, an estimate of the exact upper bound for the same, in O(n²) time. Our upper bounds are tighter than f(Γ) and β, on average and in most of the cases. For a class of trees, we prove that the new upper bounds are tighter than β and f(Γ).

A ℛ "dual" transform is introduced which relates Quantum Field Theory and String regimes, both in a curved background with D-non-compact dimensions. This operation maps the characteristic length of one regime into the other (and, as a consequence, mass domains as well). The ℛ-transform is not an assumed or a priori imposed symmetry but is revealed by the QFT and String dynamics in curved backgrounds. The Hawking–Gibbons temperature and the string maximal or critical temperature are ℛ-mapped one into the other. If back reaction of quantum matter is included, Quantum Field Theory and String phases appear, and ℛ-relations between them manifest as well. These ℛ-transformations are explicitly shown in two relevant examples: Black Hole and de Sitter spacetimes.

In this paper, we have studied particle collisions around a charged dilaton black hole in 2 + 1 dimensions. This black hole is a solution to the low energy string action in 2 + 1 dimensions. Time-like geodesics for charged particles are studied in detail. The center-of-mass energy for two charged particles colliding closer to the horizon is calculated and shown to be infinite if one of the particles has the critical charge.

In this work we explain the construction of the thermal vacuum for the bosonic string, as well that of the thermal boundary state interpreted as a D_p-brane at finite temperature. In both case we calculate the respective entropy using the entropy operator of the Thermo Field Dynamics theory. We show that the contribution of the thermal string entropy is explicitly present in the D_p-brane entropy. Furthermore, we show that the Thermo Field approach is suitable to introduce temperature in boundary states.

Using the ontological interpretation of quantum mechanics in a particular sense, we obtain the classical behavior of the scale factor and two scalar fields, derived from a string effective action for the Friedmann–Robertson–Walker (FRW) time dependent model. Besides, the Wheeler–DeWitt equation is solved exactly. We speculate that the same procedure could also be applied to S-branes.

We provide a conceptual unified description of the quantum properties of black holes (BH), elementary particles, de Sitter (dS) and Anti-de Sitter (AdS) string states.The conducting line of argument is the classical–quantum (de Broglie, Compton) duality here extended to the quantum gravity (string) regime (wave–particle–string duality). The semiclassical (QFT) and quantum (string) gravity regimes are respectively characterized and related: sizes, masses, accelerations and temperatures. The Hawking temperature, elementary particle and string temperatures are shown to be the same concept in different energy regimes and turn out the precise classical–quantum duals of each other; similarly, this result holds for the BH decay rate, heavy particle and string decay rates; BH evaporation ends as quantum string decay into pure (nonmixed) radiation. Microscopic density of states and entropies in the two (semiclassical and quantum) gravity regimes are derived and related, an unifying formula for BH, dS and AdS states is provided in the two regimes. A string phase transition towards the dS string temperature (which is shown to be the precise quantum dual of the semiclassical (Hawking–Gibbons) dS temperature) is found and characterized; such phase transition does not occurs in AdS alone. High string masses (temperatures) show a further (square root temperature behavior) sector in AdS. From the string mass spectrum and string density of states in curved backgrounds, quantum properties of the backgrounds themselves are extracted and the quantum mass spectrum of BH, dS and AdS radii obtained.

This review is devoted to the Multiple Point Principle (MPP), according to which several vacuum states with the same energy density exist in Nature. The MPP is implemented to the Standard Model (SM), Family replicated gauge group model (FRGGM) and phase transitions in gauge theories with/without monopoles. Using renormalization group equations for the SM, the effective potential in the two-loop approximation is investigated, and the existence of its postulated second minimum at the fundamental scale is confirmed. Phase transitions in the lattice gauge theories are reviewed. The lattice results for critical coupling constants are compared with those of the Higgs monopole model, in which the lattice artifact monopoles are replaced by the point-like Higgs scalar particles with magnetic charge. Considering our (3+1)-dimensional space–time as, in some way, discrete or imagining it as a lattice with a parameter a = λ_P, where λ_P is the Planck length, we have investigated the additional contributions of monopoles to the β-functions of renormalization group equations for running fine structure constants α_i(μ) (i = 1, 2, 3 correspond to the U(1), SU(2) and SU(3) gauge groups of the SM) in the FRGGM extended beyond the SM at high energies. It is shown that monopoles have N_fam times smaller magnetic charge in the FRGGM than in the SM (N_fam is a number of families in the FRGGM). We have estimated also the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. We have reviewed that, in contrast to the case of the Anti-grand-unified-theory (AGUT), there exists a possibility of unification of all gauge interactions (including gravity) near the Planck scale due to monopoles. The possibility of the [SU(5)]³ or [SO(10)]³ unification at the GUT-scale ~10¹⁸GeV is briefly considered.

We study the system formed by a gas of black holes and strings within a microcanonical formulation. The density of mass levels grows asymptotically as , (i = 1,…,N). We derive the microcanonical content of the system: entropy, equation of state, number of components N, temperature T and specific heat. The pressure and the specific heat are negative reflecting the gravitational unstability and a nonhomogeneous configuration. The asymptotic behavior of the temperature for large masses emerges as the Hawking temperature of the system (classical or semiclassical phase) in which the classical black hole behavior dominates, while for small masses (quantum black hole or string behavior) the temperature becomes the string temperature which emerges as the critical temperature of the system. At low masses, a phase transition takes place showing the passage from the classical (black hole) to quantum (string) behavior. Within a microcanonical field theory formulation, the propagator describing the string–particle–black hole system is derived and from it the interacting four-point scattering amplitude of the system is obtained. For high masses it behaves asymptotically as the degeneracy of states ρ(m) of the system (i.e. duality or crossing symmetry). The microcanonical propagator and partition function are derived from a (Nambu–Goto) formulation of the N-extended objects and the mass spectrum of the black hole–string system is obtained: for small masses (quantum behavior) these yield the usual pure string scattering amplitude and string–particle spectrum ; for growing mass the spectrum describes all the intermediate states up to the pure black hole behavior. The different black hole behaviors according to the different mass ranges: classical, semiclassical and quantum or string behaviors are present in the model.

A geometric torsion dynamics leading to an effective curvature in a second-order formalism on a D₄-brane is revisited with a renewed interest. We obtain two effective AdS₄ brane geometries on a vacuum created pair of -brane. One of them is shown to describe an AdS Schwarzschild spinning black hole and the other is shown to describe a spinning black hole bound state. It is argued that a D-instanton in a vacuum created anti-D₃-brane within a pair may describe a quintessence. It may seem to incorporate a varying vacuum energy density in a brane universe. We consider the effective curvature scalar on S¹ × S¹ to analyze torsionless geometries on a vacuum created pair of -brane. The emergent AdS₃ brane is shown to describe a Schwarzschild and a Reissner–Nordstrom (RN) geometries in the presence of extra dimension(s).

We argue that the exceptional field theory is a truncation of the nonlinear realisation of the semi-direct product of $E_{11}$ and its first fundamental as proposed in 2003. Evaluating the simple equations of the $E_{11}$ approach, and using the commutators of the $E_{11}$ algebra, we find the local variations of the fields of exceptional field theory after making a radical truncation. This procedure does not respect any of the higher level $E_{11}$ symmetries and so these are lost. We suggest that the need for the section condition in the exceptional field theory could be a consequence of the truncation.

We consider the nonlinear realisation of the semi-direct product of $E_{11}$ and its vector representation which leads to a space-time with tangent group that is the Cartan involution invariant subalgebra of $E_{11}$. We give an alternative derivation of the invariant tangent space metric that this space–time possesses and compute this metric at low levels in eleven, five and four dimensions. We show that one can gauge fix the nonlinear realisation in an $E_{11}$ invariant manner.

Using the general properties that have emerged from $E$ theory we sketch the generic features of the dynamics of branes in seven and eight dimensions. The dynamical equations are a set of duality equations involving the coordinates of the vector representation of $E_{11}$.

We study in detail the irreducible representation of $E$ theory that corresponds to massless particles. This has little algebra $I_c(E_9)$ and contains 128 physical states that belong to the spinor representation of $\mathrm{\text{SO}}(16)$. These are the degrees of freedom of maximal supergravity in eleven dimensions. This smaller number of the degrees of freedom, compared to what might be expected, is due to an infinite number of duality relations which in turn can be traced to the existence of a subaglebra of $I_c(E_9)$ which forms an ideal and annihilates the representation. We explain how these features are inherited into the covariant theory. We also comment on the remarkable similarity between how the bosons and fermions arise in $E$ theory.