Axion wormholes in AdS compactifications Euclidean wormholes [1–3] are extrema of the action in Euclidean quantum gravity that connect two distant regions, or even two disconnected asymptotic regions. Despite much work over many years it remains unclear whether wormholes can provide valid saddle point contributions to the Euclidean path integral and therefore have physical implications (see e.g. [4–8]). The Weak Gravity Conjecture (WGC) [9] adds a new dimension to this question. This is because its generalization to instantons implies the existence of super-extremal instantons which, when sourced by axions, correspond to Euclidean axion wormholes. It has been argued that such instanton contributions can destroy the flatness of the potential in models of large field inflation based on axions [10], although there is no consensus on this [11]. It is therefore important to elucidate the physical meaning - if any - of wormholes. To this end it is clearly of interest to find wormhole solutions in string theory and in particular in AdS compactifications, where the AdS/CFT dual partition function provides an alternative description of the gravitational path integral. Axionic wormholes [1] provide natural candidates for wormhole solutions in string theory. However axions are always accompanied by dilatons in string theory compactifications, and the existence of regular wormhole solutions depends delicately on the number of scalars and their couplings [4]. In a single axion-dilaton system coupled to gravity, for instance, the dilaton coupling must be sufficiently small in order for wormholes to exist. In [12] Calabi-Yau compactifications were found which allow for regular axionic wormhole solutions in flat space. The situation is more subtle however in compactifications to AdS. Type IIB on AdS5 × S5 does not admit axionic wormholes [5]. On the other hand, in [4] it was argued there are approximate wormhole solutions in Type IIB compactified on AdS3 × S3 × T4. However no clean derivation was given to determine the exact axion - dilaton content of this compactification.1 The validity of those solutions therefore remains somewhat uncertain. Specifically, their smoothness depends on the specific Wick rotation that was used in [4], but it remains unclear whether this particular Wick rotation is the one selected by AdS/CFT. The goal of this paper is to construct exact, regular axionic wormhole solutions in an AdS compactification where the Wick rotation to the Euclidean theory can be made rigorous using AdS/CFT. The wormholes we find are solutions to Euclidean IIB string theory on AdS5 × S5/Zk, whose field theory duals are certain N = 2 quiver theories [13]. The dual operators that are turned on are exactly marginal operators, which enables us to identify the Wick rotation selected by AdS/CFT and therefore rigorously determine the nature of the scalar fields in the theory. Our results further sharpen the paradox with AdS/CFT and the apparent uniqueness of quantum gravity. One is left wondering what is pathological about axionic wormholes.

String Compactification and Global Orientifolded Quivers with Inflation Abstract: We describe global embeddings of fractional D3 branes at orientifolded singularities in type IIB flux compactifications. We present an explicit Calabi-Yau example where the chiral visible sector lives on a local orientifolded quiver while non-perturbative effects, α0 corrections and a T-brane hidden sector lead to full closed string moduli stabilisation in a de Sitter vacuum. The same model can also successfully give rise to inflation driven by a del Pezzo divisor. Our model represents the first explicit Calabi-Yau example featuring both an inflationary and a chiral visible sector.

A Supersymmetric D-Brane Model of Space-Time Foam We present a supersymmetric model of space-time foam with two stacks of eight D8-branes with equal string tensions, separated by a single bulk dimension containing D0-brane particles that represent quantum fluctuations in the space-time foam. The ground-state configuration with static D-branes has zero vacuum energy. However, gravitons and other closed-string states propagating through the bulk may interact with the D0-particles, causing them to recoil and the vacuum energy to become non-zero. This provides a possible origin of dark energy. Recoil also distorts the background metric felt by energetic massless string states, which travel at less than the usual (low-energy) velocity of light. On the other hand, the propagation of chiral matter fields anchored on the D8-branes is not affected by such space-time foam effects.

“Semi-Realistic” F-term Inflation Model Building in Supergravity We describe methods for building “semi-realistic” models of F-term inflation. By semirealistic we mean that they are built in, and obey the requirements of, “semi-realistic” particle physics models. The particle physics models are taken to be effective supergravity theories derived from orbifold compactifications of string theory, and their requirements are taken to be modular invariance, absence of mass terms and stabilization of moduli. We review the particle physics models, their requirements and tools and methods for building inflation models.

Open string T-duality in double space The role of double space is essential in new interpretation of T-duality and consequently in an attempt to construct M-theory. The case of open string is missing in such approach because until now there have been no appropriate formulation of open string T-duality. In the previous paper [1], we showed how to introduce vector gauge fields ANa and ADi at the end-points of open string in order to enable open string invariance under local gauge transformations of the Kalb-Ramond field and its T-dual "restricted general coordinate transformations”. We demonstrated that gauge fields ANa and ADi are T-dual to each other. In the present article we prove that all above results can be interpreted as coordinate permutations in double space.

E(lementary)-Strings in Six-Dimensional Heterotic F-theory Using E-strings, we can analyze not only six-dimensional superconformal field theories but also probe vacua of non-perturabative heterotic string. We study strings made of D3-branes wrapped on various two-cycles in the global F-theory setup. We claim that E-strings are elementary in the sense that various combinations of E-strings can form Mstrings as well as heterotic strings and new kind of strings, called G-strings. Using them, we show that emissions and combinations of heterotic small instantons generate most of known six-dimensional superconformal theories, their affinizations and little string theories. Taking account of global structure of compact internal geometry, we also show that special combinations of E-strings play an important role in constructing six-dimensional theories of D- and E-types. We check global consistency conditions from anomaly cancellation conditions, both from five-branes and strings, and show that they are given in terms of elementary E-string combinations.

On the origin of generalized uncertainty principle from compactified M5-brane In this paper, we demonstrate that compactification in M-theory can lead to a deformation of field theory consistent with the generalized uncertainty principle (GUP). We observe that the matter fields in the M3-brane action contain higher derivative terms. We demonstrate that such terms can also be constructed from a reformulation of the field theory by the GUP. In fact, we will construct the Heisenberg algebra consistent with this deformation, and explicitly demonstrate it to be the Heisenberg algebra obtained from the GUP. Thus, we use compactification in M-theory to motivate for the existence of the GUP.

Target Space Duality in String Theory String theory (see for example [158]) assumes that the elementary particles are one dimensional extended objects rather than point like ones. String theory also comes equipped with a scale associated, nowadays, with the Planck scale (10−33 cm). The standard model describing the color and electro-weak interactions is based on the point particle notion and is successful at the Fermi scale of about 100 Gev (10−16 cm) which is 10−17 smaller than the Planck scale. The correspondence principle requires thus that string theory when applied to these low energies resembles a point particle picture. In fact string theory has an interpretation in terms of point-like field theory whose spectrum consists of an infinity of particles, all except a finite number of which have a mass of the order of the Planck scale. Integrating out the massive modes leads to an effective theory of the light particles. There exists another class of theories whose spectrum consists as well of an infinite tower of particles; this is the Kaluza-Klein type. In such a class of theories [182, 199, 12], gravity is essentially the sole basic interaction, and space-time is assumed to have, in addition to four macroscopic dimensions, extra microscopic dimensions, characterized by some small distance scale. The standard model is assumed to be a low-energy effective action of the light particles resulting from the purely gravitational higher dimensional system. String theory can also be viewed in many cases as representing a space-time with extra dimensions. Nevertheless, the effective low-energy theory, emerging from string theory, turns out to be different from that resulting from a field theory with an infinite number of particles; it possesses many more symmetries. String theory shows also differences when physics is probed at a scale much smaller than the Planck one. In fact, there are various hints that in string theory physics at a very small scale cannot be distinguished from physics at a large scale. A very striking example of that feature is that a string cannot tell if it is propagating on a space-time with one circular dimension of radius R (a dimensionless number) times the Planck scale or 1/R the Planck scale (see figure 1.A). The discrete symmetry apparent in the example is termed target space duality. Moreover, there are indications that string theory possesses an extremely large symmetry of that nature. A study of that symmetry is the subject of the review.

The Disconnect Between Quantum Mechanics and Gravity Abstract: There is a serious disconnect between quantum theory and gravity. It occurs at the level of the very foundations of quantum theory, and is far deeper than just the matter of trying to quantize a non-linear theory. We shall examine some of the physical reasons for this disconnect and show how it manifests itself at the beginning, at the level of the equivalence principle.

No Universe without Big Bang According to Einstein's theory of relativity, the curvature of spacetime was infinite at the big bang. In fact, at this point all mathematical tools fail, and the theory breaks down. However, there remained the notion that perhaps the beginning of the universe could be treated in a simpler manner, and that the infinities of the big bang might be avoided. This has indeed been the hope expressed since the 1980s by the well-known cosmologists James Hartle and Stephen Hawking with their "no-boundary proposal", and by Alexander Vilenkin with his "tunnelling proposal". Now scientists at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute/AEI) in Potsdam and at the Perimeter Institute in Canada have been able to use better mathematical methods to show that these ideas cannot work. The big bang, in its complicated glory, retains all its mystery. One of the principal goals of cosmology is to understand the beginning of our universe. Data from the Planck satellite mission shows that 13.8 billion years ago the universe consisted of a hot and dense soup of particles. Since then the universe has been expanding. This is the main tenet of the hot big bang theory, but the theory fails to describe the very first stages themselves, as the conditions were too extreme. Indeed, as we approach the big bang, the energy density and the curvature grow until we reach the point where they become infinite. As an alternative, the "no-boundary" and "tunneling" proposals assume that the tiny early universe arose by quantum tunnelling from nothing, and subsequently grew into the large universe that we see. The curvature of spacetime would have been large, but finite in this beginning stage, and the geometry would have been smooth - without boundary (see Fig. 1, left panel). This initial configuration would replace the standard big bang. However, for a long time the true consequences of this hypothesis remained unclear. Now, with the help of better mathematical methods, Jean-Luc Lehners, group leader at the AEI, and his colleagues Job Feldbrugge and Neil Turok at Perimeter Institute, managed to define the 35 year old theories in a precise manner for the first time, and to calculate their implications. The result of these investigations is that these alternatives to the big bang are no true alternatives. As a result of Heisenberg's uncertainty relation, these models do not only imply that smooth universes can tunnel out of nothing, but also irregular universes. In fact, the more irregular and crumpled they are, the more likely (see Fig. 1, right panel). "Hence the "no-boundary proposal" does not imply a large universe like the one we live in, but rather tiny curved universes that would collapse immediately", says Jean-Luc Lehners, who leads the "theoretical cosmology" group at the AEI. Hence one cannot circumvent the big bang so easily. Lehners and his colleagues are now trying to figure out what mechanism could have kept those large quantum fluctuations in check under the most extreme circumstances, allowing our large universe to unfold.  

Inflation from Supersymmetry Breaking We explore the possibility that inflation is driven by supersymmetry breaking with the superpartner of the goldstino (sgoldstino) playing the role of the inflaton. Moreover, we impose an R-symmetry that allows to satisfy easily the slow-roll conditions, avoiding the so-called η-problem, and leads to two different classes of small field inflation models; they are characterised by an inflationary plateau around the maximum of the scalar potential, where R-symmetry is either restored or spontaneously broken, with the inflaton rolling down to a minimum describing the present phase of our Universe. To avoid the Goldstone boson and remain with a single (real) scalar field (the inflaton), R-symmetry is gauged with the corresponding gauge boson becoming massive. This framework generalises a model studied recently by the present authors, with the inflaton identified by the string dilaton and R-symmetry together with supersymmetry restored at weak coupling, at infinity of the dilaton potential. The presence of the D-term allows a tuning of the vacuum energy at the minimum. The proposed models agree with cosmological observations and predict a tensor-to-scalar ratio of primordial perturbations 10 − 9 <∼ r <∼ 10 − 4 and an inflation scale 1010 GeV <∼ H ∗ <∼ 1012 GeV. H ∗ may be lowered up to electroweak energies only at the expense of fine-tuning the scalar potential.

Intrinsic Non-Commutativity of Closed String Theory One of the annoying technicalities of string theory is the presence of co-cycles in the physical vertex operators. In the standard account, these co-cycles are required in order to maintain locality on the worldsheet, i.e., to obtain mutual locality of physical vertex insertions. For example, they appear in standard discussions [1] of compactified strings, and rapidly lead to both technical and conceptual issues. In this paper, we re-analyze these issues carefully, and show that the space of string zero modes surprisingly is best interpreted as non-commutative, with the scale of non-commutativity set by α". A by-product of this realization is that the operator algebra becomes straightforward (albeit with a non-commutative product), with no need for co-cycles. This is not inconsistent with our usual notion of space-time in decompactification limits, but it does significantly impact the interpretation of compactifications in terms of local effective field theories. This is a central ingredient that has been overlooked in any of the attempts at duality symmetric formulations of string theory. Indeed, in a follow-up paper we will show that one can obtain a simple understanding of exotic backgrounds such as asymmetric orbifolds [2] and T-folds [3]. Much of the usual space-time interpretation that we use in string theory is built in from the beginning. Its origins, for example, as an S-matrix theory in Minkowski space-time is emblematic of its interpretation in terms of a collection of particle states propagating in a fixed space-time background. We typically view other solutions of string theory in a similar way, with a well-defined distinction between what is big and what is small. Each such case can be viewed as a classical or semi-classical approximation to a deeper quantum theory in which the notion of a given space-time is not built in from the beginning, but is an emergent property of a given classical limit. It is natural to ask under what circumstances a local effective field theory is obtained. Of course, we know many such instances, and we also know many examples where this does not occur, such as cases where non-commutative field theories are thought to emerge. Perhaps the avatar for the absence of a fixed space-time picture is given by duality-symmetric formulations (of which double field theories [4] and our own metastring theory [5–10], are examples). We are in fact working towards a new notion of quantum space-time, in which non-commutativity plays a central role, much as it does in ordinary quantum mechanics. In the present paper then, we uncover an important step towards such an understanding of quantum space-time.

Type IIA D-Branes, K-Theory and Matrix Theory We show that all supersymmetric Type IIA D-branes can be constructed as bound states of a certain number of unstable non-supersymmetric Type IIA D9-branes. This string-theoretical construction demonstrates that D-brane charges in Type IIA theory on spacetime manifold X are classified by the higher K-theory group K−1 (X), as suggested recently by Witten. In particular, the system of N D0-branes can be obtained, for any N, in terms of sixteen Type IIA D9-branes. This suggests that the dynamics of Matrix theory is contained in the physics of magnetic vortices on the worldvolume of sixteen unstable D9-branes, described at low energies by a U(16) gauge theory.

The standard model gauge symmetry from higher-rank unified groups in grand gauge-Higgs unification models We study grand unified models in the five-dimensional space-time where the extra dimension is compactified on S 1/ℤ 2. The spontaneous breaking of unified gauge symmetries is achieved via vacuum expectation values of the extra-dimensional components of gauge fields. We derive one-loop effective potentials for the zero modes of the gauge fields in SU(7), SU(8), SO(10), and E 6 models. In each model, the rank of the residual gauge symmetry that respects the boundary condition imposed at the orbifold fixed points is higher than that of the standard model. We verify that the residual symmetry is broken to the standard model gauge symmetry at the global minima of the effective potential for certain sets of bulk fermion fields in each model.

Superstring Field Theory and the Wess-Zumino-Witten Action We describe a notion of “higher” Wess-Zumino-Witten-like action which is natural in the context of superstring field theories formulated in the large Hilbert space. For the open string, the action is characterized by a pair of commuting cyclic A∞ structures together with a hierarchy of higher-form potentials analogous to the MaurerCartan elements which appear in the conventional Wess-Zumino-Witten action. We apply this formalism to get a better understanding of symmetries of open superstring field theory and the structure of interactions in the Ramond sector, describing an interesting connection between Ramond vertices and Feynman diagrams.  

Unification of Gauge and Yukawa Couplings The discovery of a Higgs boson at the LHC experiments opened a new era in particle physics. Aside from being the last missing particle predicted by the Standard Model (SM), it is allowing a direct probe of the electroweak (EW) symmetry breaking sector of the SM. In particular, the fact that its mass is close to the EW scale itself, has materialised the issue of naturalness. Mass terms for scalar fields are not protected by any quantum symmetry, therefore any new physics sector that couples to it will feed into the value of the mass. In the SM, the EW scale seems to be shielded from high energy scales, like the Planck one, however, no reason for this is present in the SM itself. Another intriguing hint for new physics is the unification of gauge couplings, that occurs at high energies once one takes into account the renormalisation group evolution of the couplings. This has lead to the development of Grand Unified Theories (GUT). The fact that the mass of the top quark is close to the EW scale also suggests that the Yukawa coupling of the top may have a similar origin. The emergence of low-scale extra dimensions [1], mainly supported by string theory constructions, opened new avenues for model building. One of the most interesting idea is developed in Gauge-Higgs Unification (GHU) models [2–4]. Extra dimensional models, in fact, contain a special class of scalar fields, that arise as an additional polarisation of vector gauge fields aligned with the extra compact space. If the Higgs can be identified as such a scalar, its couplings with the fermions (the top quark in particular) are also related to the gauge coupling. In addition, mass terms for the Higgs would be forbidden by gauge invariance in the bulk of the extra dimensions. If the gauge symmetry is suitably broken by boundary conditions, a massless scalar emerges in the spectrum, whose potential is then radiatively generated and finite [5, 6]. The GHU models are rather attractive as they address, at the same time, gauge-Yukawa unification and naturalness.

String Cosmology - Large-Field Inflation in String Theory This is a short review of string cosmology. We wish to connect stringscale physics as closely as possible to observables accessible to current or near-future experiments. Our possible best hope to do so is a description of inflation in string theory. The energy scale of inflation can be as high as that of Grand Unification (GUT). If this is the case, this is the closest we can possibly get in energy scales to string-scale physics. Hence, GUT-scale inflation may be our best candidate phenomenon to preserve traces of string-scale dynamics. Our chance to look for such traces is the primordial gravitational wave, or tensor mode signal produced during in- flation. For GUT-scale inflation this is strong enough to be potentially visible as a B-mode polarization of the cosmic microwave background (CMB). Moreover, a GUT-scale inflation model has a trans-Planckian excursion of the inflaton scalar field during the observable amount of inflation. Such large-field models of inflation have a clear need for symmetry protection against quantum corrections. This makes them ideal candidates for a description in a candidate fundamental theory like string theory. At the same time the need of large-field inflation models for UV completion makes them particularly susceptible to preserve imprints of their string-scale dynamics in the inflationary observables, the spectral index ns and the fractional tensor mode power r. Hence, we will focus this review on axion monodromy inflation as a mechanism of large-field inflation in string theory.

Gauge reducibility of superstring field theory and Batalin-Vilkovisky master action In this paper, we show that there exists a hidden gauge reducibility in superstring field theory based on the small dynamical string field Ψ ∈ Hβγ whose gauge variation is also small δΨ ∈ Hβγ. It requires additional ghost-antighost fields in the gauge fixed or quantum gauge theory, and thus changes the Batalin-Vilkovisky master action, which implies that additional propagating degrees of freedom appear in the loop superstring amplitudes via the gauge choice of the field theory. We present that the resultant master action can take a different and enlarged form, and that there exist canonical transformations getting it back to the canonical form. On the basis of the Batalin-Vilkovisky formalism, we obtain several exact results and clarify this underlying gauge structure of superstring field theory.

Spontaneous Supersymmetry Breaking, Negative Metric and Vacuum Energy The supersymmetric Nambu-Jona-Lasinio model proposed by Cheng, Dai, Faisel and Kong is re-analyzed by using an auxiliary superfield method in which a hidden local U(1) symmetry emerges. It is shown that, in the healthy field-space region where no negative metric particles appear, only SUSY preserving vacua can be realized in the weak coupling regime and a composite massive spin-1 supermultiplets appear as a result of spontaneous breaking of the hidden local U(1) symmetry. In the strong coupling regime, on the other hand, SUSY is dynamically broken, but it is always accompanied by negative metric particles.

High-scale Supersymmetry, the Higgs Mass and Gauge Unification Suppressing naturalness concerns, we discuss the compatibility requirements of high-scale supersymmetry breaking with the Higgs boson mass constraint and gauge coupling unification. We find that to accommodate superpartner masses significantly greater than the electroweak scale, one must introduce large non-degeneracy factors. These factors are enumerated, and implications for the allowed forms of supersymmetry breaking are discussed. We find that superpartner masses of arbitrarily high values are allowed for suitable values of tan β and the non-degeneracy factors. We also compute the large, but viable, threshold corrections that would be necessary at the unification scale for exact gauge coupling unification. Whether or not high-scale supersymmetry can be realised in this context is highly sensitive to the precise value of the top quark Yukawa coupling, highlighting the importance of future improvements in the top quark mass measurement.

Is String Interaction the Origin of Quantum Mechanics? String theory developed by demanding consistency with quantum mechanics. In this paper we wish to reverse the reasoning. We pretend open string field theory is a fully consistent definition of the theory - it is at least a self consistent sector. Then we find in its structure that the rules of quantum mechanics emerge from the non-commutative nature of the basic string joining/splitting interactions. Thus, rather than assuming the quantum commutation rules among the usual canonical variables we derive them from the physical process of string interactions. Morally we could apply such an argument to M-theory to cover quantum mechanics for all physics. If string or M-theory really underlies all physics, it seems that the door has been opened to an explanation of the origins of quantum mechanics from physical processes.

Gauge Symmetry in Phase Space Consequences for Physics and Spacetime Position and momentum enter at the same level of importance in the formulation of classical or quantum mechanics. This is reflected in the invariance of Poisson brackets or quantum commutators under canonical transformations, which I regard as a global symmetry. A gauge symmetry can be defined in phase space (XM, PM)that imposes equivalence of momentum and position for every motion at every instant of the worldline. One of the consequences of this gauge symmetry is a new formulation of physics in spacetime. Instead of one time there must be two, while phenomena described by one-time physics in 3+1 dimensions appear as various “shadows” of the same phenomena that occur in 4+2 dimensions with one extra space and one extra time dimensions (more generally, d+2). The 2T-physics formulation leads to a unification of 1T-physics systems not suspected before and there are new correct predictions from 2T-physics that 1T-physics is unable to make on its own systematically. Additional data related to the predictions, that provides information about the properties of the extra 1-space and extra 1-time dimensions, can be gathered by observers stuck in 3+1 dimensions. This is the probe for investigating indirectly the extra 1+1 dimensions which are neither small nor hidden. This 2T formalism that originated in 1998 has been extended in recent years from the worldline to field theory in d+2 dimensions. This includes 2T field theories that yield 1T field theories for the Standard Model and General Relativity as shadows of their counterparts in 4+2 dimensions. Problems of ghosts and causality in a 2T space-time are resolved automatically by the gauge symmetry, while a higher unification of 1T field theories is obtained. In this lecture the approach will be described at an elementary worldline level, and the current status of 2T-physics will be summarized.

Observational Consequences of an Interacting Multiverse The multiverse (see Ref. [1] for a general review) has become the most general scenario in modern cosmology. However, the mayor controversy of the multiverse is its observability. The physical significance of the whole multiverse proposal roots on that feature. On the one hand, it could be thought that the multiverse is unobservable because whatever the definition of the universe is, it is always associated with some notion of causal closure. Thus, either all the events that are causally connected would belong to the same definition of the universe or, on the contrary, any event of other universe is not causally connected with the observable events of our universe and thus it cannot be observed. That is so, classically. Quantum mechanically, however, the quantum states of the matter fields that propagate in two distant regions of the whole spacetime manifold can be entangled to each other [2,3] and, hence, their properties would be correlated. In that case, the boundary conditions to be applied on the scalar field should be such that they would consider the global state of the scalar field and not only particular states that correspond to each single region of the entangled pair separately. The application of the boundary conditions to the global state of the scalar field and the lack of information about the state of the scalar field in the unobservable region makes the quantum state of the scalar field in the observable region be given by a reduced density matrix that contains the effects of the quantum correlations with the scalar field of the unobservable region. Thus, distant regions of the spacetime, which are not directly observable, can however leave some imprints in the properties of the matter fields of the observable region. A similar reasoning can be made in the case of the wave function of the universe. Let us first notice that within the third quantization formalism [4,5] the wave function of the universe can formally be seen as a scalar field that propagates in the minisuperspace of homogeneous and isotropic spacetimes and matter fields. In that case, the Wheeler-DeWitt equation can formally be seen as the wave equation of a scalar field (the wave function of the universe), where the frequency of the wave equation is essentially given by the potential terms of the Wheeler-DeWitt equation. At the classical level, these terms are the potential terms of the Friedmann equation too. Therefore, the frequency of the wave equation that determines the evolution of the quantum state of the universe is ultimately related Universe 2017, 3, 49; doi:10.3390/universe3020049 www.mdpi.com/journal/universe Universe 2017, 3, 49 2 of 10 to the Friedmann equation that determines the classical evolution of the universes. An important feature is then that any interacting process that typically changes the frequency of the scalar field in a quantum field theory would change, in the parallel case of the multiverse, the potential term of the Wheeler-DeWitt equation and thus, it would have an observable consequence in the evolution of the universes. In this paper, we shall pose an interacting scheme between the wave functions of different universes. As a result of the interactions the effective value of the potential becomes discretized. It turns out then that a new whole range of cosmological processes can now be posed in the multiverse, some of them would leave different imprints in the properties of the CMB [6]. The paper is outlined as follows: in Section 2, we present the basics of the third quantization formalism, where an interaction scheme between universes can be posed in a similar way as it is done in the quantum mechanics of particle and fields. In particular, we explore in Section 2.3. the effects that different coupling functions might have in the evolutionary properties of the single universes. In Section 3, we briefly cite some ideas that have been proposed to test the multiverse. Afterwards, we explain how we expect to test the model of an interacting multiverse. We summarize and make some brief conclusions in Section 4.

STRING THEORY OR FIELD THEORY? The status of string theory is reviewed, and major recent developments - especially those in going beyond perturbation theory in the string theory and quantum field theory frameworks - are discussed. This analysis helps better understand the role and place of string theory in the modern picture of the physical world. Even though quantum field theory describes a wide range of experimental phenomena, it is emphasized that there are some insurmountable problems inherent in it - notably the impossibility to formulate the quantum theory of gravity on its basis - which prevent it from being a fundamental physical theory of the world of microscopic distances. It is this task, the creation of such a theory, which string theory, currently far from completion, is expected to solve. In spite of its somewhat vague current form, string theory has already led to a number of serious results and greatly contributed to progress in the understanding of quantum field theory. It is these developments which are our concern in this review.

String theory and non-commutative gauge theory Edward Witten -- In this paper, I will report on work with N Seiberg [1] in which we attempted to understand in a more systematic framework the relation of string theory to non-commutative Yang–Mills theory. Such a relationship was first uncovered by Connes, Douglas and Schwarz in the context of matrix theories [2]. Two of the papers most relevant to this paper are that of Schomerus [3], attempting to extract a non-commutative parameter directly from conformal field theory, and that of Nekrasov and Schwarz on instantons on non-commutative R4 [4]. For additional references to the extensive literature, I refer to [1]. Further aspects of the subject have been explained by Seiberg [5].

FOUR LECTURES ON M-THEORY Synopsis: (i) how superstring theories are unified by M-theory; (ii) how superstring and supermembrane properties follow from the D=10 and D=11 supersymmetry algebras; (iii) how D=10 and D=11 supergravity theories determine the strong coupling limit of superstring theories; (iv) how properties of Type II p-branes follow from those of M-branes.

Geometry of Strings and Branes Elementary particle physics aims to describe the fundamental constituents of Nature and their interactions. Experiments indicate that elementary particles fall into two classes: leptons, containing among others the electron and the neutrino; and quarks, which form the building blocks of protons and neutrons. The four known forces between these building blocks of matter are the gravitational, the electromagnetic, the weak, and the strong interaction. At small length scales, the gravitational interaction is many orders of magnitude weaker than all the other forces1 , and it can therefore safely be neglected. The remaining three interactions of elementary particles can be described by an elegant theory called the Standard Model. This theory is a gauge theory: it has an internal local symmetry group in which each interaction is described by the exchange of gauge fields. These gauge fields are called the photon, the W-bosons and Z-boson, and the gluons for the electromagnetic, the weak, and the strong interaction, respectively. Gauge fields are different from matter particles in several aspects: the former fall into the class of bosons, particles with integer spin and commuting statistics; the latter are called fermions, particles with half-integer spin and anti-commuting statistics. It can be shown that internal symmetry groups, such as those of the Standard Model, cannot mix bosons with fermions [1]. Microscopic physicsis described by quantum mechanics, which can be seen as a deformation of classical dynamics. It has several non-intuitive properties: one cannot simultaneously measure all observables with infinite accuracy, and many quantities can only be expressed in terms of probabilities. The Standard Model is quantum mechanically completely consistent, and the theory is in excellent agreement with experiments. At macroscopic scales, the interactions of the Standard Model are virtually absent: the strong interaction is confined to small distances; the weak interaction has an exponential decay with distance; and although the electromagnetic force has an infinite range, all large configurations of matter are approximately electrically neutral. Hence, the gravitational interaction becomes the dominant force at large length scales. Gravity is described by the theory of General Relativity. The basic ingredients of General Relativity are that space and time merge into a spacetime, that matter induces a curved geometry on spacetime, and that this geometry in turn determines the dynamics of matter. One can also try to cast General Relativity in the form of a gauge theory: in this case a gauge theory of spacetime symmetries, known as general coordinate transformations, rather than internal symmetries. The corresponding gauge field in this case is called the graviton. General Relativity is a purely classical theory. It successfully explains physics in the range of terrestrial to cosmological length scales. However, this split of physics into the macroscopic theory of General Relativity and the microscopic Standard Model is not without caveats, because General Relativity has some peculiar properties. First of all, it turns out that certain solutions to the classical field equations, known as black holes, have as a generic feature the occurrence of spacetime singularities [2] around which the gravitational field becomes infinitely large. This undermines the reason for ignoring gravitational interactions in elementary particle physics, and it becomes necessary to treat the gravitational field quantum mechanically. Most of these spacetime singularities are predicted not to be directly observable. Instead, they are conjectured always to be hidden behind event horizons – surfaces from which not even light can return. Singularities are therefore thought not to be directly observable. However, the quantum mechanical behavior of elementary particles around such event horizons is problematic, since the one-way nature of event-horizons interferes with the probabilistic interpretation of quantum mechanics. This gives rise to information paradoxes [3]. Although the energy scales necessary to probe microscopic gravitational effects are not easily obtained in laboratory experiments, they did occur in the early universe. In order to develop good cosmological models, it is therefore necessary to have a description of gravity at small length scales. As a final remark, there is the related problem of the cosmological constant, a parameter in General Relativity for which the Standard Model predicts a value many orders of magnitude larger than the value inferred from astronomical observations [4]. To solve the problems sketched above, it is necessary to construct a theory of quantum gravity. To see what problems can arise in quantizing gravity, it is instructive to compare electromagnetism and gravity since at the classical and semi-classical level there are many parallels between the two interactions, as we have summarized in table 1. They both share a characteristic long range force, although gravity can never be repulsive. Both interactions also fit into a relativistic framework, and covariant field equations for both theories were found by Maxwell, and by Einstein, respectively. Both actions are invariant under local symmetries. For electromagnetism, these symmetries form the group of phase transformations, known as U(1); for General Relativity, they form the group of general coordinate transformations. There is one particular classical effect of the gravitational interaction that has not yet been observed directly: namely the radiation of gravitational waves2 , the gravitational counterpart of optics. The quantum mechanical motion of particles in the background of classical force fields is sometimes called first quantization. For the electromagnetic force, this was studied in the first few decades of the twentieth century during which in particular the nature of blackbody radiation and the origin of the energy levels of the hydrogen atom were clarified. In the last few decades of the last century, the quantum mechanical behavior of particles in gravitational fields has been clarified: in particular, the process of Hawking radiation [6] and the microscopic origin [7] of entropy [8,9] for certain classes of black holes were discovered. To continue the discussion of quantum gravity, it is more useful to compare the gravitational with the weak or the strong interaction, as we have summarized in table 2, since electromagnetism has no self-interactions at the quantum level, in contrast to the other three interactions. For the electromagnetic interaction, one can apply quantization methods to the classical action Lem given in table 1, but this procedure fails for the action of General Relativity since it has an energy-dependent coupling constant G – this makes the theory nonrenormalizable. Some progress towards solving this non-renormalizability problem was obtained by the discovery of supergravity in 1976 [10]. Supergravity is a modified version of General Relativity having spacetime symmetries as well as internal symmetries. A characteristic property of this so-called supersymmetry is that it mixes bosons with fermions [11]. In chapter 5, we will be more precise about the structure of supersymmetry and its cousin conformal supersymmetry. Although supergravity is better behaved at high energies than General Relativity, it is still non-renormalizable. The best one can hope for is that supergravity is a low-energy effective description of a theory of quantum gravity. This is rather similar to the situation concerning the weak interaction where Fermi’s theory of beta-decay is also a non-renormalizable theory, but it can be seen to arise from the Standard Model. In order to go beyond the low-energy effective description of a theory, a prescription for calculating scattering amplitudes at higher energies is necessary. For the strong interaction, this so-called S-matrix theory was developed during the nineteen sixties, and it uses a perturbative expansion over Feynman graphs in order to calculate amplitudes. The precise prescription is fixed by a Lagrangian formulation. In the case of the strong interaction, as well as the electroweak interactions, all the Feynman rules can be derived from the Lagrangian of the Standard Model. A corresponding formalism yielding scattering amplitudes for gravity involves the concept of strings: i.e. at small length scales, particles are postulated to be tiny vibrating strings. The motivation is that the spectrum of a closed string contains the graviton, the gauge field for gravity. Since strings sweep out worldsheets rather than worldlines, as particles do, the idea of Feynman graphs has to be extended to surfaces. It was shown in the nineteen eighties that a perturbative expansion over Riemann surfaces gives quantum mechanically consistent scattering amplitudes. The string theory analog of the Standard Model was developed in the nineteen eighties, this goes under the name of string field theory. In this theory, one single string field describes all string vibrations simultaneously. For the simplest models of perturbative string theory, it can be shown that the corresponding string field theory yields the same answers for scattering amplitudes, but for more complicated perturbative string theories, there are technical complications in constructing the corresponding string field theories. The fields in the Lagrangian of the Standard Model can be rotated by two- or threedimensional unitary matrices, in which case the gauge group is called SU(2) or SU(3), respectively. Since matrices do not commute, such theories are called non-Abelian gauge theories. The quantization of the classical action of an interaction is often called second quantization, and for the weak and the strong interactions this can be consistently done using the methods of BRS-quantization [12, 13]. The symmetry groups of string field theories are much larger and much more complicated than the gauge groups of the Standard Model, and in many cases not known explicitly. This means that traditional methods of quantization fail, and one needs to use more sophisticated methods such as the BV anti-field formalism [14]. Just as the quantization of the weak interaction required more sophisticated tools than the quantization of electromagnetism, it seems also likely that the quantization of gravity will require new methods in this area. Gauge theories often have solitons – solutions of the classical field equations with fi- nite energy. In modified theories of the weak interaction there are for example magnetic monopoles. The presence of such magnetic monopoles can imply that there is a duality between electric and magnetic charges. Such dualities are powerful symmetries, since they often relate separate regimes of a given theory. String theory has higher-dimensional solitonic solutions called branes3 . In string theory, there is also a number of dualities, such as dualities between strongly and weakly coupled regimes of different versions of string theory. In all of these dualities, branes play an essential role. The overall framework of string theory and branes is called M-theory, where the M can mean anything ranging from Mystery to Membrane, according to taste. It is not clear yet whether strings are the fundamental degrees of freedom of quantum gravity, or if there is perhaps a formulation in terms of branes. The organization of this thesis is as follows. We will start in chapter 1 with a more elaborate treatment of the string theory framework, including the basic features of string theory and supergravity, as well as the various dualities and brane solutions of these theories. In chapter 2, we will describe the AdS/CFT correspondence – a recently discovered duality between theories of gravity in Anti-de-Sitter spacetimes and conformal quantum field theories. This is a remarkable duality, because several quantities within quantum gravity can be expressed in terms of concepts known from quantum field theory. A central theme in the AdS/CFT correspondence is a special brane solution of string theory: the D3-brane. In chapter 3, we will present our results [15] that show how this duality can be extended to a duality between gravity in more general curved spacetimes called domain-walls and more general quantum field theories – the DW/QFT correspondence. In particular, we will discuss a large class of brane solutions that includes the D3-brane. After choosing a suitable coordinate frame, the so-called dual frame, we will study the near-horizon geometry of these brane solutions of supergravity, and we will analyze what kinematical information can be extracted from the dual field theories. The domain-walls that appear in the analysis mentioned above describe spaces that are separated into several domains by a boundary surface – the domain-wall. Acrosssuch domainwalls, physical quantities can change their values in a discontinuous fashion. Domain-walls that have such discontinuities are sometimes called “thin” domain-walls. On the other hand, domain-walls that can be interpreted as smooth interpolations between different supergravity vacua go under the name of “thick” domain-walls. At the end of chapter 3, we will explain how these thick domain-walls have the interpretation of renormalization group flows in their dual quantum field theories. Domain-wall spacetimes have attracted renewed attention recently: they are a member of the class of brane world scenarios. In chapter 4, we will describe such brane world scenarios in more detail: the basic idea is that our four-dimensional universe is actually a hypersurface within a five-dimensional supergravity theory. The size of the extra fifth dimension transverse to the so-called brane world can be used to gain insight in the origin of some unnatural properties of four-dimensional physics. For instance, the so-called Randall-Sundrum scenarios were used to obtain a better understanding of the cosmological constant problem, as well as the unnatural ratio of the strength of the gravitational force and the remaining three interactions, the so-called hierarchy problem. Supersymmetric versions of such theories have proven to be hard to find. The main obstacle is realizing supersymmetry on the four-dimensional brane world solution: it is related to finding the vacuum structure of the corresponding five-dimensional supergravity theory. This, in turn, requires a detailed knowledge of all possible couplings of five-dimensional matter models to supergravity. The scalar fields of these matter models can be interpreted as coordinates on an abstract space. Many properties of the matter-coupled supergravity theory can then be expressed in terms of the geometrical properties of the corresponding space of scalar fields. In particular, the scalar fields generate a potential that determines the vacuum structure of the supergravity theory. For supersymmetric brane worlds to exist, this scalar potential needs to possess two different, stable minima that need to satisfy some additional constraints. Moreover, one needs to find a suitable solution that smoothly interpolates between two such minima. Such an analysis, which had been started in the nineteen eighties (albeit for different reasons), has recently been renewed, but still does not encompass the most general fivedimensional matter-coupled supergravity theory. We will take a systematic approach to construct these five-dimensional matter-couplings. This so-called superconformal program starts from the most general spacetime symmetry group, the group of superconformal transformations, which considerably simplifies the analysis of matter-couplings to supergravity. The different models possessing superconformal symmetry are called multiplets. First of all, there is the so-called Weyl multiplet: this is the smallest multiplet of the superconformal group that possess the graviton. On the other hand, there are the matter multiplets: they interact with the Weyl multiplet that forms a fixed background of conformal supergravity. Matter-couplings to non-conformal supergravity can then be obtained by breaking the conformal symmetries. In chapter 5, we will present our results [16] on the five-dimensional Weyl multiplets. We will see that there are two versions of this multiplet: the Standard Weyl multiplet and the Dilaton Weyl multiplet. Multiplets similar to the Standard Weyl multiplet also exist in four and six dimensions, but the Dilaton Weyl multiplet had so far only been found in six dimensions. We will use a well-known method to deduce the transformation rules for the different fields: the so-called Noether method. In particular, we will construct the multiplet of conserved Noether currents for the various conformal symmetries. A remarkable detail 7 is that the current multiplet that couples to the Standard Weyl multiplet contains currents that satisfy differential equations, a mechanism that so far had only been known from tendimensional conformal supergravity. Our results [17] on five-dimensional superconformal matter multiplets will be presented in chapter 6. We will discuss so-called vector multiplets: these are multiplets that contain the gauge field of the gauge group under which the multiplet transforms. We will analyze vector multiplets that transform under arbitrary transformations of the gauge group: the socalled vector-tensor multiplets. In particular, we will consider representations of the gauge group that are reducible but not completely reducible. This gives rise to previously unknown interactions between vector fields and tensor fields. The conformal symmetries can only be realized on the tensor fields if these satisfy their equations of motion. By dropping the usual restriction that the equations of motion have to follow from an action principle, we can also formulate vector-tensor multiplets with an odd number of tensor fields. Apart from vector-tensor multiplets, we will also consider hypermultiplets in chapter 6. These multiplets also possess scalar fields but not gauge fields. The scalar fields span a vector space over the quaternions. Realizing the conformal algebra on the scalar fields will induce a non-trivial geometry called hyper-complex geometry on the space of scalars. Similarly as for tensor fields, the superconformal algebra can only be realized on the fields of the hypermultiplet with the use of equations of motion. Also in this case, we will consider equations of motion that do not follow from an action principle. The special cases for which there is an action correspond to hyper-complex manifolds possessing a metric: the so-called hyper-Kähler manifolds. Furthermore, we will analyze the interaction of hypermultiplets with vector multiplets, and we will also make use of the scalar field geometry in this case. At the end of chapter 6, we will give an overview of all the geometrical concepts that we will make use of. The matter-couplings to conformal supergravity that we will construct in this way can be used as a starting point to construct matter-couplings to non-conformal supergravity. At the end of chapter 6, we will sketch some ingredients of this procedure. Whether the fivedimensional matter-couplings of supergravity that can be obtained in this way will actually modify the vacuum structure in such a way that supersymmetric brane world scenarios can be realized, remains an open question that will have to be answered by future research.

A functional perspective on emergent supersymmetry We investigate the emergence of N = 1 supersymmetry in the long-range behavior of three-dimensional parity-symmetric Yukawa systems. We discuss a renormalization approach that manifestly preserves supersymmetry whenever such symmetry is realized, and use it to prove that supersymmetry-breaking operators are irrelevant, thus proving that such operators are suppressed in the infrared. All our findings are illustrated with the aid of the ǫ-expansion and a functional variant of perturbation theory, but we provide numerical estimates of critical exponents that are based on the non-perturbative functional renormalization group.

Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of Covariant Quantum-Gravity (CQG-theory). The treatment is founded on the recently-identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly-covariant Hamilton equations and the related Hamilton-Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave-function are realized in 4−scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton-Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the 4−scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant.

Origin of time before inflation from a topological phase transition We study the origin of the universe (or pre-inflation) by suggesting that the primordial space-time in the universe suffered a global topological phase transition, from a 4D Euclidean manifold to an asymptotic 4D hyperbolic one. We introduce a complex time, τ , such that its real part becomes dominant after started the topological phase transition. Before the big bang, τ is a space-like coordinate, so that can be considered as a reversal variable. After the phase transition is converted in a causal variable. The formalism solves in a natural manner the quantum to classical transition of the geometrical relativistic quantum fluctuations: σ, which has a geometric origin.