Compactifications Of String Theory Research Articles

Supersymmetry, Supergravity, and Superstring Phenomenology

Supersymmetry, supergravity, and superstring are amongst the most popular research topics in particle physics. Supersymmetry is a generalization of the space-time symmetries of quantum field theory that links the matter particles with the force-carrying particles and implies that there are additional superparticles necessary to complete the symmetry. Supergravity is the theory that combines the principles of supersymmetry and general relativity. It naturally includes gravity along with the other fundamental forces (the electromagnetic force, the weak nuclear force, in turn already unified in the electroweak interactions, and the strong nuclear force). String theory is the leading candidate for a theory that unifies all fundamental forces in nature in a consistent scheme. It also provides a consistent framework for the theory of quantum gravity. Compactified string/M-theories make testable predictions about our four-dimensional world. The phenomenology of supersymmetry, supergravity, and superstring is thus very rich and covers many topics: flavour physics and CP violation, Higgs and collider physics, model building beyond the Standard Model, and astroparticle physics and cosmology. Some recent developments in these theories, each with important applications to particle physics and/or cosmology, are the main theme of this special issue. One of the papers of this special issue discusses the constrained generalized Killing spinors, which characterize supersymmetric flux compactifications of supergravity theories, using geometric algebra techniques. Another paper presents a study on what are called Adinkras, which are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Another paper reviews moduli stabilization in type IIB string theory compactification with fluxes. Another paper in this special issue describes supermultiplets wherein a continuously variable “tuning parameter” modifies the supersymmetry transformations. Another paper studies the constraints imposed on the Minimal Supersymmetric Standard Model (MSSM) parameter space by the Large Hadron Collider (LHC) Higgs mass measurements and gluino mass lower bound. Another paper studies helical phase inflation which realizes “monodromy inflation” in supergravity theory. Another paper considers scalar Wilson operators of Supersymmetric Yang-Mills (SYM) theories at high spin and generic twist operators in the multicolor limit. Another paper author gives an overview about the features that the Mathematica package SARAH provides to study new supersymmetric models. Another paper reviews a possible mechanism for the spontaneous breaking of supersymmetry, based on the presence of vacuum condensates. Another paper constructs and studies a formulation of a chargeless complex vector matter field in a supersymmetric framework.

Compactifications of deformed conifolds, branes and the geometry of qubits

We present three families of exact, cohomogeneity-one Einstein metrics in $(2n+2)$ dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces $CP^{n+1}$, written in a Stenzel form, whose principal orbits are the Stiefel manifolds $V_2(R ^{n+2})=SO(n+2)/SO(n)$ divided by $Z_2$. The second family are also Einstein-K\"ahler metrics, now on the Grassmannian manifolds $G_2(R^{n+3})=SO(n+3)/((SO(n+1)\times SO(2))$, whose principal orbits are the Stiefel manifolds $V_2(R^{n+2})$ (with no $Z_2$ factoring in this case). The third family are Einstein metrics on the product manifolds $S^{n+1}\times S^{n+1}$, and are K\"ahler only for $n=1$. Some of these metrics are believed to play a role in studies of consistent string theory compactifications and in the context of the AdS/CFT correspondence. We also elaborate on the geometric approach to quantum mechanics based on the K\"ahler geometry of Fubini-Study metrics on $CP^{n+1}$, and we apply the formalism to study the quantum entanglement of qubits.

Planckian axions in string theory

We argue that super-Planckian diameters of axion fundamental domains can naturally arise in Calabi-Yau compactifications of string theory. In a theory with $N$ axions $\theta^i$, the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form $-\pi<Q^{i}_{j} \theta^j<\pi$. We compute the diameter of the fundamental domain in terms of the eigenvalues $f_1^2\le\...\le f_N^2$ of the metric on field space, and also, crucially, the largest eigenvalue of $(QQ^{\top})^{-1}$. At large $N$, $QQ^{\top}$ approaches a Wishart matrix, due to universality, and we show that the diameter is at least $N f_{N}$, exceeding the naive Pythagorean range by a factor $>\sqrt{N}$. This result is robust in the presence of $P>N$ constraints, while for $P=N$ the diameter is further enhanced by eigenvector delocalization to $N^{3/2}f_N$. We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with $h^{1,1}=51$ where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys $f_N \approx 0.013 M_{pl}$. The random matrix analysis then predicts, and we exhibit, axion diameters $>M_{pl}$ for the precise vacuum parameters found in [1]. Our results provide a framework for achieving large-field axion inflation in well-understood flux vacua.

On natural inflation and moduli stabilisation in string theory

Natural inflation relies on the existence of an axion decay constant which is super-Planckian. In string theory only sub-Planckian axion decay constants have been found in any controlled regime. However in field theory it is possible to generate an enhanced super-Planckian decay constant by an appropriate aligned mixing between axions with individual sub-Planckian decay constants. We study the possibility of such a mechanism in string theory. In particular we construct a new realisation of an alignment scenario in type IIA string theory compactifications on a Calabi-Yau where the alignment is induced through fluxes. Within field theory the original decay constants are taken to be independent of the parameters which induce the alignment. In string theory however they are moduli dependent quantities and so interact gravitationally with the physics responsible for the mixing. We show that this gravitational effect of the fluxes on the moduli can precisely cancel any enhancement of the effective decay constant. This censorship of an effective super-Planckian decay constant depends on detailed properties of Calabi-Yau moduli spaces and occurs for all the examples and classes that we study. We expand these results to a general superpotential assuming only that the axion superpartners are fixed supersymmetrically and are able to show for a large class of Calabi-Yau manifolds, but not all, that the cancellation effect occurs and is independent of the superpotential. We also study simple models where the moduli are fixed non-supersymmetrically and find that similar cancellation behaviour can emerge. Finally we make some comments on a possible generalisation to axion monodromy inflation models.

Triproducts, nonassociative star products and geometry of R-flux string compactifications

We elucidate relations between different approaches to describing the nonassociative deformations of geometry that arise in non-geometric string theory. We demonstrate how to derive configuration space triproducts exactly from nonassociative phase space star products and extend the relationship in various directions. By foliating phase space with leaves of constant momentum we obtain families of Moyal-Weyl type deformations of triproducts, and we generalize them to new triproducts of differential forms and of tensor fields. We prove that nonassociativity disappears on-shell in all instances. We also extend our considerations to the differential geometry of nonassociative phase space, and study the induced deformations of configuration space diffeomorphisms. We further develop general prescriptions for deforming configuration space geometry from the nonassociative geometry of phase space, thus paving the way to a nonassociative theory of gravity in non-geometric flux compactifications of string theory.

Atlas of solar hidden photon emission

Hidden photons, gauge bosons of a U(1) symmetry of a hidden sector, can constitute the dark matter of the universe and a smoking gun for large volume compactifications of string theory. In the sub-eV mass range, a possible discovery experiment consists on searching the copious flux of these particles emitted from the Sun in a helioscope setup à la Sikivie. In this paper, we compute in great detail the flux of HPs from the Sun, a necessary ingredient for interpreting such experiments. We provide a detailed exposition of transverse photon-HP oscillations in inhomogenous media, with special focus on resonance oscillations, which play a leading role in many cases. The region of the Sun emitting HPs resonantly is a thin spherical shell for which we justify an averaged-emission formula and which implies a distinctive morphology of the angular distribution of HPs on Earth in many cases. Low mass HPs with energies in the visible and IR have resonances very close to the photosphere where the solar plasma is not fully ionised and requires building a detailed model of solar refraction and absorption. We present results for a broad range of HP masses (from 0–1 keV) and energies (from the IR to the X-ray range), the most complete atlas of solar HP emission to date.

Towards axionic Starobinsky-like inflation in string theory

It is shown that Starobinsky-like potentials can be realized in non-geometric flux compactifications of string theory, where the inflaton involves an axion whose shift symmetry can protect UV-corrections to the scalar potential. For that purpose we evaluate the backreacted, uplifted F-term axion-monodromy potential, which interpolates between a quadratic and a Starobinsky-like form. Limitations due to the requirements of having a controlled approximation of the UV theory and of realizing single-field inflation are discussed.

Toric geometry and string theory descriptions of qudit systems

In this paper, we propose a new way to approach qudit systems using toric geometry and related topics including the local mirror symmetry used in the string theory compactification. We refer to such systems as ( n , d ) quantum systems where n and d denote the number of the qudits and the basis states respectively. Concretely, we first relate the ( n , d ) quantum systems to the holomorphic sections of line bundles on n dimensional projective spaces CP n with degree n ( d − 1 ) . These sections are in one-to-one correspondence with d n integral points on a n -dimensional simplex. Then, we explore the local mirror map in the toric geometry language to establish a linkage between the ( n , d ) quantum systems and type II D-branes placed at singularities of local Calabi–Yau manifolds. ( 1 , d ) and ( 2 , d ) are analyzed in some details and are found to be related to the mirror of the ALE space with the A d − 1 singularity and a generalized conifold respectively.

Super-geometrodynamics

We present explicit solutions of the time-symmetric initial value constraints, expressed in terms of freely specfiable harmonic functions for examples of supergravity theories, which emerge as effective theories of compactified string theory. These results are a prequisite for the study of the time-evolution of topologically non-trivial initial data for supergravity theories, thus generalising the "Geometrodynamics" program of Einstein-Maxwell theory to that of supergravity theories. Specifically, we focus on examples of multiple electric Maxwell and scalar fields, and analyse the initial data problem for the general Einstein-Maxwell-Dilaton theory both with one and two Maxwell fields, and the STU model. The solutions are given in terms of up to eight arbitrary harmonic functions in the STU model. As a by-product, in order compare our results with known static solutions, the metric in isotropic coordinates and all the sources of the non-extremal black holes are expressed entirely in terms of harmonic functions. We also comment on generalizations to time-nonsymmetric initial data and their relation to cosmological solutions of gauged so-called fake supergravities with positive cosmological constant.

The super period matrix with Ramond punctures

We generalize the super period matrix of a super Riemann surface to the case that Ramond punctures are present. For a super Riemann surface of genus g with 2 r Ramond punctures, we define, modulo certain choices that generalize those in the classical theory (and assuming a certain generic condition is satisfied), a g | r × g | r period matrix that is symmetric in the Z 2 -graded sense. As an application, we analyze the genus 2 vacuum amplitude in string theory compactifications to four dimensions that are supersymmetric at tree level. We find an explanation for a result that has been found in orbifold examples in explicit computations by D’Hoker and Phong: with their integration procedure, the genus 2 vacuum amplitude always vanishes “pointwise” after summing over spin structures, and hence is given entirely by a boundary contribution.

Off-shell Amplitudes in Superstring Theory

Computing the renormalized masses and S-matrix elements in string theory, involving states whose masses are not protected from quantum corrections, requires defining off-shell amplitude with certain factorization properties. While in the bosonic string theory one can in principle construct such an amplitude from string field theory, there is no fully consistent field theory for type II and heterotic string theory. In this paper we give a practical construction of off-shell amplitudes satisfying the desired factorization property using the formalism of picture changing operators. We describe a systematic procedure for dealing with the spurious singularities of the integration measure that we encounter in superstring perturbation theory. This procedure is also useful for computing on-shell amplitudes, as we demonstrate by computing the effect of Fayet-Iliopoulos D-terms in four dimensional heterotic string theory compactifications using this formalism.

Instantons on conical half-flat 6-manifolds

We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds reduces the instanton equation to a set of matrix equations. We finally present some of its solutions and discuss the instanton configurations obtained this way.

De Sitter vacua from a D-term generated racetrack uplift

We propose an uplift mechanism using a structure of multi-Kahler moduli dependence in the F-term potential of type IIB string theory compactifications. This mechanism requires a D-term condition that fixes one modulus to be proportional to another modulus, resulting in a trivial D-term potential. De Sitter minima are realized along with an enhancement of the volume in the Large Volume Scenario and no additional suppression of the uplift term such as warping is required. We further show the possibility to realize the uplift mechanism in the presence of more Kahler moduli such that we expect the uplift mechanism to work in many other compactifications.

Distribution of the number of generations in flux compactifications

Flux compactification of string theory generates an ensemble with a large number of vacua called the landscape. By using the statistics of various properties of low-energy effective theories in the string landscape, one can therefore hope to provide a scientific foundation to the notion of naturalness. This article discusses how to answer such questions of practical interest by using flux compactification of F-theory. It is found that the distribution is approximately in a factorized form given by the distribution of the choice of 7-brane gauge group, that of the number of generations $N_{\rm gen}$ and that of effective coupling constants. The distribution of $N_{\rm gen}$ is approximately Gaussian for the range $|N_{\rm gen}| \lesssim 10$. The statistical cost of higher-rank gauge groups is also discussed.

Comments on M24 representations and CY 3 geometries

We show using string dualities that Mathieu moonshine controls Gromov-Witten invariants and periods of the holomorphic 3-form $\Omega$ for certain $CY_3$ manifolds. We also discuss how the period vectors appear in flux compactifications on these $CY_3$ manifolds and work out the connection between the sporadic group M$_{24}$ and the Yukawa couplings in four dimensional theories that arise from heterotic string theory compactifications on these $CY_3$ manifolds.

Heavy tails in Calabi-Yau moduli spaces

We study the statistics of the metric on K\"ahler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties. We find striking hierarchies in the eigenvalues of the metric and of the Riemann curvature contribution to the Hessian matrix: both spectra display heavy tails. The curvature contribution to the Hessian is non-positive, suggesting a reduced probability of metastability compared to cases in which the derivatives of the K\"ahler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with $h^{1,1}$ as large as 25, which is difficult using algorithms internal to packages such as Sage. Our results serve as input for statistical studies of the vacuum structure in flux compactifications, and of the distribution of axion decay constants in string theory.

Holomorphic bundles and the moduli space of N=1 supersymmetric heterotic compactifications

We describe the first order moduli space of heterotic string theory compactifications which preserve N =1 supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on T X to a field, the moduli space corresponds to deformations of a holomorphic structure $$ \overline{D} $$ on a bundle $$ \mathcal{Q} $$ . The bundle $$ \mathcal{Q} $$ is constructed as an extension by the cotangent bundle T ∗ X of the bundle E = End(V )⊕End(TX)⊕TX with an extension class $$ \mathcal{H} $$ which precisely enforces the anomaly cancelation condition. The deformations corresponding to the bundle E are simultaneous deformations of the holomorphic structures on the poly-stable bundles V and T X together with those of the complex structure of X. We discuss the fact that the “moduli” corresponding to End(T X) cannot be physical, but are however needed in our mathematical structure to be able to enforce the anomaly cancelation condition. In the appendix we comment on the choice of connection on T X which has caused some confusion in the community before. It has been shown by Ivanov and others that this connection should also satisfy the instanton equations, and we give another proof of this fact.

The powers of monodromy

Flux couplings to string theory axions yield super-Planckian field ranges along which the axion potential energy grows. At the same time, other aspects of the physics remain essentially unchanged along these large displacements, respecting a discrete shift symmetry with a sub-Planckian period. After a general overview of this monodromy effect and its application to large-field inflation, we present new classes of specific models of monodromy inflation, with monomial potentials μ 4−p ϕ p . A key simplification in these models is that the inflaton potential energy plays a leading role in moduli stabilization during inflation. The resulting inflaton-dependent shifts in the moduli fields lead to an effective flattening of the inflaton potential, i.e. a reduction of the exponent from a fiducial value p 0 to p < p 0. We focus on examples arising in compactifications of type IIB string theory on products of tori or Riemann surfaces, where the inflaton descends from the NS-NS two-form potential B 2, with monodromy induced by a coupling to the R-R field strength F 1. In this setting we exhibit models with p = 2/3, 4/3, 2, and 3, corresponding to predictions for the tensor-to-scalar ratio of r ≈ 0.04, 0.09, 0.13, and 0.2, respectively. Using mirror symmetry, we also motivate a second class of examples with the role of the axions played by the real parts of complex structure moduli, with fluxes inducing monodromy.

D7-brane chaotic inflation

We analyze string-theoretic large-field inflation in the regime of spontaneously-broken supergravity with conventional moduli stabilization by fluxes and non-perturbative effects. The main ingredient is a shift-symmetric Kähler potential, supplemented by flux-induced shift symmetry breaking in the superpotential. The central technical observation is that all these features are present for D7-brane position moduli in Type IIB orientifolds, potentially allowing for a realization of the axion monodromy proposal in a string theory compactification. Furthermore, our model is explicit enough to address issues of control and moduli stabilization quantitatively. On the one hand, in the large complex structure regime the D7-brane position moduli inherit a shift symmetry from their mirror-dual Type IIA Wilson lines. On the other hand, the Type IIB flux superpotential generically breaks this shift symmetry and allows, by appealing to the large flux discretuum, to tune the relevant coefficients to be small. The shift-symmetric direction in D7-brane moduli space can then play the role of the inflaton: While the D7-brane circles a certain trajectory on the Calabi–Yau many times, the corresponding F-term energy density grows only very slowly, thanks to the above-mentioned tuning of the flux. To be successful our model requires that the dilaton, all complex structure moduli and all D7-brane moduli except the inflaton are fixed at leading order by fluxes. Then the large-field inflationary trajectory can be realized in a regime where Kähler, complex structure and other brane moduli are stabilized in a conventional manner, as we demonstrate using the example of the Large Volume Scenario.

String theoretic QCD axions in the light of PLANCK and BICEP2

The QCD axion solving the strong CP problem may originate from antisymmetric tensor gauge fields in compactified string theory, with a decay constant around the GUT scale. Such possibility appears to be ruled out now by the detection of tensor modes by BICEP2 and the PLANCK constraints on isocurvature density perturbations. A more interesting and still viable possibility is that the string theoretic QCD axion is charged under an anomalous U(1)_A gauge symmetry. In such case, the axion decay constant can be much lower than the GUT scale if moduli are stabilized near the point of vanishing Fayet-Illiopoulos term, and U(1)_A-charged matter fields get a vacuum value far below the GUT scale due to a tachyonic SUSY breaking scalar mass. We examine the symmetry breaking pattern of such models during the inflationary epoch with the Hubble expansion rate 10^{14} GeV, and identify the range of the QCD axion decay constant, as well as the corresponding relic axion abundance, consistent with known cosmological constraints. In addition to the case that the PQ symmetry is restored during inflation, there are other viable scenarios, including that the PQ symmetry is broken during inflation at high scales around 10^{16}-10^{17} GeV due to a large Hubble-induced tachyonic scalar mass from the U(1)_A D-term, while the present axion scale is in the range 10^{9}-5\times 10^{13} GeV, where the present value larger than 10^{12} GeV requires a fine-tuning of the axion misalignment angle. We also discuss the implications of our results for the size of SUSY breaking soft masses.