Compactification Geometry Research Articles

C_2 generalization of the van Diejen model from the minimal (D_5,D_5) conformal matter

We study superconformal indices of 4d compactifications of the 6d minimal (D_{N+3},D_{N+3}) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with A_N, C_N and left( A_1right) ^N global symmetries. We mainly concentrate on C_2 case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our C_2 operator and previously derived A_2 generalization of van Diejen model. Finally, we also consider compactifications with A_N-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.

Gauge-Higgs models from nilmanifolds

We consider the compactification of a Yang-Mills theory on a three-dimensional nilmanifold. The compactification generates a Yang-Mills theory in four space-time dimensions, coupled to a specific scalar sector. The compactification geometry gives rise to masses for the zero-modes, proportional to the twist parameter of the nilmanifold. We study the simple example of an SU(3) model broken by a non-trivial vacuum of the scalar potential which generates three mass scales, two being at tree level, and the third one at loop level. We point out the relevance of general twisted geometries for model building and in particular for gauge-Higgs type models, as the twist generates tree-level mass hierarchies useful for grand unification and for the Higgs sector in electroweak symmetry breaking.

Chern-Simons invariants and heterotic superpotentials

The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and non-integer quantized, generalizing previous results for Wilson lines.

T-branes and G2 backgrounds

Compactification of M- / string theory on manifolds with $G_2$ structure yields a wide variety of 4D and 3D physical theories. We analyze the local geometry of such compactifications as captured by a gauge theory obtained from a three-manifold of ADE singularities. Generic gauge theory solutions include a non-trivial gauge field flux as well as normal deformations to the three-manifold captured by non-commuting matrix coordinates, a signal of T-brane phenomena. Solutions of the 3D gauge theory on a three-manifold are given by a deformation of the Hitchin system on a marked Riemann surface which is fibered over an interval. We present explicit examples of such backgrounds as well as the profile of the corresponding zero modes for localized chiral matter. We also provide a purely algebraic prescription for characterizing localized matter for such T-brane configurations. The geometric interpretation of this gauge theory description provides a generalization of twisted connected sums with codimension seven singularities at localized regions of the geometry. It also indicates that geometric codimension six singularities can sometimes support 4D chiral matter due to physical structure "hidden" in T-branes.

Birational geometry of compactifications of Drinfeld half-spaces over a finite field

We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space Ω(V) over a finite field k that does not extend to an endomorphism of the projective space P(V). This should be compared with theorem of Rémy, Thuillier and Werner that every k-automorphism of Ω(V) extends to a k-automorphism of P(V). Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi–Yau threefold in characteristic 2 and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle.

A megaxion at 750 GeV as a first hint of low scale string theory

Low scale string models naturally have axion-like pseudoscalars which couple directly to gluons and photons (but not $W$s) at tree level. We show how they typically get tree level masses in the presence of closed string fluxes, consistent with the axion discrete gauge symmetry, in a way akin of the axion monodromy of string inflation and relaxion models. We discuss the possibility that the hints for a resonance at 750 GeV recently reported at ATLAS and CMS could correspond to such a heavy axion state (megaxion). Adjusting the production rate and branching ratios suggests the string scale to be of order $M_s$ ~ $7 - 10^4$ TeV, depending on the compactification geometry. If this interpretation was correct, one extra $Z'$ gauge boson could be produced before reaching the string threshold at LHC and future colliders.

The spectra of type IIB flux compactifications at large complex structure

We compute the spectra of the Hessian matrix, ${\cal H}$, and the matrix ${\cal M}$ that governs the critical point equation of the low-energy effective supergravity, as a function of the complex structure and axio-dilaton moduli space in type IIB flux compactifications at large complex structure. We find both spectra analytically in an $h^{1,2}_-+3$ real-dimensional subspace of the moduli space, and show that they exhibit a universal structure with highly degenerate eigenvalues, independently of the choice of flux, the details of the compactification geometry, and the number of complex structure moduli. In this subspace, the spectrum of the Hessian matrix contains no tachyons, but there are also no critical points. We show numerically that the spectra of ${\cal H}$ and ${\cal M}$ remain highly peaked over a large fraction of the sampled moduli space of explicit Calabi-Yau compactifications with 2 to 5 complex structure moduli. In these models, the scale of the supersymmetric contribution to the scalar masses is strongly linearly correlated with the value of the superpotential over almost the entire moduli space, with particularly strong correlations arising for $g_s < 1$. We contrast these results with the expectations from the much-used continuous flux approximation, and comment on the applicability of Random Matrix Theory to the statistical modelling of the string theory landscape.

Early universe cosmology, effective supergravity, and invariants of algebraic forms

The presence of light scalars can have profound effects on early universe cosmology, influencing its thermal history as well as paradigms like inflation and baryogenesis. Effective supergravity provides a framework to make quantifiable, model-independent studies of these effects. The Riemanian curvature of the Kahler manifold spanned by scalars belonging to chiral superfields, evaluated along supersymmetry breaking directions, provides an order parameter (in the sense that it must necessarily take certain values) for phenomena as diverse as slow roll modular inflation, non-thermal cosmological histories, and the viability of Affleck-Dine baryogenesis. Within certain classes of UV completions, the order parameter for theories with $n$ scalar moduli is conjectured to be related to invariants of $n$-ary cubic forms (for example, for models with three moduli, the order parameter is given by the ring of invariants spanned by the Aronhold invariants). Within these completions, and under the caveats spelled out, this may provide an avenue to obtain necessary conditions for the above phenomena that are in principle calculable given nothing but the intersection numbers of a Calabi-Yau compactification geometry. As an additional result, abstract relations between holomorphic sectional and bisectional curvatures are utilized to constrain Affleck-Dine baryogenesis on a wide class of Kahler geometries.

How well can we really determine the scale of inflation?

A detection of primordial B-modes has been heralded not only as a smoking gun for the existence of inflation, but also as a way to establish the scale at which inflation took place. In this paper we critically reinvestigate the connection between a detection of primordial gravity waves and the scale of inflation. We consider whether the presence of additional fields and non-adiabaticity during inflation may have provided an additional source of primordial B-modes competitive with those of the quasi-de Sitter vacuum. In particular, we examine whether the additional sources could provide the dominant signal, which could lead to a misinterpretation of the scale of inflation. In light of constraints on the level of non-Gaussianity coming from Planck we find that only hidden sectors with strictly gravitationally strength couplings provide a feasible mechanism. The required model building is somewhat elaborate, and so we discuss possible UV completions in the context of Type IIB orientifold compactifications with RR axions. We find that an embedding is possible and that dangerous sinusoidal corrections can be suppressed through the compactification geometry. Our main result is that even when additional sources of primordial gravity waves are competitive with the inflaton, a positive B-mode detection would still be a relatively good indicator of the scale of inflation. This conclusion will be strengthened by future constraints on both non-Gaussianity and CMB polarization.

On the spin-2 Kaluza-Klein spectrum of AdS 4 × S 2 ℬ 4 $$ {\mathrm{AdS}}_4\times {S}^2\left({\mathrm{\mathcal{B}}}_4\right) $$

We perform a numerical study of the four-dimensional spin-2 Kaluza-Klein spectrum of supersymmetric AdS$_4\times S^2(\mathcal{B}_4)$ vacua and show that they do not exhibit scale separation. Our methods are generally applicable to similar problems where the compactification geometry is not known analytically, hence an analytic treatment of the spectrum of Kaluza-Klein masses is not available.

More on dimension-4 proton decay problem in F-theory.: Spectral surface, discriminant locus and monodromy

Factorized spectral surface scenario has been considered as one of solutions to the dimension-4 proton decay problem in supersymmetric compactifications of F-theory. It has been formulated in language of gauge theory on 7 + 1 dimensions, but the gauge theory descriptions can capture physics of geometry of F-theory compactification only approximately at best. Given the severe constraint on the renormalizable couplings that lead to proton decay, it is worth studying without an approximation whether or not the proton decay operators are removed completely in this scenario. We clarify how the behavior of spectral surface and discriminant locus are related, study monodromy of 2-cycles in a Calabi–Yau 4-fold geometry, and find that the proton decay operators are likely to be generated in a simple factorization limit of the spectral surface. A list of loopholes in this study, and hence a list of chances to save the factorized spectral surface scenario, is also presented.

Codimension-3 singularities and Yukawa couplings in F-theory

F-theory is one of the frameworks where all the Yukawa couplings of grand unified theories are generated and their computation is possible. The Yukawa couplings of charged matter multiplets are supposed to be generated around codimension-3 singularity points of a base complex 3-fold, and that has been confirmed for the simplest type of codimension-3 singularities in recent studies. However, the geometry of F-theory compactifications is much more complicated. For a generic F-theory compactification, such issues as flux configuration around the codimension-3 singularities, field-theory formulation of the local geometry and behavior of zero-mode wavefunctions have virtually never been addressed before. We address all these issues in this article, and further discuss nature of Yukawa couplings generated at such singularities. In order to calculate the Yukawa couplings of low-energy effective theory, however, the local descriptions of wavefunctions on complex surfaces and a global characterization of zero-modes over a complex curve have to be combined together. We found the relation between them by re-examining how chiral charged matters are characterized in F-theory compactification. An intrinsic definition of spectral surfaces in F-theory turns out to be the key concept. As a biproduct, we found a new way to understand the Heterotic–F theory duality, which improves the precision of existing duality map associated with codimension-3 singularities.

Universal holographic hydrodynamics at finite coupling

We consider thermal plasmas in a large class of superconformal gauge theories described by a holographic dual geometry of the form AdS5×M5. In particular, we demonstrate that all of the thermodynamic properties and hydrodynamic transport parameters for a large class of superconformal gauge theories exhibit a certain universality to leading order in the inverse 't Hooft coupling and 1/Nc. In particular, we show that independent of the compactification geometry, the leading corrections are derived from the same five-dimensional effective supergravity action supplemented by a term quartic in the five-dimensional Weyl tensor.

Towards an explicit model of D-brane inflation

We present a detailed analysis of an explicit model of warped D-brane inflation,incorporating the effects of moduli stabilization. We consider the potential for D3-branemotion in a warped conifold background that includes fluxes and holomorphicallyembedded D7-branes involved in moduli stabilization. Although the D7-branes significantlymodify the inflaton potential, they do not correct the quadratic term in the potential, andhence do not cause a uniform change in the slow roll parameter eta. Nevertheless,we present a simple example based on the Kuperstein embedding of D7-branes,z1 = constant, in which the potential can be fine-tuned to be sufficiently flat for inflation. To derive thisresult, it is essential to incorporate the fact that the compactification volume changesslightly as the D3-brane moves. We stress that the compactification geometry dictatescertain relationships among the parameters in the inflaton Lagrangian, and thesemicroscopic constraints impose severe restrictions on the space of possible models. We notethat the shape of the final inflaton potential differs from projections given in earlier studies:in configurations where inflation occurs, it does so near an inflection point. Finally,we comment on the difficulty of making precise cosmological predictions in thisscenario. This is the companion paper to Baumann et al (2007 Phys. Rev. Lett. 99141601).

Observing the Geometry of Warped Compactification via Cosmic Inflation

Using Dirac-Born-Infeld inflation as an example, we demonstrate that the detailed geometry of warped compactification can leave an imprint on the cosmic microwave background. We compute cosmic microwave background observables for Dirac-Born-Infeld inflation in a generic class of warped throats and find that the results (such as the sign of the tilt of the scalar perturbations and its running) depend sensitively on the precise shape of the warp factor. In particular, we analyze the warped deformed conifold and find that the results can differ from those of other warped geometries, even when these geometries approximate well the exact metric of the warped deformed conifold.

Stringy effects during inflation and reheating

We consider inflationary cosmology in the context of string compactifications with multiple throats. In scenarios where the warping differs significantly between throats, string and Kaluza-Klein physics can generate potentially observable corrections to the cosmology of inflation and reheating. First we demonstrate that a very low string scale in the ground state compactification is incompatible with a high Hubble scale during inflation, and we propose that the compactification geometry is altered during inflation. In this configuration, the lowest scale is just above the Hubble scale, which is compatible with the effective field theory but still leads to potentially observable cosmic microwave background corrections. Also in the appropriate region of parameter space, we find that reheating leads to a phase of long open strings in the standard model sector (before the usual radiation-dominated phase). We sketch the cosmology of the long string phase and we discuss possible observational consequences.

EFFECTS OF STRINGS IN INFLATION AND REHEATING

We argue that many models of inflation in string theory require the usual ten-dimensional compactification geometry to be modified during inflation. Based on arguments from the four-dimensional effective theory, we propose a modified ten-dimensional geometry in which the four-dimensional effective theory is just consistent. We also discuss the implications of the light modes in reheating, following inflation. (Based on work in progress with Anupam Mazumdar and Robert Myers.)

Enumerating Flux Vacua with Enhanced Symmetries

We study properties of flux vacua in type IIB string theory in several simple but illustrative models. We initiate the study of the relative frequencies of vacua with vanishing superpotential W=0 and with certain discrete symmetries. For the models we investigate we also compute the overall rate of growth of the number of vacua as a function of the D3-brane charge associated to the fluxes, and the distribution of vacua on the moduli space. The latter two questions can also be addressed by the statistical theory developed by Ashok, Denef and Douglas, and our results are in good agreement with their predictions. Analysis of the first two questions requires methods which are more number-theoretic in nature. We develop some elementary techniques of this type, which are based on arithmetic properties of the periods of the compactification geometry at the points in moduli space where the flux vacua are located.

Shadows of the Planck Scale: Scale Dependence of Compactification Geometry

By studying the effects of the shape moduli associated with toroidal compactifications, we demonstrate that Planck-sized extra dimensions can cast significant "shadows" over low-energy physics. These shadows distort our perceptions of the compactification geometry associated with large extra dimensions and place a fundamental limit on our ability to probe the geometry of compactification by measuring Kaluza-Klein states. We also find that compactification geometry is effectively renormalized as a function of energy scale, with "renormalization group equations" describing the "flow" of geometric parameters such as compactification radii and shape angles as functions of energy.

Geometry of compactifications of locally symmetric spaces

Pour un espace localement symetrique M nous definissons une compactification M U M(∞) que nous appelons compactification geodesique. Elle est construite en ajoutant des points limites dans M(∞) a certaines geodesiques dans M. La compactification geodesique apparait dans d'autres cas. Les constructions generales de Gromov permettent, dans le cas des espaces symetriques, d'identifier le bord de la compactification de Gromov avec M(∞). De plus M(oo) se construit naturellement avec la theorie des groupes en utilisant l'immeuble de Tits. La compactification geodesique joue deux roles fondamentaux dans l'analyse harmonique de l'espace localement symetrique: 1) c'est la compactification de Martin minimale pour les valeurs negatives du laplacien et 2) elle peut etre utilisee pour parametrer les valeurs propres du laplacien dans le spectre continu sur L 2 .