Geometric Flux Research Articles

Revisiting the two formulations of Bianchi identities and their implications on moduli stabilization

In the context of non-geometric type II orientifold compactifications, there have been two formulations for representing the various NS-NS Bianchi-identities. In the first formulation, the standard three-form flux ($H_3$), the geometric flux ($\omega$) and the non-geometric fluxes ($Q$ and $R$) are expressed by using the real six-dimensional indices (e.g. $H_{ijk}, \omega_{ij}{}^k, Q_i{}^{jk}$ and $R^{ijk}$), and this formulation has been heavily utilized for simplifying the scalar potentials in toroidal-orientifolds. On the other hand, relevant for the studies beyond toroidal backgrounds, a second formulation is utilized in which all flux components are written in terms of various involutively even/odd $(2,1)$- and $(1,1)$-cohomologies of the complex threefold. In the lights of recent model building interests and some observations made in arXiv:0705.3410 and arXiv:0709.2186, in this article, we revisit two most commonly studied toroidal examples in detail to illustrate that the present forms of these two formulations are not completely equivalent. To demonstrate the same, we translate all the identities of the first formulation into cohomology ingredients, and after a tedious reshuffling of the subsequent constraints, interestingly we find that all the identities of the second formulation are embedded into the first formulation which has some additional constraints. In addition, we look for the possible solutions of these Bianchi identities in a detailed analysis, and we find that some solutions can reduce the size of scalar potential very significantly, and in some cases are too strong to break the no-scale structure completely. Finally, we also comment on the influence of imposing some of the solutions of Bianchi identities in studying moduli stabilization.

The Flux-Scaling scenario: De sitter uplift and axion inflation

Non-geometric flux-scaling vacua provide promising starting points to realize axion monodromy inflation via the F-term scalar potential. We show that these vacua can be uplifted to Minkowski and de Sitter by adding an -brane or a D-term containing geometric and non-geometric fluxes. These uplifted non-supersymmetric models are analyzed with respect to their potential to realize axion monodromy inflation self-consistently. Admitting rational values of the fluxes, we construct examples with the required hierarchy of mass scales.

Relating double field theory to the scalar potential of N = 2 gauged supergravity

The double field theory action in the flux formulation is dimensionally reduced on a Calabi-Yau three-fold equipped with non-vanishing type IIB geometric and non-geometric fluxes. First, we rewrite the metric-dependent reduced DFT action in terms of quantities that can be evaluated without explicitly knowing the metric on the Calabi-Yau manifold. Second, using properties of special geometry we obtain the scalar potential of N=2 gauged supergravity. After an orientifold projection, this potential is consistent with the scalar potential arising from the flux-induced superpotential, plus an additional D-term contribution.

Implementing odd-axions in dimensional oxidation of 4D non-geometric type IIB scalar potential

In a setup of type IIB superstring compactification on an orientifold of a T6/Z4 sixfold, the presence of geometric flux (ω) and non-geometric fluxes (Q, R) is implemented along with the standard NS–NS and RR three-form fluxes (H, F). After computing the F/D-term contributions to the N=1 four dimensional effective scalar potential, we rearrange the same into ‘suitable’ pieces by using a set of new generalized flux orbits. Subsequently, we dimensionally oxidize the various pieces of the total four dimensional scalar potential to guess their ten-dimensional origin.

On modular completion of generalized flux orbits

In the context of type IIB orientifold compactification with the presence of (non-)geometric fluxes, we conjecture a modular completed version of the generalized flux-orbits of various NS-NS and RR fluxes. Subsequently, considering two explicit examples with frozen complex structure moduli, we illustrate the utility of these new flux orbits in a very compact rearrangement of the four dimensional effective scalar potential.

Sigma models for genuinely non-geometric backgrounds

The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories. It is found that these models are always geometric, even when both 2-form and 2-vector fields are neither vanishing nor inverse of one another. Taking a further step, we suggest an extended class of 3D sigma models, whose world volume is embedded in phase space, which allow for genuinely non-geometric backgrounds. Adopting the doubled formalism such models can be related to double field theory, albeit from a world sheet perspective.

Helical phase inflation via non-geometric flux compactifications: from natural to starobinsky-like inflation

We show that a new class of helical phase inflation models can be simply realized in minimal supergravity, wherein the inflaton is the phase component of a complex field and its potential admits a deformed helicoid structure. We find a new unique complex-valued index χ that characterizes almost the entire region of the n s − r plane favored by new Planck observations. Continuously varying the index χ, predictions interpolate from quadratic/natural inflation parameterized by a phase/axion decay constant to Starobinsky-like inflation parameterized by the α-parameter. We demonstrate that the simple supergravity construction realizing Starobinsky-like inflation can be obtained from a more microscopic model by integrating out heavy fields, and that the flat phase direction for slow-roll inflation is protected by a mildly broken global U(1) symmetry. We study the geometrical origin of the index χ, and find that it corresponds to a linear constraint relating Kahler moduli. We argue that such a linear constraint is a natural result of moduli stabilization in Type II orientifold compactifications on Calabi-Yau threefolds with geometric and non-geometric fluxes. Possible choices for the index χ are discrete points on the complex plane that relate to the distribution of supersymmetric Minkowski vacua on moduli space. More precise observations of the inflationary epoch in the future may provide a better estimation of the index χ. Since χ is determined by the fluxes and vacuum expectation values of complex structure moduli, such observations would characterize the geometry of the internal space as well.

Heterotic moduli stabilisation and non-supersymmetric vacua

Abstract We study moduli stabilisation in four-dimensional N = 1 supergravity theories which originate from compactifications of the heterotic string on certain manifolds with SU(3) structure. These theories have a non-trivial superpotential generated from geometric flux and, in general, D-terms associated to anomalous U(1) symmetries. We show that, at the perturbative level, there are no supersymmetry preserving vacua. However, subject to a certain technical condition on the D-terms which aligns the extrema of the F-term and D-term potentials, ∂ i V F = ∂ i V D = 0, we find at the perturbative level analytic stable AdS vacua which break supersymmetry. As a result, all T-moduli and the dilaton are stabilised perturbatively with supersymmetry broken at a high scale. We also show numerically that similar vacua can be found when the technical condition on the D-term is relaxed. These vacua persist in the presence of non-perturbative effects. In all cases, the vacua remain AdS.

A flux-scaling scenario for high-scale moduli stabilization in string theory

Tree-level moduli stabilization via geometric and non-geometric fluxes in type IIB orientifolds on Calabi–Yau manifolds is investigated. The focus is on stable non-supersymmetric minima, where all moduli are fixed except for some massless axions. The scenario includes the purely axionic orientifold-odd moduli. A set of vacua allowing for parametric control over the moduli vacuum expectation values and their masses is presented, featuring a specific scaling with the fluxes. Uplift mechanisms and supersymmetry breaking soft masses on MSSM-like D7-branes are discussed as well. This scenario provides a complete effective framework for realizing the idea of F-term axion monodromy inflation in string theory. It is argued that, with all masses close to the Planck and GUT scales, one is confronted with working at the threshold of controlling all mass hierarchies.

Geometry and fluxes of SL(5) exceptional field theory

We use a geometric approach to construct a flux formulation for the SL(5) U-duality manifest exceptional field theory. The resulting formalism is well-suited for studying gauged supergravities with geometric and non-geometric fluxes. Here we describe all such fluxes for both M-theory and IIB supergravity including the Ramond-Ramond fields for compactifications to seven dimensions. We define the locally non-geometric "R-flux" and globally non-geometric "Q-flux" for M-theory and find a new locally non-geometric R-flux for the IIB theory. We show how these non-geometric fluxes can be understood geometrically and give some examples of how they fluxes can be generated by acting with dualities on solutions with geometric or field-strength flux.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

Simulation and experimental study on honeycomb-ceramic thermal energy storage for solar thermal systems

A honeycomb-ceramic thermal energy storage (TES) was proposed for thermal utilization of concentrating solar energy. A numerical model was developed to simulate the thermal performances, and TES experiments were carried out to demonstrate and improve the model. The outlet temperature difference between simulation and experimental results was within 5% at the end of a charging period, indicating the simulation model was reasonable. The simulation model was applied to predict the effects of geometric, thermo-physical parameters and flow fluxes on TES performances. The temperature dropped more quickly and decreased to a lower temperature in discharging period when the conductivity was smaller. The storage capacity increased with the growth of volumetric heat capacity. As to a TES with big channels and thin walls, the outlet temperature increased quickly and greatly in a charging process and dropped sharply in a discharging process.

N S-branes, source corrected Bianchi identities, and more on backgrounds with non-geometric fluxes

In the first half of the paper, we study in details NS-branes, including the NS5-brane, the Kaluza-Klein monopole and the exotic $5_2^2$- or Q-brane, together with Bianchi identities for NSNS (non)-geometric fluxes. Four-dimensional Bianchi identities are generalized to ten dimensions with non-constant fluxes, and get corrected by a source term in presence of an NS-brane. The latter allows them to reduce to the expected Poisson equation. Without sources, our Bianchi identities are also recovered by squaring a nilpotent $Spin(D,D) \times \mathbb{R}^+$ Dirac operator. Generalized Geometry allows us in addition to express the equations of motion explicitly in terms of fluxes. In the second half, we perform a general analysis of ten-dimensional geometric backgrounds with non-geometric fluxes, in the context of $\beta$-supergravity. We determine a well-defined class of such vacua, that are non-geometric in standard supergravity: they involve $\beta$-transforms, a manifest symmetry of $\beta$-supergravity with isometries. We show as well that these vacua belong to a geometric T-duality orbit.

Bifurcations of transition states: Morse bifurcations

A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction.Transition states diffeomorphic to are known to exist for energies just above any index-1 critical point of a Hamiltonian of m degrees of freedom, with dividing surfaces . The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations.Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show how sequences of Morse bifurcations producing various interesting forms of transition state and dividing surface are present in reacting systems, by considering a hypothetical class of bimolecular reactions in gas phase.

Dirac structures on nilmanifolds and coexistence of fluxes

We study some aspects of the generalized geometry of nilmanifolds and examine to which extent different types of fluxes can coexist on them. Nilmanifolds constitute a class of homogeneous spaces which are interesting in string compactifications with fluxes since they carry geometric flux by construction. They are generalized Calabi–Yau spaces and therefore simple examples of generalized geometry at work. We identify and classify Dirac structures on nilmanifolds, which are maximally isotropic subbundles closed under the Courant bracket. In the presence of non-vanishing fluxes, these structures are twisted and closed under appropriate extensions of the Courant bracket. Twisted Dirac structures on a nilmanifold may carry multiple coexistent fluxes of any type. We also show how dual Dirac structures combine to Courant algebroids and work out an explicit example where all types of generalized fluxes coexist. These results may be useful in the context of general flux compactifications in string theory.

Gauged supergravities and non-geometric Q/R-fluxes from asymmetric orbifold CFT’s

Abstract We investigate the orbifold limits of string theory compactifications with geometric and non-geometric fluxes. Exploiting the connection between internal fluxes and structure constants of the gaugings in the reduced supergravity theory, we can identify the types of fluxes arising in certain classes of freely-acting symmetric and asymmetric orbifolds. We give a general procedure for deriving the gauge algebra of the effective gauged supergravity using the exact CFT description at the orbifold point. We find that the asymmetry is, in general, related to the presence of non-geometric Q- and R- fluxes. The action of T-duality is studied explicitly on various orbifold models and the resulting transformation of the fluxes is derived. Several explicit examples are provided, including compactifications with geometric fluxes, Q-backgrounds (T-folds) and R-backgrounds. In particular, we present an asymmetric $ {{\mathbb{Z}}_4} $ × $ {{\mathbb{Z}}_2} $ orbifold in which all geometric and non-geometric fluxes ω, H, Q, R are turned on simultaneously. We also derive the corresponding flux backgrounds, which are not in general T-dual to geometric ones, and may even simul-taneously depend non-trivially on both the coordinates and their winding T-duals.

Heterotic moduli stabilisation

Abstract We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactified on internal manifolds which are smooth Calabi-Yau three-folds up to α′ effects. We first review how to stabilise all the geometric and gauge bundle moduli in a supersymmetric way by including fractional fluxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to the superpotential as well as non-perturbative and threshold effects. We then show that the inclusion of α′ corrections to the Kähler potential leads to new stable Minkowski (or de Sitter) vacua where the complex structure moduli and the dilaton are fixed supersymmetrically at leading order, while the stabilisation of the Kähler moduli at a lower scale leads to spontaneous breaking supersymmetry. The minimum lies at moderately large volumes of all the geometric moduli, at perturbative values of the string coupling and at the right phenomenological value of the GUT gauge coupling. We also provide a dynamical derivation of anisotropic compactifications with stabilised moduli which allow for perturbative gauge coupling unification around 1016 GeV. The value of the gravitino mass can be anywhere between the GUT and the TeV scale depending on the stabilisation of the complex structure moduli. In general, these are fixed by turning on background fluxes, leading to a gravitino mass around the GUT scale since the heterotic three-form flux does not contain enough freedom to tune the superpotential to small values. Moreover accommodating the observed value of the cosmological constant is a challenge. Low-energy supersymmetry could instead be obtained by focusing on particular Calabi-Yau constructions where the gauge bundle is holomorphic only at a point-like sub-locus of complex structure moduli space, or situations with a small number of complex structure moduli (like orbifold models), since in these cases one may fix all the moduli without turning on any quantised background flux. However obtaining the right value of the cosmological constant is even more of a challenge in these cases. Another option would be to focus on compactifications on non-complex manifolds, since these allow for new geometric fluxes which could be used to tune the superpotential as well as the cosmological constant, even if the moduli space of these manifolds is presently only poorly understood.

Exploring double field theory

We consider a flux formulation of Double Field Theory in which fluxes are dynamical and field-dependent. Gauge consistency imposes a set of quadratic constraints on the dynamical fluxes, which can be solved by truly double configurations. The constraints are related to generalized Bianchi Identities for (non-)geometric fluxes in the double space, sourced by (exotic) branes. Following previous constructions, we then obtain generalized connections, torsion and curvatures compatible with the consistency conditions. The strong constraint-violating terms needed to make contact with gauged supergravities containing duality orbits of non-geometric fluxes, systematically arise in this formulation.

Dynamic control of quantum geometric heat flux in a nonequilibrium spin-boson model

We study the quantum geometric heat flux in the nonequilibrium spin-boson model. By adopting the noninteracting-blip approximation that is able to accommodate the strong system-bath coupling, we show that there exists a nonzero geometric heat flux only when the two-level system is nondegenerate. Moreover, the pumping, no pumping, and dynamic control of geometric heat flux are discussed in detail, compared to the results with Redfield weak-coupling approximation. In particular, the geometric energy transfer induced by modulation of two system-bath couplings is identified, which is exclusive to quantum transport in the strong system-bath coupling regime.

A Lorentz invariant doubled world-sheet theory

We propose a Lorentz invariant version of Tseytlin's doubled worldsheet theory that makes T-duality covariance of the string manifest. This theory can be derived as a gauge fixed version of Buscher's gauging procedure, in which the left-over gauge field component acts as a Lagrange multiplier. This description can naturally account for fractional linear O(D,D) transformations of the metric and b-field. It is capable of describing non-geometric backgrounds; geometric and non-geometric fluxes are encoded in the doubled anti-symmetric tensor field strength.