Five-form Flux Research Articles

A 4D IIB flux vacuum and supersymmetry breaking. Part II. Bosonic spectrum and stability

We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux Φ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length ℓ, which is spanned by a conformal coordinate in a finite range 0 < z < zm. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schrödinger operators on a finite interval, characterized by pairs of real constants μ and μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}, with μ equal to 1/3 or 2/3 in all cases and different values of μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}. The potentials behave as μ2−1/4z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\\mu^2-1/4}{z^2} $$\\end{document} and μ~2−1/4zm−z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{{\\overset{\\sim }{\\mu}}^2-1/4}{{\\left({z}_m-z\\right)}^2} $$\\end{document} near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza-Klein excitations of non-singlet vectors and scalars the Schrödinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio Φ/ℓ4 decreases.

Metastable vacua from torsion and machine learning

By implementing an error function on a Machine Learning algorithm we look for minimal conditions to construct stable Anti de Sitter and de Sitter vacua from dimensional type IIB String theory compactification on Kähler manifolds with torsion. This allows to have contributions to the scalar potential from the five-form flux and from D-branes wrapping torsional cycles, interpreted as non-BPS states. The former implies the possibility to construct stable AdS vacua while the latter constitutes a mechanism to uplift AdS to dS vacua. Particularly we consider {widehat{D5}} non-BPS states to uplift the stable AdS vacua to an (apparently) stable dS minimum. Both results – the generation of an AdS vacuum and the corresponding uplifting to a dS one – are restricted to a certain type of configurations, specifically with the number of O3 orientifolds bounded from below by the number of D3-branes and fluxes. Under these conditions, we report over 170 dS (classical) stable vacua. In all of them, the uplifted effective potential becomes very flat indicating the presence of possible sources of instabilities. We comment on their relationship with the Swampland Conjectures.

The AdS5 × S5 superstring.

The duality between the type IIB superstring theory in an AdS5 × S5 background with N units of five-form flux and super Yang-Mills theory with a U(N) gauge group has been studied extensively. My version of the construction of the superstring world-sheet action is reviewed here.

Toric geometry and the dual of \u2110-extremization

We consider d = 3, mathcal{N}=2 gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, Σg , with a topological twist that preserves two supersymmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a D = 11 supergravity dual of the form AdS2 × Y9, with electric four-form flux and where Y9 is topologically a fibration of a Sasakian Y7 over Σg . These D = 11 solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS4 × Y7, with a Sasaki-Einstein metric on Y7. We show that an off-shell entropy function for the dual AdS2 solutions may be computed using the toric data and Kähler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of Y7 over Σg and a Kähler class parameter for Σg . We also discuss the class of supersymmetric AdS3 × Y7 solutions of type IIB supergravity with five-form flux only in the case that Y7 is toric, and show how the off-shell central charge of the dual field theory can be obtained from the toric data. We illustrate with several examples, finding agreement both with explicit supergravity solutions as well as with some known field theory results concerning ℐ-extremization.

F-theory and AdS3/CFT2 (2, 0)

We continue to develop the program initiated in [1] of studying supersymmetric AdS3 backgrounds of F-theory and their holographic dual 2d superconformal field theories, which are dimensional reductions of theories with varying coupling. Imposing 2d mathcal{N}=left(0,2right) supersymmetry,wederivethegeneralconditionsonthegeometryforTypeIIB AdS3 solutions with varying axio-dilaton and five-form flux. Locally the compact part of spacetime takes the form of a circle fibration over an eight-fold {mathcal{Y}}_8^{tau } , which is elliptically fibered over a base {tilde{mathrm{mathcal{M}}}}_6 . We construct two classes of solutions given in terms of a product ansatz tilde{{mathrm{mathcal{M}}}_6}=varSigma times {mathrm{M}}_4 , where Σ is a complex curve and {tilde{mathrm{mathcal{M}}}}_4 is locally a Kähler surface. In the first class {tilde{mathrm{mathcal{M}}}}_4 is globally a Kähler surface and we take the elliptic fibration to vary non-trivially over either of these two factors, where in both cases the metrics on the total space of the elliptic fibrations are not Ricci-flat. In the second class the metric on the total space of the elliptic fibration over either curve or surface are Ricci-flat. This results in solutions of the type AdS3 × K3 × ℳ5τ, dual to 2d (0, 2) SCFTs, and AdS3 × S3/Γ × CY3, dual to 2d (0, 4) SCFTs, respectively. In all cases we compute the charges for the dual field theories with varying coupling and find agreement with the holographic results. We also show that solutions with enhanced 2d mathcal{N}=left(2,2right) supersymmetry must have constant axio-dilaton. Allowing the internal geometry to be non-compact leads to the most general class of Type IIB AdS5 solutions with varying axio-dilaton, i.e. F-theoretic solutions, that are dual to 4d mathcal{N}=1 SCFTs.

Supersymmetric AdS5 solutions of type IIB supergravity without D3 branes

We analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space (AdS5) with a five-dimensional Riemannian manifold M5, where the five-form flux vanishes, while all remaining fluxes are allowed to be non-vanishing consistent with SO(4,2) symmetry. This completes the program of classifying all supersymmetric solutions of ten and eleven-dimensional supergravity with an AdS5 factor. We investigate the supersymmetry conditions in some special cases, and demonstrate how these are satisfied by a solution originally found in [13], utilising the method of non-Abelian T-duality.

Superconformal symmetry in the Kaluza-Klein spectrum of warped AdS(3)

We study the Kaluza-Klein spectrum of warped $AdS_3$ compactifications of type IIB with five-form flux which are dual to $\mathcal{N}=(0,2)$ SCFTs in two dimensions. We prove that the spectra of fluctuations of both the spin 2 sector of the graviton and the axio-dilaton are bounded. At the bound the modes have the correct quantum numbers to be chiral primaries and descendants thereof respectively. Moreover, we prove that the same modes give rise to superpartners in the dilatino spectrum. Our results show that a subset of the mesonic chiral ring of the dual SCFT is isomorphic to the first Kohn-Rossi cohomology groups. As an example, we consider the compactification of four-dimensional $Y^{p,q}$ theories on Riemann surfaces for the case of the universal twist. We conclude by studying fluctuations of the three-form, where we are able to identify Betti multiplets after imposing some mild assumptions.

Type IIB supergravity solutions with AdS5 from Abelian and non-Abelian T dualities

We present a large class of new backgrounds that are solutions of type IIB supergravity with a warped AdS5 factor, non-trivial axion-dilaton, B-field and three-form Ramond-Ramond flux but yet have no five-form flux. We obtain these solutions and many of their variations by judiciously applying non-Abelian and Abelian T-dualities, as well as coordinate shifts to AdS5 × X 5 IIB supergravity solutions with X 5 = S 5 , T 1,1 , Y p,q . We address a number of issues pertaining to charge quantization in the context of non-Abelian T-duality. We comment on some properties of the expected dual super conformal field theories by studying their CFT central charge holographically. We also use the structure of the supergravity Page charges, central charges and some probe branes to infer aspects of the dual super conformal field theories.

IIB black hole horizons with five-form flux and extended supersymmetry

We classify under some assumptions the IIB black hole horizons with 5-form flux preserving more than 2 supersymmetries. We find that the spatial horizon sections with non-vanishing flux preserving 4 supersymmetries are locally isometric either to S^1 * S^3 * T^4 or to S^1 * S^3 * K_3 and the associated near horizon geometries are locally isometric to AdS_3 * S^3 * T^4 and AdS_3 * S^3 * K_3$, respectively. The near horizon geometries preserving more than 4 supersymmetries are locally isometric to R^{1,1} * T^8.

IIB black hole horizons with five-form flux and KT geometry

We investigate the near horizon geometry of IIB supergravity black holes with non-vanishing 5-form flux preserving at least two supersymmetries. We demonstrate that there are three classes of solutions distinguished by the choice of Killing spinors. We find that the spatial horizon sections of the class of solutions with an SU(4) invariant pure Killing spinor are hermitian manifolds and admit a hidden Kahler with torsion (KT) geometry compatible with the SU(4) structure. Moreover the Bianchi identity of the 5-form, which also implies the field equations, can be expressed in terms of the torsion H as d(\omega\wedge H)=\partial\bar\partial \omega^2=0, where omega is a Hermitian form. We give several examples of near horizon geometries which include group manifolds, group fibrations over KT manifolds and uplifted geometries of lower dimensional black holes. Furthermore, we show that the class of solutions associated with a Spin(7) invariant spinor is locally a product R^{1,1} x S, where S is a holonomy Spin(7) manifold.

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of Kahler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.

Quantum corrections toη/s

We consider corrections to the ratio of the shear viscosity to the entropy density in strongly coupled non-Abelian plasmas using the AdS/CFT correspondence. In particular, higher-derivative terms with the five-form Ramond-Ramond flux, which have been ignored in all previous calculations, are included. This provides the first reliable calculation of the leading order correction in the inverse 't Hooft coupling to the celebrated result $\ensuremath{\eta}/s=1/4\ensuremath{\pi}$. The leading correction in inverse powers of the number of colors is computed. Our results hold very generally for quiver gauge theories with an internal manifold ${L}_{pqr}$ in the holographic dual. Our analysis implies that any thermal properties, e.g., all hydrodynamic quantities in these theories, will not be affected by the five-form flux terms at this order.

AdS solutions through transgression

We present new classes of explicit supersymmetric AdS_3 solutions of type IIB supergravity with non-vanishing five-form flux and AdS_2 solutions of D=11 supergravity with electric four-form flux. The former are dual to two-dimensional SCFTs with (0,2) supersymmetry and the latter to supersymmetric quantum mechanics with two supercharges. We also investigate more general classes of AdS_3 solutions of type IIB supergravity and AdS_2 solutions of D=11 supergravity which in addition have non-vanishing three-form flux and magnetic four-form flux, respectively. The construction of these more general solutions makes essential use of the Chern-Simons or "transgression" terms in the Bianchi identity or the equation of motion of the field strengths in the supergravity theories. We construct infinite new classes of explicit examples and for some of the type IIB solutions determine the central charge of the dual SCFTs. The type IIB solutions with non-vanishing three-form flux that we construct include a two-torus, and after two T-dualities and an S-duality, we obtain new AdS_3 solutions with only the NS fields being non-trivial.

IIB backgrounds with five-form flux

We investigate all N = 2 supersymmetric IIB supergravity backgrounds with non-vanishing five-form flux. The Killing spinors have stability subgroups Spin ( 7 ) ⋉ R 8 , SU ( 4 ) ⋉ R 8 and G 2 . In the SU ( 4 ) ⋉ R 8 case, two different types of geometry arise depending on whether the Killing spinors are generic or pure. In both cases, the backgrounds admit a null Killing vector field which leaves invariant the SU ( 4 ) ⋉ R 8 structure, and an almost complex structure in the directions transverse to the lightcone. In the generic case, the twist of the vector field is trivial but the almost complex structure is non-integrable, while in the pure case the twist is non-trivial but the almost complex structure is integrable and associated with a relatively balanced Hermitian structure. The G 2 backgrounds admit a time-like Killing vector field and two spacelike closed one-forms, and the seven directions transverse to these admit a co-symplectic G 2 structure. The Spin ( 7 ) ⋉ R 8 backgrounds are pp-waves propagating in an eight-dimensional manifold with holonomy Spin ( 7 ) . In addition we show that all the supersymmetric solutions of simple five-dimensional supergravity with a time-like Killing vector field, which include the AdS 5 black holes, lift to SU ( 4 ) ⋉ R 8 pure Killing spinor IIB backgrounds. We also show that the LLM solution is associated with a co-symplectic co-homogeneity one G 2 manifold which has principal orbit S 3 × S 3 .

AdS spacetimes from wrapped D3-branes

We derive a geometrical characterization of a large class of AdS3 and AdS2 supersymmetric spacetimes in type IIB supergravity with non-vanishing five-form flux using G-structures. These are obtained as special cases of a class of supersymmetric spacetimes with an or (time) factor that are associated with D3 branes wrapping calibrated two or three cycles, respectively, in manifolds with SU(2), SU(3), SU(4) and G2 holonomy. We show how two explicit AdS solutions, previously constructed in gauged supergravity, satisfy our more general G-structure conditions. For each explicit solution, we also derive a special holonomy metric which, although singular, has an appropriate calibrated cycle. After analytic continuation, some of the classes of AdS spacetimes give rise to known classes of BPS bubble solutions with and symmetry. These have 1/2, 1/4 and 1/8 supersymmetry, respectively. We present a new class of 1/8 BPS geometries with symmetry, obtained by analytic continuation of the class of AdS spacetimes associated with D3-brane wrapped on associative three cycles.

A microscopic limit on gravitational waves from D-brane inflation

We derive a microscopic bound on the maximal field variation of the inflaton during warped D-brane inflation. By a result of Lyth, this implies an upper limit on the amount of gravitational waves produced during inflation. We show that a detection at the level $r&gt;0.01$ would falsify slow roll D-brane inflation. In DBI (Dirac-Born-Infeld) inflation, detectable tensors may be possible in special compactifications, provided that $r$ decreases rapidly during inflation. We also show that for the special case of DBI inflation with a quadratic potential, current observational constraints imply strong upper bounds on the five-form flux.

New supersymmetricAdS3solutions

We construct infinite new classes of supersymmetric solutions of $D=11$ supergravity that are warped products of ${\mathrm{AdS}}_{3}$ with an eight-dimensional manifold ${\mathcal{M}}_{8}$ and have nonvanishing four-form flux. In order to be compact, ${\mathcal{M}}_{8}$ is constructed as an ${S}^{2}$ bundle over a six-dimensional manifold ${B}_{6}$ which is either K\"ahler-Einstein or a product of K\"ahler-Einstein spaces. In the special cases that ${B}_{6}$ contains a two-torus, we also obtain new ${\mathrm{AdS}}_{3}$ solutions of type IIB supergravity, with constant dilaton and only five-form flux. Via the anti-de Sitter (AdS)-conformal field theory (CFT) correspondence the solutions with compact ${\mathcal{M}}_{8}$ will be dual to two-dimensional conformal field theories with $N=(0,2)$ supersymmetry. Our construction can also describe noncompact geometries and we discuss examples in type IIB which are dual to four-dimensional $N=1$ superconformal theories coupled to stringlike defects.

Supersymmetric 3D Anti–de Sitter Space Solutions of Type IIB Supergravity

For every positively curved Kähler-Einstein manifold in four dimensions, we construct an infinite family of supersymmetric solutions of type IIB supergravity. The solutions are warped products of AdS3 with a compact seven-dimensional manifold and have nonvanishing five-form flux. Via the anti-de Sitter/conformal field theory correspondence, the solutions are dual to two-dimensional conformal field theories with (0,2) supersymmetry. The corresponding central charges are rational numbers.

Spinning strings in [formula omitted]: A worldsheet perspective

We examine the leading Regge string states relevant for semi-classical spinning string solutions. Using elementary RNS techniques, quadratic terms in an effective Lagrangian are constructed which describe massive NSNS strings in a space–time with five-form flux. We then examine the specific case of AdS 5 × S 5 , finding the dependence of AdS “energy” ( E 0 ) on spin in AdS ( S), spin on the sphere ( J), and orbital angular momentum on the sphere ( ∇ a ∇ a ).

Penrose Limits, Deformed pp-Waves and the String Duals of Script N = 1 Large-NGauge Theory

A certain conformally invariant N=1 supersymmetric SU(n) gauge theory has a description as an infra-red fixed point obtained by deforming the N=4 supersymmetric Yang-Mills theory by giving a mass to one of its N=1 chiral multiplets. We study the Penrose limit of the supergravity dual of the large n limit of this N=1 gauge theory. The limit gives a pp-wave with R-R five-form flux and both R-R and NS-NS three-form flux. We discover that this new solution preserves twenty supercharges and that, in the light-cone gauge, string theory on this background is exactly solvable. Correspondingly, this latter is the stringy dual of a particular large charge limit of the large n gauge theory. We are able to identify which operators in the field theory survive the limit to form the string's ground state and some of the spacetime excitations. The full string model, which we exhibit, contains a family of non-trivial predictions for the properties of the gauge theory operators which survive the limit.