G2 Holonomy Research Articles

The complete Kaluza-Klein spectra of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 0 M-theory on AdS4× (squashed S7)

The squashed seven-sphere operator spectrum is completed by deriving the spectrum of the spin-3/2 operator. The implications of the results for the AdS4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supermultiplets obtained from compactification of eleven-dimensional supergravity are analysed. The weak G2 holonomy plays an important role when solving the eigenvalue equations on the squashed sphere. Here, a novel and more universal algebraic approach to the whole eigenvalue problem on coset manifolds is provided. Having obtained full control of all the operator spectra, we can finally determine the irreps D(E0, s) for all supermultiplets in the left-squashed vacuum. This includes an analysis of possible boundary conditions. By performing an orientation flip on the seven-sphere, we also obtain the full spectrum for the non-supersymmetric right-squashed compactification which is of interest in the swampland context and in particular for the AdS swampland conjecture. Here, a number of boundary condition ambiguities arise making the analysis of dual marginal operators somewhat involved. This work is a direct continuation of [1] and [2].

Evidence for an algebra of G2 instantons

In this short note, we present some evidence towards the existence of an algebra of BPS G2 instantons. These are instantonic configurations that govern the partition functions of 7d SYM theories on local G2 holonomy manifolds mathcal{X} . To shed light on such structure, we begin investigating the relation with parent 4d mathcal{N} = 1 theories obtained by geometric engineering M-theory on mathcal{X} . The main point of this paper is to substantiate the following dream: the holomorphic sector of such theories on multi-centered Taub-NUT spaces gives rise to an algebra whose characters organise the G2 instanton partition function. As a first step towards this program we argue by string duality that a multitude of geometries mathcal{X} exist that are dual to well-known 4d SCFTs arising from D3 branes probes of CY cones: all these models are amenable to an analysis along the lines suggested by Dijkgraaf, Gukov, Neitzke and Vafa in the context of topological M-theory. Moreover, we discuss an interesting relation to Costello’s twisted M-theory, which arises at local patches, and is a key ingredient in identifying the relevant algebras.

On the squashed seven-sphere operator spectrum

We derive major parts of the eigenvalue spectrum of the operators on the squashed seven-sphere that appear in the compactification of eleven-dimensional supergravity. These spectra determine the mass spectrum of the fields in AdS4 and are important for the corresponding mathcal{N} = 1 supermultiplet structure. This work is a continuation of the work in [1] where the complete spectrum of irreducible isometry representations of the fields in AdS4 was derived for this compactification. Some comments are also made concerning the G2 holonomy and its implications on the structure of the operator equations on the squashed seven-sphere.

Generalising G2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski mathcal{N} = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

Bryant–Salamon [formula omitted] manifolds and coassociative fibrations

Bryant–Salamon constructed three 1-parameter families of complete manifolds with holonomy G2 which are asymptotically conical to a holonomy G2 cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on R7: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of ℂ3 with a line, and the Harvey–Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere S4, and the cone on ℂP3, whose smooth fibres are T∗S2, and whose singular fibres are R4∕{±1}. We relate these fibrations to hypersymplectic geometry, Donaldson’s work on Kovalev–Lefschetz fibrations, harmonic 1-forms and the Joyce–Karigiannis construction of holonomy G2 manifolds, and we construct vanishing cycles and associative “thimbles” for these fibrations.

\u201cLagrangian disks\u201d in M-theory

While the study of bordered (pseudo-)holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G2 holonomy spaces and to Spin(7) metrics on 8-manifolds with T2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T[M4] on the other.

A circle quotient of a G2 cone

A study is made of R6 as a singular quotient of the conical space R+×CP3 with holonomy G2, with respect to an obvious action by U(1) on CP3 with fixed points. Closed expressions are found for the induced metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of SO(3) on R6, which can be used effectively to describe the resulting geometrical features.

Mathcal{SW}left(frac{3}{2},2right) subsymmetry in G2, Spin(7) and mathcal{N} = 2 CFTs

Spectral flow, spacetime supersymmetry, topological twists, chiral primaries related to marginal deformations, mirror symmetry: these are important consequences of the worldsheet mathcal{N} = 2 superconformal symmetry of strings on Calabi-Yau manifolds. To various degrees of certainty, these features were also established when the target is either 7d or 8d with exceptional holonomy G2 or Spin(7) respectively. We show that these are more than mere analogies. We exhibit an underlying symmetry mathcal{SW}left(frac{3}{2},2right) making a bridge between the latter cases and K3 target spaces. Reviewing unitary representations of mathcal{SW}left(frac{3}{2},2right) leads us to speculate on further roles of this algebra in string theory compactifications and on the existence of topologically twisted versions of mathcal{SW}left(frac{3}{2},2right) theories.

No-scale and scale-separated flux vacua from IIA on G2 orientifolds

We discuss flux compactifications of IIA string theory on G2 holonomy spaces with O2/O6-planes to three dimensions and find two classes of solutions: (1) No-scale Minkowski vacua from NSNS 3-form fluxes and RR 4-form fluxes. (2) By adding Romans mass we find AdS_3 vacua for which the AdS scale can be decoupled completely from the KK scale while the solution is at tunable weak coupling and large volume. For the AdS_3 vacuum we only have a proper 3D description (i.e. smeared orientifold description) of the solution like the 4D analogue of IIA moduli stabilization from fluxes and O6-planes. This 3D description reveals that moduli with non-compact moduli spaces can be stabilized at the classical level. For both types of vacua we can have supersymmetry or not.

M-theoretic Lichnerowicz formula and supersymmetry

A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric operators to the effective action, the Bianchi identities for fluxes, and some equations of motion. Recently, such formulae have also been shown to underlie the (generalised) geometry of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that describes eleven-dimensional supergravity together with its higherderivative couplings. The first corrections to the action appear at eight-derivative level, and the construction yields two different supersymmetric invariants, each with a free coefficient. We discuss the restriction of our construction to seven-dimensional internal spaces, and implications for compactifications on manifolds of G2 holonomy. Inclusion of fluxes and computation of contributions with higher than eight derivatives are also discussed.

Compact G2 holonomy spaces from SU(3) structures

We construct novel classes of compact G2 spaces from lifting type IIA flux backgrounds with O6 planes. There exists an extension of IIA Calabi-Yau orientifolds for which some of the D6 branes (required to solve the RR tadpole) are dissolved in F2 fluxes. The backreaction of these fluxes deforms the Calabi-Yau manifold into a specific class of SU(3)-structure manifolds. The lift to M-theory again defines compact G2 manifolds, which in case of toroidal orbifolds are a twisted generalisation of the Joyce construction. This observation also allows a clear identification of the moduli space of a warped compactification with fluxes. We provide a few explicit examples, of which some can be constructed from T-dualising known IIB orientifolds with fluxes. Finally we discuss supersymmetry breaking in this context and suggest that the purely geometric picture in M-theory could provide a simpler setting to address some of the consistency issues of moduli stabilisation and de Sitter uplifting.

Superconformal algebras for twisted connected sums and G2 mirror symmetry

We realise the Shatashvili-Vafa superconformal algebra for G2 string compactifications by combining Odake and free conformal algebras following closely the recent mathematical construction of twisted connected sum G2 holonomy manifolds. By considering automorphisms of this realisation, we identify stringy analogues of two mirror maps proposed by Braun and Del Zotto for these manifolds.

G2 Holonomy Manifolds are Superconformal

We study the chiral de Rham complex (CDR) over a manifold M with holonomy G2. We prove that the vertex algebra of global sections of the CDR associated to M contains two commuting copies of the Shatashvili-Vafa G2 superconformal algebra. Our proof is a tour de force, based on explicit computations.

Infinitely many M2-instanton corrections to M-theory on G2-manifolds

We consider the non-perturbative superpotential for a class of four-dimensional mathcal{N}=1 vacua obtained from M-theory on seven-manifolds with holonomy G2. The class of G2-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over S3. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of [1], which relates M-theory on TCS G2-manifolds to E8 × E8 heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the G2-compactification.

Framed M-branes, corners, and topological invariants

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.

A 4D gravity theory and $G_2$-holonomy manifolds

Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bundle. We make essentially the same ansatz for the calibrating 3-form, but use the curvature 2-forms instead of the ASD ones. We show that the resulting 3-form defines a metric of G2 holonomy if the connection satisfies a certain second-order PDE. This is exactly the same PDE that arises as the field equation of a certain 4-dimensional gravity theory formulated as a diffeomorphism-invariant theory of SO(3) connections. Thus, every solution of this 4-dimensional gravity theory can be lifted to a G2-holonomy metric. Unlike all previously known constructions, the theory that we lift to 7 dimensions is not topological. Thus, our construction should give rise to many new metrics of G2 holonomy. We describe several examples that are of cohomogeneity one on the base.

Mirror symmetry for G2-manifolds: twisted connected sums and dual tops

Recently, at least 50 million of novel examples of compact G2 holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on G2 manifolds obtained from building blocks constructed from dual pairs of tops, which are the closest to toric CY hypersurfaces, and formulate the analogue of the Batyrev mirror map for this class of G2 holonomy manifolds, thus obtaining several millions of novel dual superstring backgrounds. In particular, this leads us to conjecture a plethora of novel exact dualities among the corresponding 2d mathcal{N} = 1 sigma models.

Self-duality in higher dimensions

Let ω be a 2-form on a 2n dimensional manifold. In previous work, we called ω “strong self-dual, if the eigenvalues of its matrix with respect to an orthonormal frame are equal in absolute value. In a series of papers, we showed that strong self-duality agrees with previous definitions; in particular if ω is strong self-dual, then, in 2n dimensions, ωn is proportional to its Hodge dual ω and in 4n dimensions, ωn is Hodge self-dual. We also obtained a local expression of the Bonan 4-form on 8 manifolds with Spin7 holonomy, as the sum of the squares of any orthonormal basis of a maximal linear subspace of strong self-dual 2-forms. In the present work we generalize the notion of strong self-duality to odd dimensional manifolds and we express the dual of the Fundamental 3-form 7 manifolds with G2 holonomy, as a sum of the squares of an orthonormal basis of a maximal linear subspace of strong self-dual 2-forms.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carri\'on and Fern\'andez and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the instanton condition. The latter piece is contained in $H^1_{\check{d}_\theta}(Y,TY)$, and is given by the kernel of a map ${\cal\check F}$ which generalises the concept of the Atiyah map for holomorphic bundles on complex manifolds to the case at hand. In fact, the map ${\cal\check F}$ is given in terms of the curvature of the bundle and maps $H^1_{\check{d}_\theta}(Y,TY)$ into $H^2_{\check{d}_A}(Y,\mathrm{End}(V))$, and moreover can be used to define a cohomology on an extension bundle of $TY$ by $\mathrm{End}(V)$. We comment further on the resemblance with the holomorphic Atiyah algebroid and connect the story to physics, in particular to heterotic compactifications on $(Y,V)$ when $\alpha'=0$.

On aspects of holographic thermal QCD at finite coupling

In the context of string theoretic dual of thermal QCD-like theories at finite gauge/string coupling of [1] (as part of the ‘MQGP’ limit of [2]), we obtain the QCD deconfinement temperature compatible with lattice results for the right number of light flavors Nf=3, and the correct mass scale of the light (first generation) quarks. The type IIB background of [1] is also shown to be thermodynamically stable. Further, we show that the temperature dependence of DC electrical conductivity mimics a one-dimensional Luttinger liquid, and the requirement of the Einstein relation (ratio of electrical conductivity and charge susceptibility equal to the diffusion constant) to be satisfied requires a specific dependence of the Ouyang embedding parameter on the horizon radius. These results arise due to the non-Kählerity and non-conformality of the type IIB background. On the geometrical side we quantify the former (non-Kählerity) by evaluating the SU(3)/G2-structure torsion classes of the local type IIA mirror/M-theory uplift. Analogous to what was shown for the type IIB background in [5], we first show that the type IIA delocalized SYZ mirror (after fine tuning) can also be approximately supersymmetric. We then work out the G2-structure torsion classes of the local M-theory uplift of the mirror type IIA metric – in the large-N limit at finite coupling, G2 structure approaches G2 holonomy.