G2 Manifolds Research Articles

G2 mirrors from Calabi-Yau mirrors

We study the worldsheet CFTs of type II strings on compact G2 orbifolds obtained as quotients of a product of a Calabi-Yau threefold and a circle. For such models, we argue that the Calabi-Yau mirror map implies a mirror map for the associated G2 varieties by examining how anti-holomorphic involutions behave under Calabi-Yau mirror symmetry. The mirror geometries identified by the worldsheet CFT are consistent with earlier proposals for twisted connected sum G2 manifolds.

Electric-magnetic duality in a class of G2-compactifications of M-theory

We study electric-magnetic duality in compactifications of M-theory on twisted connected sum (TCS) G2 manifolds via duality with F-theory. Specifically, we study the physics of the D3-branes in F-theory compactified on a Calabi-Yau fourfold Y, dual to a compactification of M-theory on a TCS G2 manifold X. mathcal{N} = 2 supersymmetry is restored in an appropriate geometric limit. In that limit, we demonstrate that the dual of D3-branes probing seven-branes corresponds to the shrinking of certain surfaces and curves, yielding light particles that may carry both electric and magnetic charges. We provide evidence that the Minahan-Nemeschansky theories with En flavor symmetry may be realized in this way. The SL(2, ℤ) monodromy of the 3/7-brane system is dual to a Fourier-Mukai transform of the dual IIA/M-theory geometry in this limit, and we extrapolate this monodromy action to the global compactification. Away from the limit, the theory is broken to mathcal{N} = 1 supersymmetry by a D-term.

Quantum gravity bounds on mathcal{N} = 1 effective theories in four dimensions

We propose quantum gravitational constraints on effective four-dimensional theories with mathcal{N} = 1 supersymmetry. These Swampland constraints arise by demanding consistency of the worldsheet theory of a class of axionic, or EFT, strings whose existence follows from the Completeness Conjecture of quantum gravity. Modulo certain assumptions, we derive positivity bounds and quantization conditions for the axionic couplings to the gauge and gravitational sector at the two- and four-derivative level, respectively. We furthermore obtain general bounds on the rank of the gauge sector in terms of the gravitational couplings to the axions. We exemplify how these bounds rule out otherwise consistent effective supergravity theories as theories of quantum gravity. Our derivations of the quantum gravity bounds are tested and further motivated in concrete string theoretic settings. In particular, this leads to a sharper version of the bound on the gauge group rank in F-theory on elliptic four-folds with a smooth base, which improves the known geometrical Kodaira bounds. We furthermore provide a detailed derivation of the EFT string constraints in heterotic string compactifications including higher derivative corrections to the effective action and apply the bounds to M-theory compactifications on G2 manifolds.

Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds

A natural approach to the construction of nearly G_2 manifolds lies in resolving nearly G_2 spaces with isolated conical singularities by gluing in asymptotically conical G_2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G_2 manifolds, whose endpoint is the original nearly G_2 conifold and whose parameter is the scale of the glued in asymptotically conical G_2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G_2 manifolds: if the rate of the metric is below -3, then the G_2 4-form is exact if and only if the manifold is Euclidean mathbb R^7. A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below -3, then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean mathbb R^6.

A compact non‐formal closed G2 manifold with b1=1$b_1=1$

AbstractWe construct a compact manifold with a closed G2 structure not admitting any torsion‐free G2 structure, which is non‐formal and has first Betti number . We develop a method of resolution for orbifolds that arise as a quotient with M a closed G2 manifold under the assumption that the singular locus carries a nowhere‐vanishing closed 1‐form.

Linear instability of Sasaki Einstein and nearly parallel G2 manifolds

In this paper, we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel [Formula: see text] manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly, we prove that nearly parallel [Formula: see text] manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space [Formula: see text] which is a [Formula: see text]-dimensional homology sphere with a proper nearly parallel [Formula: see text] structure.

M-theory on G2 Manifolds Moduli to Phenomenology

M-theory on G2 Manifolds Moduli to Phenomenology

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.

On TCS G2 manifolds and 4D emergent strings

In this note, we study the Swampland Distance Conjecture in TCS G2 manifold compactifications of M-theory. In particular, we are interested in testing a refined version — the Emergent String Conjecture, in settings with 4d N = 1 supersymmetry. We find that a weakly coupled, tensionless fundamental heterotic string does emerge at the infinite distance limit characterized by shrinking the K3-fiber in a TCS G2 manifold. Such a fundamental tensionless string leads to the parametrically leading infinite tower of asymptotically massless states, which is in line with the Emergent String Conjecture. The tensionless string, however, receives quantum corrections. We check that these quantum corrections do modify the volume of the shrinking K3-fiber via string duality and hence make the string regain a non-vanishing tension at the quantum level, leading to a decompactification. Geometrically, the quantum corrections modify the metric of the classical moduli space and are expected to obstruct the infinite distance limit. We also comment on another possible type of infinite distance limit in TCS G2 compactifications, which might lead to a weakly coupled fundamental type II string theory.

M-theory and orientifolds

We construct the M-Theory lifts of type IIA orientifolds based on K3-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum G2 manifolds. Beautifully, the two building blocks forming the G2 manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of G2 manifolds with different inequivalent TCS realizations.

G2 holonomy, Taubes\u2019 construction of Seiberg-Witten invariants and superconducting vortices

Using a reformulation of topological mathcal{N} = 2 QFT’s in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a G2 manifold constructed from the space of self-dual 2-forms over a four-fold X, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes’ construction of the Seiberg-Witten invariants when X is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT’s arising from mathcal{N} = 2 QFT’s from all Gaiotto theories on arbitrary 4-manifolds.

Superpotential of three dimensional mathcal{N} = 1 heterotic supergravity

We dimensionally reduce the ten dimensional heterotic supergravity action on spacetimes of the form mathcal{M} (1, 2) x Y, where mathcal{M} (1, 2) is three dimensional maximally symmetric Anti de Sitter or Minkowski space, and Y is a compact seven dimensional manifold with G2 structure. In doing so, we derive the real superpotential functional of the corresponding three dimensional mathcal{N} = 1 theory. We confirm that extrema of this functional correspond to supersymmetric heterotic compacti:fications on manifolds of G2 structure in the large volume, weak coupling limit to first order in a'. We make some comments on the role of the superpotential functional with respect to the coupled moduli problem of instanton bundles over G2 manifolds.

Toric geometry of G2–manifolds

We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

On mirror maps for manifolds of exceptional holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

Lagrangian-type submanifolds of G2 and Spin(7) manifolds and their deformations

Three-dimensional Harvey-Lawson submanifolds were introduced in an earlier paper by Akbulut and Salur [1], as examples of Lagrangian-type manifolds inside a G2 manifold. In this paper, we discuss these as well as two other similar types of submanifolds of G2 and Spin(7) manifolds and their deformations. We first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. We then introduce a new class of Lagrangian-type four-dimensional submanifolds of M, call them RS-submanifolds, and prove that the space of deformations of a smooth, compact, orientable RS-submanifold of a G2 manifold M can be identified with the space of closed 3-forms on RS. Finally, we describe an analogous setting for Spin(7) manifolds by defining a new class of Lagrangian-type four-dimensional submanifolds of a Spin(7) manifold N, which we call L-submanifolds. We show that the space of deformations of a smooth, compact, orientable L-submanifold of N can be identified with the space of closed 3-forms on L.

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.

Einstein Warped G2 and Spin(7) Manifolds

In this paper most of the classes of G2-structures with Einstein induced metric of negative, null or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the torsion forms of the SU(3)-structure and the warping function, which allows to give characterizations of the principal classes of Einstein warped G2 manifolds. Similar results are obtained for Einstein warped Spin(7) manifolds with fiber a G2 manifold.

Minimal hypersurfaces in nearly [formula omitted] manifolds

We study hypersurfaces in a nearly G2 manifold. We define various quantities associated to such a hypersurface using the G2 structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the G2 structure of the ambient manifold, to be nearly Kähler. Then using the nearly G2 structure on the round sphere S7, we prove that for a compact minimal hypersurface M6 of constant scalar curvature in S7 with the shape operator A satisfying |A|2>6, there exists an eigenvalue λ>12 of the Laplace operator on M such that |A|2=λ−6, thus giving the next discrete value of |A|2 greater than 0 and 6, thus generalizing the result of Deshmukh (2010) about nearly Kähler S6.

Topological M-theory, self-dual gravity and the Immirzi parameter

Working in the first order formalism of gravity, we propose an action that combines the self and (anti)self-dual parts of the curvature and comprises all the diffeomorphism invariant Lagrangians that one can consider in this formalism. We use this action to derive the (2+1)-dimensional version of the Immirzi parameter. Our derivation relates explicitly the Immirzi parameter to the existence of two classically equivalent actions for the description of gravity in (2+1) dimensions, namely the standard and exotic actions introduced by Witten in the description of (2+1) gravity as a gauge theory. Furthermore, we identify the two decompositions of the seven dimensional manifold of topological M-theory that lead to three dimensional Chern–Simons actions which are the building blocks of several gravitational theories, including the standard and exotic actions, as well as the action for topologically massive AdS gravity. Finally, we motivate the three dimensional Immirzi ambiguity from the structure of topological M-theory. We conclude that this ambiguity originates in the S-duality of the theory, and can be geometrically interpreted as a choice of the volume of the 7 dimensional G2 manifold.