Special Holonomy Research Articles

Cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

Coulomb and Higgs phases of G2-manifolds

Ricci flat manifolds of special holonomy are a rich framework as models of the extra dimensions in string/M-theory. At special points in vacuum moduli space, special kinds of singularities occur and demand a physical interpretation. In this paper we show that the topologically distinct G2-holonomy manifolds arising from desingularisations of codimension four orbifold singularities due to Joyce and Karigiannis correspond physically to Coulomb and Higgs phases of four dimensional gauge theories. The results suggest generalisations of the Joyce-Karigiannis construction to arbitrary ADE-singularities and higher order twists which we explore in detail in explicitly solvable local models. These models allow us to derive an isomorphism between moduli spaces of Ricci flat metrics on these non-compact G2-manifolds and flat ADE-connections on compact flat 3-manifolds which we establish explicitly for SU(n).

Smooth Maps Minimizing the Energy and the Calibrated Geometry

We generalize the notion of calibrated submanifolds to smooth maps and show that several kinds of smooth maps appearing in the differential geometry are applicable to our situation. Moreover, we apply this notion to give the lower bound to some energy functionals of smooth maps in the given homotopy class between Riemannian manifolds and consider the energy functional which is minimized by the identity maps on the Riemannian manifolds with special holonomy groups.

On the formality problem for manifolds with special holonomy

In ??1 and 2 we follow our online talk at the 21st Geometrical Seminar (Beograd, Serbia) on June 30, 2022 by giving a survey of the formality problem for manifold with special holonomy and exposing recent results by M. Amann and the author on the formality of Joyce?s examples of G2-manifolds. In ?3 we expose the approach to establishing the formality by using the intersection Massey products.

Geometric Structures on $$G_2$$ Manifolds

One promising direction for future investigation is the study of almost contact structures on \(G_2\) manifolds. In this paper, we review the properties of special holonomy geometries and almost contact structures. We show that every closed, oriented, smooth 3-manifold has a trivial normal bundle inside a Riemannian 7 manifold with spin structure. We also study the 3-dimensional associative and Harvey–Lawson submanifolds of \(G_2\) manifolds and show that their almost contact structures can be extended to the ambient space.

Mirror of volume functionals on manifolds with special holonomy

We can define the “volume” V for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold X, which can be considered to be the “mirror” of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics.In this paper, (1) we introduce the negative gradient flow of V, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When X is Kähler, we relate the negative gradient of V to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau.(2) We relate the functional V to a deformed Hermitian Yang–Mills (dHYM) connection, a deformed Donaldson–Thomas connection for a G2-manifold (a G2-dDT connection), a deformed Donaldson–Thomas connection for a Spin(7)-manifold (a Spin(7)-dDT connection), which are considered to be the “mirror” of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When X is a compact Spin(7)-manifold, we prove the “mirror” of the Cayley equality, which implies the following. (a) Any Spin(7)-dDT connection is a global minimizer of V and its value is topological. (b) Any Spin(7)-dDT connection is flat on a flat line bundle. (c) If X is a product of S1 and a compact G2-manifold Y, any Spin(7)-dDT connection on the pullback of the Hermitian complex line bundle over Y is the pullback of a G2-dDT connection modulo closed 1-forms.We also prove analogous statements for G2-manifolds and Kähler manifolds of dimension 3 or 4.

Tachibana-type theorems and special Holonomy

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and -nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence-free Bochner tensors. For quaternion Kähler manifolds, we obtain a partial result towards the LeBrun–Salamon conjecture.

Topological G2 and Spin(7) strings at 1-loop from double complexes

We study the topological G2 and Spin(7) strings at 1-loop. We define new double complexes for supersymmetric NSNS backgrounds of string theory using generalised geometry. The 1-loop partition function then has a target-space interpretation as a particular alternating product of determinants of Laplacians, which we have dubbed the analytic torsion. In the case without flux where these backgrounds have special holonomy, we reproduce the worldsheet calculation of the G2 string and give a new prediction for the Spin(7) string. We also comment on connections with topological strings on Calabi-Yau and K3 backgrounds.

Stringy tachyonic instabilities of non-supersymmetric Ricci flat backgrounds

Superstring/M-theory compactified on compact Ricci flat manifolds have recently been conjectured to exhibit instabilities whenever the metrics do not have special holonomy. We use worldsheet conformal field theory to investigate instabilities of Type II superstring theories on compact, Ricci flat, spin 3-manifolds including a worldsheet description of their spin structures. The instabilities are signalled by the appearance of stringy tachyons at small radius and a negative (1-loop) vacuum energy density at large radius. We briefly discuss the extension to higher dimensions.

Global perturbation potential function on complete special holonomy manifolds

In this article, we introduce and study the notion of a complete special holonomy manifold $(X,\omega)$ which is given by a global perturbation potential function, i.e., there is a function $f$ on $X$ such that $\omega'=\omega-\mathcal{L}_{\nabla f}\omega$ is sufficiently small in $L^{\infty}$-norm. We establish some vanishing theorems on the $L^{2}$ harmonic forms under some conditions on the global perturbation potential function.

Yang-Mills flow on special-holonomy manifolds

This paper develops Yang-Mills flow on Riemannian manifolds with special holonomy. By analogy with the second-named author's thesis, we find that a supremum bound on a certain curvature component is sufficient to rule out finite-time singularities. Assuming such a bound, we prove that the infinite-time bubbling set is calibrated by the defining (n−4)-form.

Supersymmetry, Ricci flat manifolds and the String Landscape

A longstanding question in superstring/M theory is does it predict supersymmetry below the string scale? We formulate and discuss a necessary condition for this to be true; this is the mathematical conjecture that all stable, compact Ricci flat manifolds have special holonomy in dimensions below eleven. Almost equivalent is the proposal that the landscape of all geometric, stable, string/M theory compactifications to Minkowski spacetime (at leading order) are supersymmetric. For simply connected manifolds, we collect together a number of physically relevant mathematical results, emphasising some key outstanding problems and perhaps less well known results. For non-simply connected, non-supersymmetric Ricci flat manifolds we demonstrate that many cases suffer from generalised Witten bubble of nothing instabilities.

Hodge theory of SKT manifolds

We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holonomy group causes a decomposition of the twisted cohomology. For SKT manifolds this decomposition is accompanied by an identity between different Laplacian operators and equates different cohomologies defined in terms of the SKT structure. We illustrate our theory with examples based on Calabi–Eckmann manifolds, instantons, Hopf surfaces and Lie groups.

Special holonomy manifolds, domain walls, intersecting branes and T-folds

We discuss the special holonomy metrics of Gibbons, Lu, Pope and Stelle, which were constructed as nilmanifold bundles over a line by uplifting supersymmetric domain wall solutions of supergravity to 11 dimensions. We show that these are dual to intersecting brane solutions, and considering these leads us to a more general class of special holonomy metrics. Further dualities relate these to non-geometric backgrounds involving intersections of branes and exotic branes. We discuss the possibility of resolving these spaces to give smooth special holonomy manifolds.

Self-duality and associated parallel or cocalibrated G_2 structures

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional geometry. These are applied to the Bryant-Salamon cons\-truction of complete $\mathrm{G}_2$-holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on $M$. We then discover new examples of that special holonomy on disk bundles over ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$, respectively, the real and complex hyperbolic space. Only in the end we present the new $\mathrm{G}_2$ structures on principal bundles.

Wrapped brane solutions in romans F(4) gauged supergravity

We explore the spectrum of lower-dimensional anti-de Sitter (AdS) solutions in F(4) gauged supergravity in six dimensions. The ansatz employed corresponds to D4-branes partially wrapped on various supersymmetric cycles in special holonomy manifolds. Re-visiting and extending previous results, we study the cases of two, three, and four-dimensional supersymmetric cycles within Calabi-Yau threefold, fourfold, G2, and Spin(7) holonomy manifolds. We also report on non-supersymmetric AdS vacua, and check their stability in the consistently truncated lower-dimensional effective action, using the Breitenlohner-Freedman bound. We also analyze the IR behavior and discuss the admissibility of singular flows.

Unitarization from geometry

We study the perturbative unitarity of scattering amplitudes in general dimensional reductions of Yang-Mills theories and general relativity on closed internal manifolds. For the tree amplitudes of the dimensionally reduced theory to have the expected high-energy behavior of the higher-dimensional theory, the masses and cubic couplings of the Kaluza-Klein states must satisfy certain sum rules that ensure there are nontrivial cancellations between Feynman diagrams. These sum rules give constraints on the spectra and triple overlap integrals of eigenfunctions of Laplacian operators on the internal manifold and can be proven directly using Hodge and eigenfunction decompositions. One consequence of these constraints is that there is an upper bound on the ratio of consecutive eigenvalues of the scalar Laplacian on closed Ricci-flat manifolds with special holonomy. This gives a sharp bound on the allowed gaps between Kaluza-Klein excitations of the graviton that also applies to Calabi-Yau compactifications of string theory.

The doubled geometry of nilmanifold reductions

A class of special holonomy spaces arise as nilmanifolds fibred over a line interval and are dual to intersecting brane solutions of string theory. Further dualities relate these to T-folds, exotic branes, essentially doubled spaces and spaces with R-flux. We develop the doubled geometry of these spaces, with the various duals arising as different slices of the doubled space.

Spinorially Twisted Spin Structures. II: Twisted Pure Spinors, Special Riemannian Holonomy and Clifford Monopoles

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special K\"ahler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also introduce the Clifford monopole equations as a natural geometric generalization of the Seiberg-Witten equations. We show that they restrict to the Seiberg-Witten equations in 4 dimensions, and that they admit non-trivial solutions on manifolds with special Riemannian holonomy.

Ricci flow, Killing spinors, and T-duality in generalized geometry

We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the dilaton shift in string theory and prove that the Hull-Strominger system is preserved by T-duality.