G-structures Research Articles

A type of nearly parallel G-structures

We study properties of a certain symmetric tensor r induced by the intrinsic torsion of a Riemannian G–structure. This tensor naturally arises in the context of nearly Kähler manifolds and is parallel with respect to the canonical Hermitian connection. In general, we call a G-structure a second order parallel if ∇Gr=0 for a minimal G–connection ∇G. We show correlation with harmonicity of a G–structure and with G–structures with parallel torsion. An example of second order parallel G–structure is, apart from nearly Kähler manifolds and nearly parallel G2 structures, an α-Kenmotsu manifold.

N = (2, 0) AdS3 solutions of M-theory

We consider the most general solutions of eleven-dimensional supergravity preserving N = 2 supersymmetry whose metrics are warped products of three-dimensional anti-de Sitter space with an eight-dimensional manifold, focusing on those realising (2,0) superconformal symmetry. We give a set of necessary and sufficient conditions for a solution to be supersymmetric, which can be phrased, in the general case, in terms of a local SU(2) structure and its intrinsic torsion. We show that these supergravity backgrounds always admit a nowhere-vanishing Killing vector field that preserves the solution and encodes the U(1) R-symmetry of the dual field theory. We illustrate our results with examples which have appeared in the literature, including those with SU(4), G2 and SU(3) structures, and discuss new classes of Minkowski solutions.

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.

Deformations of nearly G2 structures

We describe the second order obstruction to deformation for nearly G 2 structures on compact manifolds. Building on work of Alexandrov and Semmelmann, this allows proving rigidity under deformation for the proper nearly G 2 structure on the Aloff–Wallach space N ( 1 , 1 ) .

Three-dimensional flux vacua from IIB on co-calibrated G2 orientifolds

We derive the 3D N = 1 superpotential for the closed string sector of type IIB supergravity on toroidal O5 orientifolds with co-calibrated G2 structure and RR background flux. We find that such compactifications can provide full closed string moduli stabilization on supersymmetric hbox {AdS}_3 vacua, and once we include brane-supersymmetry-breaking we also find indication for the existence of classical 3D de Sitter solutions. The latter however are rather difficult to reconcile with the “shape” moduli stabilization and flux quantization. We also discuss the possibility of achieving scale separation in hbox {AdS}_3 and hbox {dS}_3 vacua, but such effects seem to be hindered by the geometric flux quantization.

On the vortex filament in 3-spaces and its generalizations

In this article, we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations. We shall present some effective geometric tools applied in the study, such as the Schrodinger flow, the geometric Korteweg-de Vries (KdV) flow and the generalized bi-Schrodinger flow, as well as the complex and para-complex structures. It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating, since it relates to almost complex structures and the G2 structure on $${\mathbb{S}^6}$$ . As a new result in this survey, we describe the equation of generalized bi-Schrodinger flows from ℝ1 into a Riemannian surface.

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.

High-resolution crystal structure of gelsolin domain 2 in complex with the physiological calcium ion

The second domain of gelsolin (G2) hosts mutations responsible for a hereditary form of amyloidosis. The active form of gelsolin is Ca2+-bound; it is also a dynamic protein, hence structural biologists often rely on the study of the isolated G2. However, the wild type G2 structure that have been used so far in comparative studies is bound to a crystallographic Cd2+, in lieu of the physiological calcium. Here, we report the wild type structure of G2 in complex with Ca2+ highlighting subtle ion-dependent differences. Previous findings on different G2 mutations are also briefly revised in light of these results.

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

We develop some consequences of the connection between Calabi–Yau structures and torsion-free G2 structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi–Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi–Yau structures on a six-dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi–Yau threefolds, extended between these moduli spaces, is a local diffeomorphism: that is, that every deformation of a glued Calabi–Yau threefold arises from an essentially unique deformation of the asymptotically cylindrical pieces.

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.

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.

On the Curvature Properties of Manifolds with Deformed G2 Structures

In this paper, we study the curvature properties of a manifold with structure group G2 whose fundamental 3-form is deformed by a Killing vector field of unit length. We obtain some results concerning conditions under which this manifold is flat, Einstein, or isometric to the unit sphere.

Shi-type estimates and finite time singularities of flows of G2 structures

In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $\kappa$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.

The Infinitesimal Moduli Space of Heterotic G2 Systems

Heterotic string compactifications on integrable G2 structure manifolds Y with instanton bundles {(V,A), (TY,tilde{theta})} yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative {mathcal{D}} and show that it is equivalent to a heterotic G2 system encoding the geometry of the heterotic string compactifications. This operator {mathcal{D}} acts on a bundle {mathcal{Q}=T^*Y oplus {rm End}(V) oplus {rm End}(TY)} and satisfies a nilpotency condition {check{{mathcal{D}}}^2=0} , for an appropriate projection of {mathcal D}. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group {check H^1_{check{{mathcal{D}}}}(mathcal{Q})}. We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the {alpha'} expansion.

Eleven-dimensional supergravity in 4D, N = 1 superspace

We give a formulation of linearized 11D supergravity in 4D, N = 1 superspace keeping all eleven bosonic coordinates. The fields are fluctuations around M = R4|4 × Y, where Y is a background Riemannian 7-manifold admitting a G2 structure. We embed the 11D fields into superfield representations of the 4D, N = 1 superconformal algebra. These consist of the conformal graviton superfield, seven conformal gravitino superfields, a tensor hierarchy of superfields describing the 11D 3-form, and a non-abelian Kaluza-Klein vector multiplet gauging the tensor hierarchy by diffeomorphisms on Y. The quadratic action consists of the linearization of a superspace volume term and a Chern-Simons action for the gauged hierarchy coupled to the supergravity and gravitino superfields, and the full structure is fixed by superconformal and gauge invariance. When this action is projected to components, we recover the full linearized action of 11D supergravity.

Noether's theorem in multisymplectic geometry

We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multi-Hamiltonian system. We show that a homotopy co-momentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry.We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. We also apply our results to manifolds with a torsion-free G2 structure.

Accessing the nucleon transverse structure in inclusive deep inelastic scattering

We revisit the standard analysis of inclusive Deep Inelastic Scattering off nucleons taking into account the fact that on-shell quarks cannot be present in the final state, but they rather decay into hadrons – a process that can be described in terms of suitable “jet” correlators. As a consequence, a spin-flip term associated with the invariant mass of the produced hadrons is generated nonperturbatively and couples to the target's transversity distribution function. In inclusive cross sections, this provides an hitherto neglected and large contribution to the twist-3 part of the g2 structure function, that can explain the discrepancy between recent calculations and fits of this quantity. It also provides an extension of the Burkhardt–Cottingham sum rule, providing new information on the transversity function, as well as an extension of the Efremov–Teryaev–Leader sum rule, suggesting a novel way to measure the tensor charge of the proton.

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound onΛ(x,t)=|∇T(x,t)|g(t)2+|Rm(x,t)|g(t)212\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Lambda(x,t)=\\left(|\\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\\right)^{\\frac 12}$$\\end{document}will imply bounds on all covariant derivatives of Rm and T. (2). We show that {Lambda(x,t)} will blow up at a finite-time singularity, so the flow will exist as long as {Lambda(x,t)} remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

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.

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

In this paper, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost 3-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the 3-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact (qc) structure of the quaternionic Heisenberg groups. We focus on the 7-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated G2 structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.