arXiv subjects

Sergey Grigorian

Publications and source records attributed to Sergey Grigorian.

17 recordsLinked to original sources

arXiv2026

Octonionic structure operator and its right spectrum

We study a canonical $G_2$-equivariant operator $h:\mathbb{O}\otimes_{\mathbb{R}}V\to \mathbb{O}\otimes_{\mathbb{R}}V$ defined using only octonion multiplication, where $V$ is the standard $7$-dimensional $G_2$-module. We first compute its ordinary real spectrum using the $G_2$-decomposition of $\mathbb{O}\otimes_{\mathbb{R}}V$. We then analyze the octonionic right-eigenvalue problem $$ h(\widehat w)=\widehat w\lambda, \qquad \lambda\in\mathbb{O}. $$ After fixing a complex slice $\mathbb{R}\oplus\mathbb{R}\widehat u\subset\mathbb{O}$, the problem becomes a real spectral problem for $H_{u,Q}=h-Q R_{\widehat u}$, whose residual symmetry is $\mathrm{SU}(3)$. The resulting $\mathrm{SU}(3)$-block decomposition yields two explicit spectral loci in each slice: a quartic curve and a circle. The equations defining these loci are independent of the slice, and the full right spectrum is obtained by allowing $\widehat u$ to vary over the unit sphere in $\operatorname{Im}\mathbb{O}$.

math.RA↗

arXiv2024

Algebraic structures on parallelizable manifolds

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map $\rho_p:\mathfrak{l} \longrightarrow T_p \mathbb{L}$ for each point $p \in \mathbb{L}$, where $\mathfrak{l}$ is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of $\mathfrak{l}$. Furthermore, flows of these vector fields give rise to a product between elements of $% \mathfrak{l}$ and $\mathbb{L}$, which in turn induces a local loop structure (i.e. a non-associative analog of a group). Furthermore, we also define a generalization of a Lie algebra structure on $\mathfrak{l}$. We will describe the properties and examples of these constructions.

math.RA↗

arXiv2023

The Coulomb gauge in non-associative gauge theory

The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop $\mathbb{L}$, its tangent algebra $\mathfrak{l},$ a finite-dimensional Lie group $\Psi $, that is the pseudoautomorphism group of $\mathbb{L}$, a smooth manifold $M$ with a principal $\Psi $-bundle $\mathcal{P}$, and associated bundles $\mathcal{Q}$ and $\mathcal{A}$ with fibers $\mathbb{L}$ and $\mathfrak{l}$, respectively. A configuration in this theory is defined as a pair $\left( s,\omega \right) $, where $s$ is a section of $\mathbb{Q}$ and $\omega $ is a connection on $\mathcal{P}$. The torsion $T^{\left( s,\omega \right) }$ is the key object in the theory, with a role similar to that of a connection in standard gauge theory. The original motivation for this study comes from $G_{2}$-geometry, and the questions of existence of $G_{2}$-structures with particular torsion types. In particular, given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm.

math.DG↗

arXiv2020

Smooth loops and loop bundles

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Mauer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also consider the critical points of some related functionals. Throughout, we see how some of the known properties of $G_{2}$-structures can be seen from this more general setting.

math.DG↗

arXiv2020

Isometric flows of $G_2$-structures

We survey recent progress in the study of flows of isometric $G_2$-structures on 7-dimensional manifolds, that is, flows that preserve the metric, while modifying the $G_2$-structure. In particular, heat flows of isometric $G_2$-structures have been recently studied from several different perspectives, in particular in terms of $3$-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives.

math.DG↗

arXiv2019

Estimates and monotonicity for a heat flow of isometric G2-structures

Given a $7$-dimensional compact Riemannian manifold $\left( M,g\right) $ that admits $G_{2}$-structure, all the $G_{2}$-structures that are compatible with the metric $g$ are parametrized by unit sections of an octonion bundle over $M$. We define a natural energy functional on unit octonion sections and consider its associated heat flow. The critical points of this functional and flow precisely correspond to $G_{2}$-structures with divergence-free torsion. In this paper, we first derive estimates for derivatives of $V\left( t\right) $ along the flow and prove that the flow exists as long as the torsion remains bounded. We also prove a monotonicity formula and and an $\varepsilon $-regularity result for this flow. Finally, we show that within a metric class of $G_{2}$-structures that contains a torsion-free $G_{2}$-structure, under certain conditions, the flow will converge to a torsion-free $G_{2}$-structure.

math.DG↗

arXiv2018

Flows of co-closed $G_{2}$-structures

We survey recent progress in the study of $G_{2}$-structure Laplacian coflows, that is, heat flows of co-closed $G_{2}$-structures. We introduce the properties of the original Laplacian coflow of $G_{2}$-structures as well as the modified coflow, reviewing short-time existence and uniqueness results for the modified coflow and well as recent Shi-type estimates that apply to a more general class of $G_{2}$-structure flows.

math.DG↗

arXiv2018

G2-structures for N=1 supersymmetric AdS4 solutions of M-theory

We study the N=1 supersymmetric solutions of D=11 supergravity obtained as a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold M. Using the octonion bundle structure on M we reformulate the Killing spinor equations in terms of sections of the octonion bundle on M. The solutions then define a single complexified G2-structure on M or equivalently two real G2-structures. We then study the torsion of these G2-structures and the relationships between them.

hep-th↗

arXiv2017

G2-structures and octonion bundles

We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. We also interpret the choice of a particular G2-structure within the same metric class as a choice of gauge and show that under a change of this gauge, the torsion does transform as an octonion-valued connection 1-form. Finally, we also show an explicit relationship between the octonion bundle and the spinor bundle, define an octonionic Dirac operator and explore an energy functional for octonion sections. We then prove that critical points correspond to divergence-free torsion, which is shown to be an octonionic analog of the Coulomb gauge.

math.DG↗

arXiv2015

Modified Laplacian coflow of $G_{2}$-structures on manifolds with symmetry

We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly Kähler manifold. We show that in these cases the $G_{2}$-structures are determined by their torsion components up to a phase factor. We then study the modified Laplacian coflow $\frac{dψ}{dt}=Δ_{ψ}ψ+2d\left( \left( C-Tr T\right) φ\right) $ of these $G_{2}$-structures, where $φ$ and $ψ$ are the fundamental $3$-form and $4 $-form which define the $G_{2}$-structure and $Δ_{ψ}$ is the Hodge Laplacian associated with the $G_{2}$-structure. This flow is known to have short-time existence and uniqueness. We analyse the soliton equations for this flow and obtain new compact soliton solutions.

math.DG↗

arXiv2013

Short-time behaviour of a modified Laplacian coflow of G2-structures

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $φ$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified flow.

math.DG↗

arXiv2012

G2-structure deformations and warped products

We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimensional and seven-dimensional components (so-called conformally nearly parallel G2-manifolds) to G2-structures with just a one-dimensional torsion component (nearly parallel G2-manifolds). We find that deformations between such structures exist if and only if the metric is a particular warped product metric.

math.DG↗

arXiv2011

Deformations of G2-structures with torsion

We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive an exact expression for the torsion of the deformed G2-structure. We then specialize to a case when the deformation is defined by a vector v and we explicitly derive the expressions for the different torsion components of the new G2-structure in terms of the old torsion components and derivatives of v. In particular this gives a set of differential equations for the vector v which have to be satisfied for a transition between G2-structures with particular torsions. For some specific torsion classes we find that these equations have no solutions.

math.DG↗

arXiv2010

Moduli spaces of G2 manifolds

This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G2 manifolds, local deformation theory of G2 structures and the local geometry of the moduli spaces of G2 structures.

math.DG↗

arXiv2009

Betti numbers of a class of barely G2 manifolds

We calculate explicitly the Betti numbers of a class of barely G2 manifolds - that is, G2 manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces and the involutions are free acting.

math.DG↗

arXiv2008

Local geometry of the G2 moduli space

We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $ϕ$ and compute the expansion of the Hodge star of $ϕ$ to fourth order in the deformations of $ϕ$. By considering M-theory compactified on a G2 manifold, the G2 moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $ϕ$ we work out the full curvature of this metric and relate it to the Yukawa coupling.

hep-th↗

arXiv2007

Minisuperspace Models in M-theory

We derive the full canonical formulation of the bosonic sector of 11-dimensional supergravity, and explicitly present the constraint algebra. We then compactify M-theory on a warped product of homogeneous spaces of constant curvature, and construct a minisuperspace of scale factors. First classical behaviour of the minisuperspace system is analysed, and then a quantum theory is constructed. It turns out that there similarities with the "pre-Big Bang" scenario in String Theory.

hep-th↗