Introduction to $$\mathrm {G}_2$$ Geometry

Karigiannis, Spiro

doi:10.1007/978-1-0716-0577-6_1

Cited by 12 publications

(16 citation statements)

References 17 publications

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“…We need to compute the integrand (Tr gt h t )vol (M,gt) = (g t ) ab (h t ) ab vol (M,gt) of (2). Using (13) and our adapted frame we compute…”

Section: Almost Complex Submanifolds Of Manifolds With U(m)-structurementioning

confidence: 99%

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A variational characterization of calibrated submanifolds

Cheng¹,

Karigiannis²,

Madnick³

2022

Preprint

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Let M be a fixed compact oriented embedded submanifold of a manifold M . Consider the volume V(g) = M vol (M,g) as a functional of the ambient metric g on M , where g = g| M . We show that g is a critical point of V with respect to a special class of variations of g, obtained by varying a calibration µ on M in a particular way, if and only if M is calibrated by µ. We do not assume that the calibration is closed. We prove this for almost complex, associative, coassociative, and Cayley calibrations, generalizing earlier work of Arezzo-Sun in the almost Kähler case. The Cayley case turns out to be particularly interesting, as it behaves quite differently from the others. We also apply these results to obtain a variational characterization of Smith maps.

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“…We need to compute the integrand (Tr gt h t )vol (M,gt) = (g t ) ab (h t ) ab vol (M,gt) of (2). Using (13) and our adapted frame we compute…”

Section: Almost Complex Submanifolds Of Manifolds With U(m)-structurementioning

confidence: 99%

“…It also induces an orientation, and thus we have a Hodge dual 4-form ψ = ⋆ϕ. See [4,9,10,13] for more about G 2 -structures. We do not assume anything about the torsion ∇ϕ of the G 2 -structure.…”

Section: Generalities On Manifolds With G 2 -Structure and Their Dist...mentioning

confidence: 99%

A variational characterization of calibrated submanifolds

Cheng¹,

Karigiannis²,

Madnick³

2022

Preprint

Self Cite

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“…These are the same conventions as [14] but opposite to [15]. Dually, again using only the incidence geometry of F and the composition factor ǫ, we can define the four-form…”

Section: Remark 32mentioning

confidence: 99%

Commutation relations of $\mathfrak g\_2$ and the incidence geometry of the Fano plane

de Traubenberg,

Slupinski

2022

Preprint

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We continue our study and classification of structures on the Fano plane F and its dual F * involved in the construction of octonions and the Lie algebra g 2 (F) over a field F. These are a "composition factor" ǫ : F × F → {−1, 1},hhhh inducing an octonion multiplication, and a function δ * : Aut(F) × F * → {−1, 1} such that g ∈ Aut(F) can be lifted to an automorphism of the octonions iff δ * (g, •) is the Radon transform of a function on F. We lift the action of Aut(F) on F to the action of a non-trivial eight-fold covering Aut( Fǫ ) on a two-fold covering Fǫ of F contained in the octonions. This extends tautologically to an action on the octonions by automorphism. Finally, we associate to incident point-line pairs a generating set of g 2 (F) and express brackets in terms of the incidence geometry of F and ǫ.

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“…Type-H: The natural 3-form extending (3.11) This is indeed the result of (6.5) with the supercharge preserving all Type-H surfaces (3.11). This form is closed, so defines a calibration, and as in the Type-C above (6.10), this is a G 2 -structure [76][77][78][79] (see also [80] for a gentle introduction). Let us prove this implies that the first order equation (6.6) is satisfied.…”

Section: Holographymentioning

confidence: 99%

Observations on BPS observables in 6d

Drukker,

Trépanier

2020

Preprint

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We study possible geometries and R-symmetry breaking patterns that lead to globally BPS surface operators in the six dimensional N = (2, 0) theory. We find four main classes of solutions in different subspaces of R 6 and a multitude of subclasses and specific examples. We prove that these constructions lead to supersymmety preserving observables and count the number of preserved supercharges. We discuss the underlying geometry, calculate their anomalies and present analogous structures for the holographic dual M2-branes in AdS 7 × S 4 . We also comment on the dimensional reduction of these observables to line and surface operators in 4d and 5d theories. This rich spectrum of operators are proposed as the simplest and most natural observables of this mysterious theory.

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Introduction to $$\mathrm {G}_2$$ Geometry

Cited by 12 publications

References 17 publications

A variational characterization of calibrated submanifolds

A variational characterization of calibrated submanifolds

Commutation relations of $\mathfrak g\_2$ and the incidence geometry of the Fano plane

Observations on BPS observables in 6d

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