it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio...

manifold G2 manifold Kähler manifold Calabi–Yau manifold Hyperkähler manifold Quaternionic Kähler manifold Riemannian symmetric space Spin(7) manifold The Wikibook...

uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact...

K3 surface is Kähler (by Siu). Almost complex manifold Hyperkähler manifold Quaternion-Kähler manifold K-energy functional Cannas da Silva (2001), Definition...

differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness...

super-space extensions, was developed by J. Harnad and S. Shnider. Hyperkähler manifolds of dimension 4 k {\displaystyle 4k} also admit a twistor correspondence...

describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold  T. The non-linear σ-model was introduced by Gell-Mann &...

A hyperkähler manifold is a quaternionic manifold with a torsion-free Sp ⁡ ( n ) {\displaystyle \operatorname {Sp} (n)} -structure. A hyperkähler manifold...

compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of...

self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise. Higher-dimensional...

complex Lagrangian submanifolds of hyperkähler manifolds, fixed points of a real structure of Calabi–Yau manifolds. The SYZ conjecture deals with the...

In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind...

mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group G is the quotient of a fiber of a hyperkähler moment map M → g...

Roger; Dancer, Andrew S. (2000), "The geometry and topology of toric hyperkähler manifolds" (PDF), Communications in Analysis and Geometry, 8 (4): 727–760...

Forms and Geometric Invariants," from which the theory arose. Given a manifold and a Lie algebra valued 1-form A {\displaystyle \mathbf {A} } over it...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

dimensions spacetime is folded up to form a small torus or other compact manifold. This would leave only the familiar four dimensions of spacetime visible...

and Andrew Neitzke, which used physical ideas to derive new results in hyperkähler geometry. Another realization of the AdS/CFT correspondence states that...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

SO(4n), so every hyperkähler manifold is a Calabi–Yau manifold, every Calabi–Yau manifold is a Kähler manifold, and every Kähler manifold is orientable....

}_{2}} . Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by...

algebraic surfaces, and via the ADHM construction, or hyperkähler reduction (see hyperkähler manifold), a geometric invariant theory procedure. The groundbreaking...

sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of a completely different Calabi–Yau manifold. This equivalence provides...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...

Calabi–Yau manifold into simpler pieces and considering the effects of T-duality on these pieces. The simplest example of a Calabi–Yau manifold is a torus...

compact extra dimensions must be shaped like a Calabi–Yau manifold. A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional...

the form of a Calabi–Yau manifold. Within the more complete framework of M-theory, they would have to take form of a G2 manifold. A particular exact symmetry...

not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Every hyperkähler manifold is also hypercomplex. The converse...

between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are...

Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold...