17,888 research outputs found
Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We
observe that the irreducible spinor representation of the Lorentz group Spin(6)
requires us to consider the vector spaces of two-forms and four-forms on an
equal footing. The doubling of the two-form vector space due to the Hodge
duality doubles the variety of six-dimensional spin manifolds. We explore how
the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the
gauge theory formulation of six-dimensional Riemannian manifolds, we show that
the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian
Yang-Mills equations on the Calabi-Yau manifold. Therefore the mirror symmetry
of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian
Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory
perspective.Comment: v5; 49 pages, version to appear in Advances in High Energy Physic
Heterotic-Heterotic String Duality and Multiple K3 Fibrations
A type IIA string compactified on a Calabi-Yau manifold which admits a K3
fibration is believed to be equivalent to a heterotic string in four
dimensions. We study cases where a Calabi-Yau manifold can have more than one
such fibration leading to equivalences between perturbatively inequivalent
heterotic strings. This allows an analysis of an example in six dimensions due
to Duff, Minasian and Witten and enables us to go some way to prove a
conjecture by Kachru and Vafa. The interplay between gauge groups which arise
perturbatively and nonperturbatively is seen clearly in this example. As an
extreme case we discuss a Calabi-Yau manifold which admits an infinite number
of K3 fibrations leading to infinite set of equivalent heterotic strings.Comment: 13 pages, LaTe
D-branes as a Bubbling Calabi-Yau
We prove that the open topological string partition function on a D-brane
configuration in a Calabi-Yau manifold X takes the form of a closed topological
string partition function on a different Calabi-Yau manifold X_b. This
identification shows that the physics of D-branes in an arbitrary background X
of topological string theory can be described either by open+closed string
theory in X or by closed string theory in X_b. The physical interpretation of
the ''bubbling'' Calabi-Yau X_b is as the space obtained by letting the
D-branes in X undergo a geometric transition. This implies, in particular, that
the partition function of closed topological string theory on certain bubbling
Calabi-Yau manifolds are invariants of knots in the three-sphere.Comment: 32 pages; v.2 reference adde
Isometric embeddings of families of special Lagrangian submanifolds
We prove that certain Riemannian manifolds can be isometrically embedded
inside Calabi-Yau manifolds. For example we prove that given any real-analytic
one parameter family of Riemannian metrics on a 3-dimensional manifold
with volume form independent of and with a real-analytic family of
nowhere vanishing harmonic one forms , then can be
realized as a family of special Lagrangian submanifolds of a Calabi-Yau
manifold . We also prove that certain principal torus bundles can be
equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus
action. We use this to construct examples of -parameter families of special
Lagrangian tori inside -dimensional Calabi-Yau manifolds with torus
symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian
fibrations with -symmetry.Comment: 27 page
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