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462
Phases of N = 2 theories in two dimensions
 NUCL. PHYS. B
, 1993
"... By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on CalabiYau hypersurfaces in weighted projective spaces and LandauGinzburg models. The construction permits one to recover the known corres ..."
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Cited by 235 (1 self)
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By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on CalabiYau hypersurfaces in weighted projective spaces and LandauGinzburg models. The construction permits one to recover the known correspondence between these types of models and to greatly extend it to include new classes of manifolds and also to include models with (0, 2) worldsheet supersymmetry. The construction also predicts the possibility of certain physical processes involving a change in the topology of spacetime.
CalabiYau moduli space, mirror manifolds and spacetime topology change in string theory
 Nucl. Phys. B
, 1994
"... We analyze the moduli spaces of CalabiYau threefolds and their associated conformally invariant nonlinear σmodels and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such CalabiYau conformal theories admits a decom ..."
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Cited by 141 (20 self)
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We analyze the moduli spaces of CalabiYau threefolds and their associated conformally invariant nonlinear σmodels and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such CalabiYau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular CalabiYau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological CalabiYau space — the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kähler and complex structure moduli spaces of a CalabiYau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kähler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized by the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.
The Orbifold Chow Ring of Toric DeligneMumford Stacks
 JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
, 2004
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Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 99 (15 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
Quantum cohomology rings of toric manifolds
, 1993
"... We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary ddimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties of q ..."
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Cited by 95 (2 self)
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We compute the quantum cohomology ring H ∗ ϕ(PΣ,C) of an arbitrary ddimensional smooth projective toric manifold PΣ associated with a fan Σ. The multiplicative structure of H ∗ ϕ (PΣ,C) depends on the choice of an element ϕ in the ordinary cohomology group H 2 (PΣ,C). There are many properties of quantum cohomology rings H ∗ ϕ (PΣ,C) which are supposed to be valid for quantum cohomology rings of Kähler manifolds.
Orbifold resolution by Dbranes
 Nucl. Phys. B
, 1997
"... We study topological properties of the Dbrane resolution of threedimensional orbifold singularities, C3 /Γ, for finite abelian groups Γ. The Dbrane vacuum moduli space is shown to fill out the background spacetime with Fayet–Iliopoulos parameters controlling the size of the blowups. This Dbrane ..."
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Cited by 91 (6 self)
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We study topological properties of the Dbrane resolution of threedimensional orbifold singularities, C3 /Γ, for finite abelian groups Γ. The Dbrane vacuum moduli space is shown to fill out the background spacetime with Fayet–Iliopoulos parameters controlling the size of the blowups. This Dbrane vacuum moduli space can be classically described by a gauged linear sigma model, which is shown to be nongeneric in a manner that projects out nongeometric regions in its phase diagram, as anticipated from a number of perspectives.
On the Hodge structure of projective hypersurfaces in toric varieties
 Duke Math. J
, 1994
"... The purpose of this paper is to explain one extension of the ideas of the GriffithsDolgachevSteenbrink method for describing the Hodge theory of smooth (resp. quasismooth) hypersurfaces in complex projective spaces (resp. in weighted projective spaces). The main idea of this method is the represen ..."
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Cited by 82 (6 self)
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The purpose of this paper is to explain one extension of the ideas of the GriffithsDolgachevSteenbrink method for describing the Hodge theory of smooth (resp. quasismooth) hypersurfaces in complex projective spaces (resp. in weighted projective spaces). The main idea of this method is the representation of the Hodge components H d−1−p,p (X) in the middle cohomology group of projective hypersurfaces X = {z ∈ P d: f(z) = 0} in P d = ProjC[z1,...,zd+1] using homogeneous components of the quotient of the polynomial ring C[z1,...,zd+1] by the ideal J(f) = 〈∂f/∂z1,...,∂f/∂zd+1〉. Basic references are [13, 14, 24, 29]. In this paper, we consider hypersurfaces X in compact ddimensional toric varieties PΣ associated with complete rational polyhedral fan Σ of simplicial cones R d. According to the theory of toric varieties [12, 22, 9, 23], PΣ is defined by glueing together of affine toric varieties Aσ = SpecC[ˇσ ∩ Z d] (σ ∈ Σ) where ˇσ denotes the dual to σ cone. Weighted projective spaces are examples of toric varieties. M. Audin [1] first noticed that there exists another approach to the definition of the