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161
Orientifolds and Mirror symmetry
, 2003
"... We study parity symmetries and crosscap states in classes of N = 2 supersymmetric quantum field theories in 1+1 dimensions, including nonlinear sigma models, gauged WZW models, LandauGinzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of t ..."
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Cited by 271 (11 self)
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We study parity symmetries and crosscap states in classes of N = 2 supersymmetric quantum field theories in 1+1 dimensions, including nonlinear sigma models, gauged WZW models, LandauGinzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of them. The case of the N = 2 minimal model are studied in RCFT, and LG models. We also identify mirror pairs of orientifolds, extending the correspondence between symplectic geometry and algebraic geometry by including unorientable worldsheets. Through the analysis in various models and comparison in the overlapping regimes, we obtain a global picture of orientifolds and Dbranes.
Mirror principle I
 I. ASIAN J. MATH
, 1997
"... We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruc ..."
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Cited by 125 (13 self)
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We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles – including any direct sum of line bundles – on Pn. This includes proving the formula of Candelasde la OssaGreenParkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential function for the quintic in P4. We derive, among many other examples, the multiple cover formula for GromovWitten invariants of P1, computed earlier by MorrisonAspinwall and by Manin in different approaches. We also prove a formula for enumerating Euler classes which arise in the socalled local mirror symmetry for some noncompact CalabiYau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma
Hypergeometric functions and mirror symmetry in toric varieties
, 1999
"... We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in ..."
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Cited by 72 (4 self)
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We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. We compare the monodromy calculations for the Picard–Fuchs system associated with the periods of a Calabi–Yau manifold M with the algebrogeometric computations of the cohomology action of Fourier– Mukai functors on the bounded derived category of coherent sheaves on the mirror Calabi–Yau manifold W. We obtain the complete dictionary between the two sides for the one complex parameter case of Calabi–Yau complete intersections in weighted projective spaces, as well as for some two parameter cases. We also find the complex of sheaves on W × W that corresponds to a loop in the moduli space of complex structures on M induced by a phase transition of W.
Dbranes on stringy CalabiYau manifolds
, 2000
"... We argue that Dbranes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau–Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states wi ..."
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Cited by 56 (1 self)
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We argue that Dbranes corresponding to rational B boundary states in a Gepner model can be understood as fractional branes in the Landau–Ginzburg orbifold phase of the linear sigma model description. Combining this idea with the generalized McKay correspondence allows us to identify these states with coherent sheaves, and to calculate their Ktheory classes in the large volume limit, without needing to invoke mirror symmetry. We check this identification against the mirror symmetry results for the example of the Calabi–Yau hypersurface in WIP 1,1,2,2,2.
On Dbranes from Gauged Linear Sigma Models
, 2000
"... We study both Atype and Btype Dbranes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the largevolume phase/nonlinear sigma model limit of the corresponding CalabiYau manifold, w ..."
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Cited by 45 (10 self)
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We study both Atype and Btype Dbranes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the largevolume phase/nonlinear sigma model limit of the corresponding CalabiYau manifold, where we also find that we need to add a contact term on the boundary. These considerations enable to us to derive the boundary conditions in the full gauged linear sigma model, including the addition of the appropriate boundary contact terms, such that these boundary conditions have the correct nonlinear sigma model limit. Most of our results are derived for the quintic CalabiYau manifold, though we comment on possible generalisations.
Quantum Cohomology Of Flag Varieties
 Internat. Math. Res. Notices
, 1995
"... Introduction The quantum cohomology ring of a Kahler manifold X is a deformation of the usual cohomology ring which appears naturally in theoretical physics in the study of the supersymmetric nonlinear sigma models with target X. In [W], Witten introduces the quantum multiplication of cohomology cl ..."
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Cited by 41 (2 self)
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Introduction The quantum cohomology ring of a Kahler manifold X is a deformation of the usual cohomology ring which appears naturally in theoretical physics in the study of the supersymmetric nonlinear sigma models with target X. In [W], Witten introduces the quantum multiplication of cohomology classes on X as a certain deformation of the usual cupproduct, obtained by adding to it the socalled instanton corrections (see also [V]). These can be in turn interpreted as intersection numbers on a sequence of moduli spaces of (holomorphic) maps P ! X. To make this interpretation rigorous according to mathematical standards, one encounters severe problems, mainly because these moduli spaces are not compact and they may have the wrong dimension. Recently, substantial efforts have been made to put the theory on firm mathematical footing, and a proof for the existence of the quantum cohomology ring, using methods of symplectic topology, has been given by Ruan and Tian [RT], for a large cla
Notes on certain (0,2) correlation functions
"... In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton correc ..."
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Cited by 31 (11 self)
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In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.