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Symplectic surgery and GromovWitten invariants of CalabiYau 3folds
 I, Invent. Math
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Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
, 1995
"... We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi–Yau hypersurface M ⊂ V. In the linear model the instanton moduli spaces are relatively simple objects and ..."
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Cited by 160 (11 self)
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We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi–Yau hypersurface M ⊂ V. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth V, our results reproduce and clarify an algebraic solution of the V model due to Batyrev. In addition, we find an algebraic relation determining the solution for M in terms of that for V. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured “monomialdivisor mirror map” of Aspinwall, Greene, and Morrison.
Mirror symmetry, mirror map and applications to complete . . .
 EXPERIMENTAL NUCLEAR PHYSICS B
, 1995
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Mirror Symmetry for Lattice Polarized K3 Surfaces
 J. Math. Sci
, 1996
"... Introduction. There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of CalabiYau threefolds made by a group of theoretical physicists (see [11,26] for references). Roughly speaking this duality, called mirror sym ..."
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Cited by 144 (4 self)
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Introduction. There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of CalabiYau threefolds made by a group of theoretical physicists (see [11,26] for references). Roughly speaking this duality, called mirror symmetry, pairs two families F and F ∗ of CalabiYau threefolds in such a way that the following properties are satisfied:
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 142 (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.
Mirror principle
 I. Asian J. Math
, 1997
"... Abstract. 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 ..."
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Cited by 127 (13 self)
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Abstract. 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
Equivariant GromovWitten invariants
 INTERNAT. MATH. RES. NOTICES
, 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a co ..."
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Cited by 126 (10 self)
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The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (GW) theory, i.e., intersection theory on spaces of (pseudo) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a compact Kähler manifold X, the equivariant GWtheory provides, as we will show in Section 3, the equivariant cohomology space H ∗ G (X) with a Frobenius structure (see [11]). We will discuss applications of the equivariant theory to the computation ([15], [18]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cupproduct operators (Sections 7, 8), to the S1equivariant Floer homology theory on the loop space LX (see Section 6 and [14], [13]), and to a “quantum ” version of the Serre duality theorem (Section 12). In Sections 9–11 we combine the general theory developed in Sections 1–6 with the fixedpoint localization technique [21], in order to prove the mirror conjecture (in the form suggested in [14]) for projective complete intersections. By the mirror conjecture, one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kähler CalabiYau nfold and, respectively, complex and symplectic geometry on another CalabiYau nfold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on CalabiYau 3folds (1991, see the theorem below) raised a number of new mathematical problems—challenging tests of maturity for modern methods of symplectic topology. On the other hand, in 1993 I suggested that the relation between symplectic and complex geometry predicted by the mirror conjecture can be extended from the class of CalabiYau manifolds to more general compact symplectic manifolds if one admits non
Building a better racetrack
 JHEP 0406
"... We find IIb compactifications on CalabiYau orientifolds in which all Kähler moduli are stabilized, along lines suggested by Kachru, Kallosh, Linde and Trivedi. ..."
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Cited by 118 (8 self)
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We find IIb compactifications on CalabiYau orientifolds in which all Kähler moduli are stabilized, along lines suggested by Kachru, Kallosh, Linde and Trivedi.
Generalized Hypergeometric Functions and Rational Curves on CalabiYau complete intersections in Toric Varieties
 COMMUN. MATH. PHYS
, 1995
"... We formulate general conjectures about the relationship between the Amodel connection on the cohomology of a ddimensional CalabiYau complete intersection V of r hypersurfaces V1,...,Vr in a toric variety PΣ and the system of differential operators annihilating the special generalized hypergeometr ..."
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Cited by 98 (14 self)
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We formulate general conjectures about the relationship between the Amodel connection on the cohomology of a ddimensional CalabiYau complete intersection V of r hypersurfaces V1,...,Vr in a toric variety PΣ and the system of differential operators annihilating the special generalized hypergeometric function Φ0 depending on the fan Σ. In this context, the mirror symmetry phenomenon can be interpreted as the twofold characterization of the series Φ0. First, Φ0 is defined by intersection numbers of rational curves in PΣ with the hypersurfaces Vi and their toric degenerations. Second, Φ0 is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the homolomorphic differential dform on an another CalabiYau dfold V ′ called the mirror of V. Using this generalized hypergeometric series, we propose a general construction for mirrors V ′ of V and canonical qcoordinates on the moduli spaces of CalabiYau manifolds.