Results 1  10
of
52
The Witten equation and its virtual fundamental cycle
, 2007
"... We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory. ..."
Abstract

Cited by 31 (5 self)
 Add to MetaCart
We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory.
ORBIFOLD QUANTUM RIEMANNROCH, LEFSCHETZ AND SERRE
, 2009
"... Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted i ..."
Abstract

Cited by 30 (9 self)
 Add to MetaCart
Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus0 orbifold GromovWitten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
FJRW rings and LandauGinzburg Mirror Symmetry. arXiv:0906.0796v1
"... for BG. ii ACKNOWLEDGEMENTS Thanks are due to several people, without whom this work would not be appearing in its present form. I benefited greatly from an invitation of Tyler Jarvis to visit Brigham Young University. While there, I met several students working on similar material, with whom I coll ..."
Abstract

Cited by 28 (1 self)
 Add to MetaCart
(Show Context)
for BG. ii ACKNOWLEDGEMENTS Thanks are due to several people, without whom this work would not be appearing in its present form. I benefited greatly from an invitation of Tyler Jarvis to visit Brigham Young University. While there, I met several students working on similar material, with whom I collaborated to produce [KP+]. I enjoyed fruitful discussions with Huijin Fan, Takashi Kimura, and Ralph Kaufmann, and am grateful for the extended contact I have had with Alessandro Chiodo, whose enthusiasm and expertise were invaluable in producing this work. My studies at the University of Michigan have been generously supported by the Rackham School of Graduate Studies and the National Research Foundation of South Africa. Moral support has also been readily available, and deeply appreciated. I will cherish the wonderful friends who have left me with such happy memories of my time in Ann Arbor. Finally, I owe an inestimable debt to Yongbin Ruan. He has been remarkably generous with his time and energy, endlessly encouraging, and extremely supportive throughout.
LANDAUGINZBURG/CALABIYAU CORRESPONDENCE, GLOBAL MIRROR SYMMETRY AND ORLOV EQUIVALENCE
, 2013
"... ..."
QUANTUM DMODULES AND GENERALIZED MIRROR TRANSFORMATIONS
, 2004
"... In the previous paper [Iri1], we constructed equivariant Floer cohomology for complete intersections in toric variety and showed that it is isomorphic to the small quantum Dmodule after a mirror transformation when the first Chern class of the tangent bundle is nef. In this paper, we show that in ..."
Abstract

Cited by 24 (7 self)
 Add to MetaCart
In the previous paper [Iri1], we constructed equivariant Floer cohomology for complete intersections in toric variety and showed that it is isomorphic to the small quantum Dmodule after a mirror transformation when the first Chern class of the tangent bundle is nef. In this paper, we show that in nonnef case, equivariant Floer cohomology reconstructs the big quantum Dmodule using mirror theorem by Coates and Givental [CG]. This reconstruction procedure gives a generalized mirror transformation first observed by Jinzenji in low degrees [Jin1, Jin2].
Invariance of tautological equations I: conjectures and applications
"... Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
(Show Context)
Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications include the proofs of Witten’s conjecture on the relations between higher spin curves and Gelfand– Dickey hierarchy and Virasoro conjecture for target manifolds with conformal semisimple quantum cohomology, both for genus up to two. 1.
Witten’s conjecture, Virasoro conjecture, and semisimple Frobenius manifolds
, 2002
"... Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
(Show Context)
Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action. 1.
The abelian/nonabelian correspondence and Frobenius manifolds
, 2007
"... We propose an approach via Frobenius manifolds to the study (began in [BCK2]) of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, forT a maximal torus in G. The ensuing conjecture expresses the Gromov–Wi ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
We propose an approach via Frobenius manifolds to the study (began in [BCK2]) of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, forT a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.
An−1 singularities and nKdV hierarchies
"... Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally genera ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
(Show Context)
Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov – Witten invariants of symplectic manifolds. The papers [5, 4] contain two equivalent constructions, motivated by some results in Gromov – Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito’s Frobenius structure [14] on the miniversal deformation of the An−1singularity, the total descendent potential is a taufunction of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n − 1 solutions of the KdV hierarchy. 1. Introduction: Singularities and Frobenius