For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar 1. We also resolve two outstanding conjectures of Witten. The rst conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.

http://annals.math.princeton.edu/wp-content/uploads/annals-v178-n1-p01-p.pdf

The Witten equation, mirror symmetry, and quantum singularity theory

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Moduli of representations of the fundamental group of a smooth projective variety, II . Finite group actions and étale cohomology . A rank theorem for analytic maps between power series spaces . Some groups whose reduced C[*]-algebra is simple . Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rahm

In [23, 24], Y.-P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.-P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.-P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.-P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semi-simple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov–Witten potentials coincide.

/pdf/tautological-relations-and-the-r-spin-witten-conjecture-185ig8o851.pdf

Tautological relations and the r-spin Witten conjecture

In this article, we study the Berglund--Hubsch transpose construction W^T for invertible quasihomogeneous potential W. We introduce the dual group G^T and establish the state space isomorphism between the Fan-Jarvis-Ruan-Witten A-model of W/G and the orbifold Milnor ring B-model of W^T/G^T. Furthermore, we prove a mirror symmetry theorem at the level of Frobenius algebra structure for G^max. Then, we interpret Arnol'd strange duality of exceptional singularities W as mirror symmetry between W/J and its strange dual W^SD.

FJRW rings and Landau-Ginzburg mirror symmetry

We study a system of nonlinear elliptic PDEs associated with a quasi-homogeneous polynomial. These equations were proposed by Witten as the replacement for the Cauchy-Riemann equation in the singularity (Landau-Ginzburg) setting. We introduce a perturbation to the equation and construct a virtual cycle for the moduli space of its solutions. Then, we study the wall-crossing of the deformation of the virtual cycle under perturbation and match it to classical Picard-Lefschetz theory. An extended virtual cycle is obtained for the original equation. Finally, we prove that the extended virtual cycle satisfies a set of axioms similar to those of Gromov-Witten theory and r-spin theory.

The Witten equation and its virtual fundamental cycle

We compute the recently introduced Fan–Jarvis–Ruan–Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov–Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan–Jarvis–Ruan–Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau–Ginzburg/Calabi–Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental’s quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan–Jarvis–Ruan–Witten theory.

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

L'objet, de cet, article est la notion de structure r-spin: un fibre en droites dont la puissance r-ieme est isomorphe au fibre canonique. Au-dessus du champ Mg des courbes lisses de genre g, les structures r-spin forment un torseur fini sous le groupe des fibres de r-torsion. Au-dessus du champ M, des courbes stables de genre g, les structures r-spin forment un champ etale, mais la finitude et la structure de torseur ne sont pas preservees. On ameliore drastiquement cet etat de choses si on residue le probleme dans la categorie des courbes champetres ("twisted curves" au "sens d'Abramovich et Vistoli). On trouve d'abord que. dans cette categorie, il existe plusieurs compactifications de Mg; chacune correspond a un multi- indice ī = (l 0 ,l 1 ,... ) identifiant une notion de stabilite: la ī- stabilite. On determine par la suite tout choix convenable de ī pour lequel les structures r-spin forment, un torseur fini au-dessus du champ des courbes ī-stables.

/pdf/stable-twisted-curves-and-their-r-spin-structures-1oapdnq1sv.pdf

Stable twisted curves and their $r$-spin structures

We compute the recently introduced Fan-Jarvis-Ruan-Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov-Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan-Jarvis-Ruan-Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau-Ginzburg/Calabi-Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental's quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan-Jarvis-Ruan-Witten theory.

Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations

The enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

/pdf/towards-an-enumerative-geometry-of-the-moduli-space-of-45dnfnmrzt.pdf

Towards an enumerative geometry of the moduli space of twisted curves and rth roots

The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In math.AG/0603687, we construct a compact stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford's formula for the Chern character of the Hodge bundle to the direct image of the universal r-th root in K-theory.

/pdf/towards-an-enumerative-geometry-of-the-moduli-space-of-p9sph3pctc.pdf

Alessandro Chiodo

Papers

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

Stable twisted curves and their $r$-spin structures

Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations

Towards an enumerative geometry of the moduli space of twisted curves and rth roots

Towards an enumerative geometry of the moduli space of twisted curves and r-th roots