A B model quantum differential system and application to mirror symmetry

Samuel Boissiere'S

Description

Niveau: Supérieur, Doctorat, Bac+8
A B-model quantum differential system and application to mirror symmetry A. Douai 19 mars 2010 1 Introduction The aim of this talk is to define a correspondance between quantum cohomology (A-side) and singularity theory (B-side) : this is an aspect of mirror symmetry. The main idea is to attach to each A-model (resp. B-model) a quantum differential system (and not only a quantum D-module in the sense of [9]), that is a trivial bundle on P1?M (M depends on the situation : it can be the germ (Cµ, 0) or C?...), equipped with a meromorphic connection with prescribed poles and a flat “metric” : two models will be mirror partners if their quantum differential systems are isomorphic. On the A-side, such quantum differential systems arise canonically (see Samuel Boissiere's talk). On the B-side, the situation is a priori less clear : starting with a B-model (a regular, tame function on an affine manifold), it is indeed possible to construct several quantum differential systems, which can be difficult to compare. A canonical construction, which fits very well with mirror symmetry (at least on the known examples !), is given by Hodge theory (“M. Saito's method”). These notes are organized as follows : we first define quantum differential systems (for which Frobenius manifolds are a good motivation).

• gauss-manin system


• global section


• connection ?


• p1 ?m ?


• connection


• quantum differential


• model quantum

Sujets

• Sheaf (mathematics)
• Connection

Informations

• Publié par: profil-zyak-2012
• Publié le: 01 mars 2010
• Nombre de lectures: 11
• Langue: English

Extrait

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AB-model quantum dierential system and application to mirror symmetry

Introduction

A. Douai

19 mars 2010

The aim of this talk is to dene a correspondance between quantum cohomology (A-side) and singularity theory (B-side) : this is an aspect of mirror symmetry. The main idea is to attach to eachA-model (resp.B-model) aquantum dierential system(and not only a 1 quantumD-modulein the sense of [9]), that is atrivialbundle onPM(Mdepends on the  situation : it can be the germ (C,0) orC...), equipped with a meromorphic connection with prescribed poles and a at “metric” : two models will be mirror partners if their quantum dierential systems are isomorphic. On theA-side, such quantum dierential systems arise canonically(seeSamuelBoissiere’stalk).OntheB-side, the situation isa prioriless clear : starting with aB-model (a regular, tame function on an ane manifold), it is indeed possible to construct several quantum dierential systems, which can be dicult to compare. A canonical construction, which ts very well with mirror symmetry (at least on the known examples !), is given by Hodge theory (“M. Saito’s method”). These notes are organized as follows : we rst dene quantum dierential systems (for which Frobenius manifolds are a good motivation). We then explain how to attach (cano- nically) such a system to a regular, tame, function on an ane manifold. The last part is devoted mirror symmetry : a good test is provided by the small quantum cohomology of weighted projective spaces. We end with few words about the “J-function” of a quantum dierential system.

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Quantum dierential systems (or Saito structures)

2.1 A motivation : Frobenius manifolds LetMbe a complex manifold. AFrobenius structureonM(see for instance [10], [13]) is dened by the following data ( M denotes the sheaf of holomorphic vector elds onM) : 1 1. a commutative and associative produ ldi.e :  cton  M (a Higgs e M → M  M , O M -linear, such that ∧ = 0),

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