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A new approach in curve counting theories and mirror symmetry

Description du projet

Un nouveau regard dans le miroir: cordes vibrantes et comptage des courbes

La théorie des cordes postule que l’Univers est constitué de particules quantiques qui sont des cordes vibrantes unidimensionnelles plutôt que des points. Cette théorie permet d’unifier la mécanique quantique et la gravité, en étendant la théorie générale de la relativité d’Einstein, qui décrit les effets gravitationnels comme résultant de la courbure du tissu de l’espace-temps, à des distances plus courtes et à des échelles d’énergie plus élevées. La symétrie miroir est un aspect important de la théorie des cordes, étroitement liée au comptage des courbes, où une courbe complexe représente la surface d’univers d’une corde se propageant dans l’espace-temps. Avec le soutien du programme Actions Marie Skłodowska-Curie, le projet NCMS développe une nouvelle approche mathématique de la symétrie miroir et d’autres théories de comptage des courbes.

Objectif

A Marie Sklodowska-Curie IF at ETH-Zurich will lead to major developments of the PI's current research.

The PI's research focuses on better understanding the mathematical implications of the physical dualities that arise in the study of string theory. Mirror symmetry, which is a kind of duality in string theory, equates two physical theories called the A-model and B-model. Mirror symmetry predicts that the A-model (resp. the B-model) of a space/variety is equivalent to the B-model (resp. the A-model) of its
mirror space/variety. In mathematics, the A-model corresponds to Gromov--Witten (GW) theory, which is one of the first modern curve counting theories in enumerative geometry. To a physicist, a complex curve represents the worldsheet of a string propagating through space-time.

In 2018, the PI initiated a new research program which provides a novel approach of using orbifold techniques to count curves in algebraic varieties with tangency conditions along co-dimension one sub-varieties (divisors). This novel approach defines a generalization of relative GW theory which plays a central role in mirror symmetry. Several major advances have been achieved in the past two years and a new research direction has been created.

This proposal focuses on this new research program and its applications to curve counting theories and mirror symmetry. We expect to build a firm foundation for our new theory and expand this program along various directions. The proposal is divided into three main projects. The first project focuses on structural properties of the new GW theory and its relation with punctured GW theory. The second project explores applications of the new theory to several aspects of mirror symmetry including Gross--Siebert program, the Strominger--Yau--Zaslow (SYZ) conjecture and the Doran--Harder--Thompson (DHT) conjecture. The third application focuses on its connections with other curve counting theories.

Coordinateur

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Contribution nette de l'UE
€ 203 149,44
Adresse
Raemistrasse 101
8092 Zuerich
Suisse

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Région
Schweiz/Suisse/Svizzera Zürich Zürich
Type d’activité
Higher or Secondary Education Establishments
Liens
Coût total
€ 203 149,44