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A new approach in curve counting theories and mirror symmetry

Descrizione del progetto

Un nuovo sguardo allo specchio: stringhe vibranti e conteggio delle curve

La teoria delle stringhe postula che l’universo sia costituito di particelle quantistiche caratterizzate da stringhe vibranti unidimensionali, piuttosto che da punti. È un modo per unificare la meccanica quantistica e la gravità, ampliando la teoria della relatività generale di Einstein, che descrive gli effetti gravitazionali come derivanti dalla curvatura del tessuto dello spaziotempo, a distanze più brevi e su scale di energia più elevate. La simmetria speculare è un aspetto importante della teoria delle stringhe, intrinsecamente legato al conteggio delle curve, dove una curva complessa rappresenta il foglio del mondo di una stringa che si propaga nello spaziotempo. Con il supporto del programma di azioni Marie Skłodowska-Curie, il progetto NCMS sta sviluppando un nuovo approccio matematico alla simmetria speculare e ad altre teorie di conteggio delle curve.

Obiettivo

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.

Coordinatore

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

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Regione
Schweiz/Suisse/Svizzera Zürich Zürich
Tipo di attività
Higher or Secondary Education Establishments
Collegamenti
Costo totale
€ 203 149,44