AN D-T INVARIANTS

Analytic Donaldson-Thomas invariants

Coordinatore	THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD    more  Organization address address: University Offices, Wellington Square city: OXFORD postcode: OX1 2JD contact info Titolo: Ms. Nome: Valerie Cognome: Timms Email: send email Telefono: -275369 Fax: -275404
Nazionalità Coordinatore	United Kingdom [UK]
Totale costo	0 €
EC contributo	164˙269 €
Programma	FP7-PEOPLE Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)
Code Call	FP7-PEOPLE-IIF-2008
Funding Scheme	MC-IIF
Anno di inizio	2010
Periodo (anno-mese-giorno)	2010-01-14   -   2012-01-13

 Partecipanti

#	participant	country	role	EC contrib. [€]
1	THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD  Organization address address: University Offices, Wellington Square city: OXFORD postcode: OX1 2JD contact info Titolo: Ms. Nome: Valerie Cognome: Timms Email: send email Telefono: -275369 Fax: -275404	UK (OXFORD)	coordinator	164˙269.69

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 Obiettivo del progetto (Objective)

'Calabi-Yau 3-folds are 6-dimensional spaces with a rich geometrical structure. In Mathematics, Calabi-Yau 3-folds are interesting to Algebraic, Symplectic and Differential Geometers. In Physics, they are essential ingredients for building a universe: String Theory claims the universe has 10 dimensions, and is the product of a large 4-dimensional space-time with a small Calabi-Yau 3-fold. String Theorists made some strange and exciting conjectures about Calabi-Yau 3-folds, known as "Mirror Symmetry". Many of these concern "invariants", numbers associated to the Calabi-Yau 3-fold, which for deep reasons depend on only part of the geometric structure. This proposal concerns "Donaldson-Thomas (D-T) invariants" of Calabi-Yau 3-folds M. These are integers which "count" geometric objects called coherent sheaves on M. The definition of D-T invariants uses algebraic geometry, and requires both a symplectic structure (polarization) and a complex structure, but the invariants are unchanged by deformations of the complex structure. Our goal is to find a new symplectic definition of D-T invariants using gauge theory. Given a compact symplectic 6-manifold with c1=0 we choose a compatible generic almost complex structure J and define new "analytic D-T invariants" which "count" solutions of a gauge-theory equation generalizing Hermitian-Einstein connections. This is a substitute for counting holomorphic vector bundles, the simplest kind of coherent sheaf. The difficult issues concern compactness of the moduli spaces, and understanding limits of solutions. We aim to show these analytic D-T invariants are independent of J, and depend only on M as a symplectic manifold. We aim to formulate a "generalized MNOP conjecture" which expresses usual D-T invariants in terms of our analytic D-T invariants and the Gromov-Witten invariants and Betti numbers of M. This brings D-T invariants into symplectic geometry, and also reveals new symmetries and structure in D-T invariants.'