http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

Basic Algebraic Geometry

A review of the relation between Chern-Simons gauge theory and topological string theory on noncompact Calabi-Yau spaces is given. This relation has made it possible to give an exact solution of topological string theory on these spaces to all orders in the string coupling constant. Here the focus is on the construction of this solution, which is encoded in the topological vertex, and the implications of the physics of string/gauge theory duality for knot theory and for the geometry of Calabi-Yau manifolds.

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Chern-Simons theory and topological strings

We argue the connection of Nekrasov's partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of = 2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters 1,2 of toric action on 2 factorizes correctly as the character of SU(2)L × SU(2)R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T2 action allows us to obtain the generating functions of equivariant χy and elliptic genera of the Hilbert scheme of n points on 2 by the method of topological vertex.

https://iopscience.iop.org/article/10.1088/1126-6708/2005/05/039/pdf

Instanton counting, Macdonald function and the moduli space of D-branes

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Revetements Etales Et Groupe Fondamental Sga 1

Geometric Transitions

The purpose of the present paper is threefold. First: giving a sur- vey on weighted projective spaces by the toric point of view, which seems to be missing in the literature. Second: providing characterizations of fans and poly- topes giving (polarized, in case) weighted projective spaces, so inaugurating a kind of recognition process of toric data like fans and polytopes. Third: build- ing a mathematical framework for an algorithmic and computational approach to wps's, actually realized in (21, 22).

Weighted projective spaces from the toric point of view with computational applications.

Z-linear Gale duality and poly weighted spaces (PWS)

We prove that every Q-factorial complete toric variety is a finite quotient of a poly weighted space (PWS), as defined in our previous work arXiv:1501.05244. This generalizes the Batyrev-Cox and Conrads description of a Q-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) \cite[Lemma~2.11]{BC} and \cite[Prop.~4.7]{Conrads}, to every possible Picard number, by replacing the covering WPS with a PWS. As a consequence we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every Q-factorial complete toric variety the description given in arXiv:1501.05244, Thm. 2.9, for a PWS.

/pdf/a-q-factorial-complete-toric-variety-is-a-quotient-of-a-poly-ivg3386o2j.pdf

A Q--factorial complete toric variety is a quotient of a poly weighted space

The focus of these lectures is the Gopakumar-Vafa's insight that ``Large N dualities'' (relating gauge theories and closed strings) are realized, in certain cases, by "transition in geometry". In their pivotal 1998 example, the gauge theory is SU(N) Chern-Simons theory on S^3, for large N, and the transition is the "conifold" transition between two Calabi--Yau varieties. Much progress has been made to support Gopakumar and Vafa's conjecture, including the lift of the transition to a transformation between 7-manifolds with G_2 holonomy. In another direction, this set up brings us to consider the uncharted territory of "open Gromov-Witten invariants". The lectures, hence the notes, were prepared for an audience of beginning graduate students, in mathematics and physics, whom we hope to get interested in this subject. Because most of the material presented in these lectures comes from the physics literature, we aimed to build a bridge for the mathematicians towards the physics papers on the subject.

Michele Rossi

Papers

Geometric Transitions

Weighted projective spaces from the toric point of view with computational applications.

Z-linear Gale duality and poly weighted spaces (PWS)

A Q--factorial complete toric variety is a quotient of a poly weighted space

Large N dualities and transitions in geometry