Geometric Transitions 25 Giugno 2007 Michele Rossi

Let Y be a smooth, complex, projective variety with Calabi—Yau Varieties Y will be called a Calabi-Yau variety if : Remark. This definition of Calabi-Yau variety is a generalization of 1-dimensional smooth elliptic curves, 2-dimensional smooth K3 surfaces.

Geometric Transition Let Y be a Calabi-Yau 3-fold and be a birational contraction onto a normal variety. If there exists a complex deformation (smoothing) of to a Calabi-Yau 3-fold, then the process of going from to is called a geometric transition (for short transition or g.t.) and denoted by or by the diagram A conifold transition is a g.t. admitting singular locus composed at most by ordinary double points (nodes).

The basic example: the conifold in. Let be the singular hypersurface with generic is the generic quintic 3-fold containing the plane Its singular locus is then given by 1. is composed by 16 nodes. 2. The resolution: can be simultaneously resolved by a small blow up such that Y is a smooth Calabi-Yau 3-fold. 3. The smoothing: can be obviously smoothed to the generic quintic 3-fold

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983) Let be a conifold transition and (i.e. p is a node). Let be a local chart such that and has local equation

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983) Blow up along the plane and look at the proper transform of U, which is described in by

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983)

The local smoothing of the node is given by the 1-parameter family where Let for some real

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983)

Topologically is a cone over

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983) => admits a natural complex structure.

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983) admits a natural symplectic structure.

Local meaning of a conifold transition: a surgery in topology (H. Clemens 1983)

Theorem 1 (Local conifold as a surgery) Let be the closed unit ball and consider Then we get compact tubular neighborhoods and of the vanishing cycle and of the exceptional respectively. Consider the standard diffeomorphism Then induces a surgery from to and can be obtained from by removing and pasting in, by means of the diffeomorphism

Local transition as a topological surgery

Global geometry of a conifold transition Let be a conifold transition. Then where is a node, there exists a simultaneous resolution which is a birational morphism contracting N rational curves, admits N vanishing cycles which are 3- spheres. Example: the conifold Then:, contains 16 exceptional rational curves, contains 16 vanishing spheres. On the other hand:, and,, and

Theorem 2 (Clemens 1983, Reid 1987, Werner-vanGeemen 1990, Tian 1992, Namikawa-Steenbrink 1995, Morrison- Seiberg 1997,...) Let be a conifold transition and let N be the number of nodes composing, k be the maximal number of homologically independent exceptional rational curves in Y, c be the maximal number of homologically independent vanishing cycles in. 1. ; 2.(Betti numbers) for and where vertical equalities are given by Poincaré Duality; 3.(Hodge numbers) = homological type of

Applications: g.t. in algebraic geometry (the Reid’s Fantasy) The problem: What about the moduli space of C.-Y. 3-folds ? Since there are plenty of topologically distinct well known examples of Calabi-Yau 3-folds, it should be wildly reducible, on the contrary of moduli spaces of elliptic curves and K3 surfaces. M. Reid in 1987: Use g.t.’s instead of analytic deformations to apply to C.-Y. 3-folds the same idea employed by Kodaira for K3 surfaces: find the right category to work with ! Conjecture (the Reid's fantasy): Up to some kind of inductive limit over r, the birational classes of projective C.-Y. 3-folds can be fitted together, by means of geometric transitions, into one irreducible family parameterized by the moduli space of complex structures over suitable connected sum of copies of solid hypertori

Applications: g.t. in physics (I) (the vacuum degeneracy problem) Consistent 10-dimensional super-string theories: Low energy limits: 10-dimensional super-gravity locally modelled on T-duality Mirror Symmetry S-duality Minkovsky space- time Compact C.-Y. 3-fold Calabi-Yau web conjecture. (P.Candelas & C. 1988) C.-Y. 3- folds could be (mathematically) connected each other by means of geometric (conifold) transitions. A. Strominger in 1995: at least for a conifold transition, the topological change is physically explained by the condensation of massive black holes to massless ones.

Applications: g.t. in physics (II) (set-up of open/closed string dualities) G. t'Hooft in 1974 conjectured that large N limit of gauge theories are equivalent (dual) to some kind of closed string theories. E. Witten in 1992 showed that a SU(N) (or U(N)) Chern-Simons gauge theory on the 3-sphere, is equivalent to an open type II-A string theory on with D-branes wrapped on the lagrangian. R. Gopakumar and C. Vafa in 1998 proposed evidences to the t'Hooft conjecture showing that large N, SU(N) Chern-Simons gauge theory on (or equivalently open type II-A string theory on, after Witten) are dual to type II-A closed string theory “compactified” on the resolution of the local conifold. The geometric set up of this duality is given by the local conifold transition The same for type II-B theories after Mirror Symmetry.

Open Problems. 1. G.T.’s as transformations: a bridge from complex to symplectic. Understand the local behaviour of a g.t. Smith-Thomas-Yau 2003: symplectic resolutions symplectic Reid’s fantasy (??) 3.“Analytic classification” of g.t.’s: under which conditions a g.t. can be “deformed” to a simpler one? Homological type of g.t.’s. Are the equivalence classes of g.t.’s described by conifolds ones? If no, which are the further simplest g.t.’s? 2. Lifting g.t.’ s to special holonomy 7-manifolds. Physical motivation (Witten 1995): M- theory Acharya, Atiyah, Maldacena, Vafa, Witten ( ): lifts the local conifold transition to a “flop” between 7-dimensional manifolds with holonomy, by means of suitable -actions.