Geometric Transitions

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Geometric Transitions
25 Giugno 2007
Michele Rossi
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CalabiYau Varieties
Let Y be a smooth, complex, projective variety
with
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.
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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).
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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

• is composed by 16 nodes.
• The resolution can be
  simultaneously resolved by a small blow up
• such that Y is a smooth Calabi-Yau 3-fold.
• The smoothing can be obviously smoothed
  to
• the generic quintic 3-fold

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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
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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
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
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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
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
Topologically is a cone over
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
gt admits a natural complex structure.
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
admits a natural symplectic structure.
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Local meaning of a conifold transition a surgery
in topology (H. Clemens 1983)
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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
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(No Transcript)
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Local transition as a topological surgery
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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

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• 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 .

• (Betti numbers) for
  and
• where vertical equalities are given by
  Poincaré Duality
• (Hodge numbers)

homological type of
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Applications g.t. in algebraic geometry(the
Reids 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

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Applications g.t. in physics (I)(the vacuum
degeneracy problem)
Consistent 10-dimensional super-string theories
T-duality
T-duality
Mirror Symmetry
S-duality
Low energy limits 10-dimensional super-gravity
locally modelled on
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.
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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.
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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 Reids fantasy (??)

2. Lifting g.t. s to special holonomy
7-manifolds.

• Physical motivation (Witten 1995) M-theory

• Acharya, Atiyah, Maldacena, Vafa, Witten
• (2000-2001) lifts the local conifold
  transition to
• a flop between 7-dimensional manifolds with
  holonomy , by means of
• suitable -actions.

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?