Superstrings on CalabiYau manifolds

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Superstrings on Calabi-Yau manifolds

• Matthias Gaberdiel
• ETH Zürich

DPG Tagung Dortmund
30 March 2006
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String basics
String theory is currently one of the most
promising candidates for a theory of quantum
gravity.
Roughly speaking, the idea behind string
theory is that the fundamental objects, in terms
of which the theory is being formulated, are
one-dimensional strings.
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String basics
The strings can either be
or
closed
open
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String basics
They propagate (initially) in a given fixed
background.
The consistency conditions (conformal beta
equations) require that this background
satisfies a modification of the
Einstein equations
Thus string theory incorporates general
relativity.
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String basics
Furthermore, the excitation spectrum of the
string contains the graviton that describes
fluctuations of the background.
Thus string theory may in fact be background
independent.
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String basics
The other vibrational excitations of the string
contain the quanta that correspond to the
familiar elementary particles.
Thus string theory incorporates (some
extension of) the standard model of particle
physics.
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String basics
The interaction of these excitations is described
geometrically by the joining and splitting of
strings
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String basics
instead of
This smooth description suggests that string
theory has in fact better UV properties than
conventional quantum field theories.
On the other hand, string field theory is
conceptually not that well developed in
particular one does not yet understand how to
derive the Feynman rules of string field theory
from first principles.
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String basics
So far I have described the standard string
folklore....
But how does one actually describe this theory
explicitly?
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Quantitative strings
In order to describe strings quantitatively think
of them in terms of a 2-dimensional field
theory that is defined on the world-sheet of the
string
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Sigma model
The propagation of the string is then
described by specifying to which point in the
target space (space-time) every world-sheet point
is mapped to.
This map is controlled by the sigma-model action
where G is the space-time metric (that may
depend on the position X). h is the world-sheet
metric.
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Sigma model
This sigma model is classically invariant under
reparametrisations and Weyl rescalings of the
world-sheet.
We can use this symmetry to go to conformal
gauge in which we take the world-sheet metric to
be proportional to the usual 2d Minkowski metric.
Then the residual symmetry is the
conformal (Weyl) symmetry. The resulting 2d field
theory is therefore a conformal field theory.
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Conformal transformations
The conformal transformations preserve the
metric up to rescalings. They therefore preserve
angles, but not necessarily lengths.
For a conformally invariant theory it therefore
does not matter whether one looks like this
or like
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Virasoro algebra
The conformal symmetry in 2 dimensions is thus
very powerful!
In fact, the algebra of infinitesimal local
conformal transformations is infinite
dimensional
Virasoro algebra
c central charge
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Conformal symmetry
Because of this large symmetry 2d conformal field
theories can be solved essentially based on
symmetry considerations alone.
In fact, 2d conformal field theories have a
very rich mathematical structure they define
vertex operator algebras, and have had a
significant impact on many areas in
modern mathematics, such as group theory, number
theory, geometry, etc.
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No ghost theorem
From the point of view of string theory, the
conformal symmetry is a gauge symmetry. Just as
in electrodynamics, this gauge symmetry can then
remove the negative norm states (ghosts) that the
covariant theory initially has.
In the present context this requires that
Goddard, Thorn
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Critical dimension
If the target space is flat space, then
cdimension.
Thus D26 is the critical dimension of
(bosonic) string theory!
In order to describe 4d physics, the idea is then
that
i.e. that 22 dimensions are compactified on
some internal manifold compactification.
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Compactification
The internal theory may again be defined in
terms of a non-linear sigma model, but in
general it may be any conformal field theory with
c22.
It is therefore not really appropriate to talk
about an internal manifold since the relevant
conformal field theory may not have a geometric
interpretation.
There are however interesting situations
where it does!
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Fermions
The analysis for the (world-sheet) fermionic
string is similar in this case the residual
gauge symmetry is the N1 superconformal
symmetry that contains in addition to the
Virasoro generators a superfield generator G
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No ghost theorem (ii)
In this case the no-ghost theorem requires that
Since each flat direction contributes c11/23/2
to the central charge, this corresponds to a
critical dimension
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Supersymmetry
The fermionic string is not necessarily
(space-time) supersymmetric (not even in D10
dimensions).
Many of the most interesting (and best
understood) string theories are
however spacetime supersymmetric.
However, it is not really true that string
theory predicts supersymmetry.
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N2 algebra
Spacetime supersymmetry requires that the
world-sheet theory has an N2 superconformal
symmetry
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Geometry
For theories with N2 superconformal symmetry, we
can at least partially identify the
corresponding geometry.
Topological twist
the operator
satisfies
and thus defines a cohomology.
Lerche, Vafa, Warner Witten
This cohomology is the cohomology of the
corresponding geometry!
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Mirror symmetry
However, equivalent superconformal field
theories may give rise to different
cohomologies and thus different geometries
Mirror symmetry
Candelas
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Calabi-Yau manifolds
From a geometrical point of view, the
condition for a compactification to preserve
spacetime supersymmetry is that the internal
manifold has covariantly constant spinors.
For N2 superconformal field theories with c9
the corresponding manifolds should thus be D6
Calabi-Yau manifolds.
In fact, a more careful analysis has revealed
that in general the relevant manifolds are only
generalised CY. Hitchin
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Gepner models
In certain examples the relevant Calabi-Yau
can actually be identified from the
superconformal description.
In particular, this is possible for a class
of superconformal models that can be
constructed by tensoring the so-called N2
minimal models
Gepner models
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The Quintic
The corresponding Calabi-Yau manifolds are
hypersurfaces in (weighted) projective space.
The simplest example is the quintic that is
defined by the equation
in complex projective space
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D-branes
So far we have (implicitly) only discussed
closed strings. For open strings we need in
addition to specify the boundary conditions at
the end-points of the open string.
In flat space, the relevant boundary
conditions are for every direction either
Neumann (endpoint can freely move)
or Dirichlet (endpoint has fixed position).
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D-branes
Thus can describe the boundary
conditions geometrically.
The brane is the hypersurface on which open
strings can end.
Polchinski
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D-branes on Calabi-Yaus
Even if the geometry is not flat, one may think
of D-branes as hypersurfaces that wrap
certain cycles in the internal manifold.
However, this description may be misleading.
In order to understand the geometry that
the string sees we should really analyse
these D-branes in conformal field theory.
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D-branes in Gepner models
In particular, we should try to do so for the
Calabi-Yau compactifications that are of
primary phenomenological interest.
From a string theory point of view we should
therefore study D-branes in Gepner models. Some
results in this direction were already obtained
some time ago.
Recknagel, Schomerus Brunner et.al.
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Matrix factorisations
Recently, Kontsevich has proposed that the
superconformal D-branes of the Gepner model
corresponding to the CY surface W0 can be
characterised in terms of matrix factorisations
of
as
where E and J are polynomial matrices in the
variables
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Simple factorisations
The simplest factorisations are the
factorisations of a single polynomial
as
with
If W is a sum of different such
polynomials, these separate factorisations can be
tensored together to give a factorisation of W.
Ashok et al
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Quintic factorisations
For example, for the case of the quintic where
we have a superconformal D-brane for each
tensor product of five such factorisations.
These factorisations, however, do not
describe the fundamental D-branes of this
theory.
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Permutation factorisations
The fundamental factorisations are tensor
products of the rank 1 factorisations that come
from writing
where the product runs over the dth roots of -1.
Tensoring this factorisation with three
one-variable factorisations describes then for
example the D0-brane on the quintic.
Ashok et al
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D-branes from factorisations
We have recently managed to identify these
matrix factorisations with explicit D-brane
constructions in conformal field theory.
Brunner, MRG
Furthermore, we have also managed to identify all
the fundamental supersymmetric D-branes for a
large class of Gepner models.
Caviezel, Fredenhagen, MRG
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D-branes for Gepner models
Having explicit conformal field
theory constructions for D-branes in Gepner
models will open the way to analyse the relation
between geometry and conformal field theory in
more detail.
It may also have interesting applications for
string phenomenology.
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Conclusions

• Described some basic properties of string theory
• Geometry from N2 superconformal field theory
• Calabi-Yaus and Gepner models
• D-branes in Gepner models from matrix
  factorisations