Matrix Cosmology

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Matrix Cosmology An Introduction
Miao Li University of
Science and Technology Institute of
Theoretical Physics Chinese Academy of
Science 1st Asian Winter School, Phoenix Park
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• Contents
• A toy model
• Matrix description
• A class of generalizations
• More generalizations
• Quantum computations

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• Motivations
• String theory faces the following challenges
  posed
• by cosmology
• Formulate string theory in a time-dependent
• background in general.
• 2. Explain the origin of the universe, in
  particular, the
• nature of the big bang singularity.
• 3. Understand the nature of dark energy.

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None of the above problems is easy.
1. A toy model One and half years ago, in
paper hep-th/0506180, Craps, Sethi and
Verlinde consider the simple background
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This background is not as simple as it appears,
since the Einstein metric has a null
singularity at . The spacetime Looks
like a cone
lightcone time
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CSV shows that perturbative string description
breaks down near the null singularity. In fact,
the scattering amplitudes diverge at any finite
order. I suspect that string S-matrix does not
exist. Nevertheless, CSV shows that a variation
of matrix Theory can be a good effective
description.
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In the 11 dimensional perspective, the metric
is locates in a finite distance
away in terms of the affine parameter if we
follow a null geodesic. If ,
then These quantities blow up at
.
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More comments on the singularity later. ?
String vertex operators With a constant dilaton,
a vertex operator assumes the form with the
on-shell condition
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With , we need to attach a factor
to the vertex operator The on-shell
condition for k is the same as before. The
vertex operator blows up at
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? Scattering amplitudes Blows up whenever
2g-2ngt0. Thus, the string perturbative S-matrix
is ill-defined.
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2. Matrix description CSV propose to use a
matrix model to describe the Physics in this
background, since ? The background preserves
half of all SUSY
if ? There is a decoupling argument a la Seiberg
and Sen.
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? The proposal In the IIA matrix string model,
for a sector with a Fixed longitudinal momentum
Where , the matrix action
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The Yang-Mills coupling constant is related to
the string couping constant through Now
with We simply have Thus,
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So near the big bang singularity, the SYM is a
free nonabelian theory. On the other hand,
near the theory tends to a CFT with an
orbifold target space. Strings are in the
twisted sectors.
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More details ? For ,
in the gauge ? For Residual gauge
symmetry is permutations of the eigen-values of
the matrices
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3. A class of generalizations In hep-th/0506260,
I showed that the CSV model is a special case of
a large class of models. In terms of the 11
dimensional M theory picture, the metric assumes
the form where there are 9 transverse
coordinates, grouped into 9-d and d .
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This metric in general breaks half of
supersymmetry. Next we specify to the special
case when both f and g are linear function of
If d9 and one takes the minus sign in
the above, we get a flat background. The null
singularity still locates at .
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Again, perturbative string description breaks
down near the singularity. To see this,
compacitfy one spatial direction, say , to
obtain a string theory. Start with the light-cone
world-sheet action We use the light-cone
gauge in which , we see that there are
two effective string tensions
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As long as d is not 1, there is in general no
plane wave vertex operator, unless we restrict
to the special situation when the vertex operator
is independent of . For instance, consider a
massless scalar satisfying The momentum
component contains a imaginary Part thus the
vertex operator contains a factor diverging
near the singularity.
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Since each vertex operator is weighted by the
string coupling constant, one may say that the
effective string coupling constant diverges. In
fact, the effective Newton constant also
diverges We conjecture that in this class of
string background, there is no S-matrix at
all. However, one may use D0-branes to describe
the theory, since the Seiberg decoupling
argument applies.
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We shall not present that argument here,
instead, We simply display the matrix action. It
contains the bosonic part and fermionic
part This action is quite rich. Lets
discuss the general conclusions one can draw
without doing any calculation.
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Case 1.
The kinetic term of is
always simple, but the kinetic term of
vanishes at the singularity, this implies that
these coordinates fluctuate wildly.
Also, coefficient of all other terms vanish, so
all matrices are fully nonabelian. As ,
the coefficients of interaction terms blow up, so
all bosonic matrices are forced to be Commuting.
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Case 2.
At the big bang, are
independent of time, and are nonabelian moduli if
dgt4. There is no constraint on other commutators
of bosonic matrices. As , if dgt4, all
matrices have to be commuting. For dlt4,
are nonabelian.
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4. More generalizations Bin Chen in
hep-th/0508191 considers the following More
general background where This class of
backgrounds all preserve half of SUSY
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Bin Chens background has to satisfy only one
diff. equation. However, it is not clear whether
one can write down a matrix model. Das and
Michelson in hep-th/0508068 study a background
appears to be a special case of Bin Chen
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Das and Michelson claim that one can write down a
matrix model for this background. It is
interesting that these authors noted that, a
String which appears to be weakly coupled at
later times is actually a fuzzy cylinder at early
times. Das, Michelson, Narayan and Trivedi in
hep-th/0602107 constructed a model in IIB
string Theory which is a deformation of
, this work overlaps with the work of Chu and
Ho.
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Ishino and Ohta in hep-th/0603215 study the
matrix string description of the following
background Again, all functions are
functions of only u. They are subject to a single
equation
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Finally, Chu and Ho in hep-th/0602054
consider the following class of time-dependent
deformation of the AdS solution
Where Also subject to a single equation.
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Chu and Ho propose that string theory in this
background is dual to a generalized super
Yang- Mills theory in 31 dimension with both
time- dependent metric and time-dependent
coupling.
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5. Quantum computations To check whether these
matrix descriptions are really correct, we need
to compute at least the interaction between two
D0-branes. This is done in hep-th/0507185 by
myself and my student Wei Song. There, we use
the shock wave to represent the background
generated by a D0-brane which carries a net
stress tensor .
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In fact, the most general ansatz is for
multiple D0-branes localized in the transverse
space , but smeared in the transverse space
. The background metric of the shock wave
is with
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The probe action of a D0-brane in such a
background is with We see that in the big
bang, the second term in the square root blows
up, thus the perturbative expansion in terms of
small v and large r breaks down.
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The breaking-down of this expansion implies the
breaking-down the loop perturbation in the
matrix calculation. This is not surprising,
since for instance, some nonabelian degrees of
freedom become light at the big bang as the
term in the CSV model shows.
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Therefore, it is of no surprising that some
Computations done so far have not correctly
reproduce the previous result. In
hep-th/0512335, Wei Song and myself used Matrix
model to compute interaction between
two D0-branes, we find a null static potential,
however there is a complex term, signaling
an instability.
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Craps, Rajaraman and Sethi in hep-th/0601062
also computed the interaction at the one loop
level, and found a different result. They found
a static potential decays at later times. Why
these results are different? Possible
answers 1. Results depend sensitively on the
method of calculation initial conditions can be
subtle. 2. D0-branes and associated potential
are not good observables.
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Conclusion Time-dependent backgrounds are
beasts hard to tame in string theory.