Transcript Document

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
Contents:
1. A toy model
2. Matrix description
3. A class of generalizations
4. More generalizations
5. Quantum computations
Motivations:
String theory faces the following challenges posed
by cosmology:
1. 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.
……
None of the above problems is easy.
1. A toy model
Recently, in paper
hep-th/0506180,
Craps, Sethi and Verlinde consider the “simple”
background:
This background is not as simple as it appears, since
the Einstein metric
has a null singularity at
Looks like a cone:
. The spacetime
lightcone time
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.
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
.
More comments on the singularity later.
★ String vertex operator
With a constant dilaton, a vertex operator assumes
The form
with the on-shell condition:
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
★ Scattering amplitudes
Blows up whenever 2g-2+n>0.
Thus, the string perturbative S-matrix is ill-defined.
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.
●
★ The proposal
In the IIA matrix string model, for a sector with a
Fixed longitudinal momentum
Where
, the matrix action
The Yang-Mills coupling constant is related to the
string couping constant through
Now with
We simply have
Thus,
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.
More details:
●
For
●
For
,
in the gauge
Residual gauge symmetry is permutations of the
eigen-values of the matrices
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
.
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 d=9 and one takes the minus sign in the above, we
get a flat background.
The null singularity still locates at
.
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:
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.
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.
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. Let’s discuss the general
conclusions one can draw without doing any
calculation.
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.
Case 2.
At the big bang,
are independent of time, and
are nonabelian moduli if d>4. There is no constraint
on other commutators of bosonic matrices.
As
, if d>4, all matrices have to be commuting.
For d<4,
are nonabelian.
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
Bin Chen’s 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:
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.
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
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.
Chu and Ho propose that string theory in this
background is dual to a generalized super YangMills theory in 3+1 dimension with both timedependent metric and time-dependent coupling.
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
.
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
.
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.
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.
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.
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.
Conclusion:
Time-dependent backgrounds are beasts hard to
tame in string theory.