Transcript Document

Matrix Cosmology
Miao Li
Institute of Theoretical Physics
Chinese Academy of Science
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.
Recently, in paper
hep-th/0506180,
Craps, Sethi and Verlinde consider the “simple”
background in which the string frame metric is flat,
While the dilaton has a linear profile in a light-like
direction:
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 pertubative 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 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 matrice
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.
To check whether these matrix descriptions are really
correct, we need to compute at least the interaction
between two D0-branes. This calculation is carried
out only on the supergravity side 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 localizedin 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.
Conclusions:
We are only seeing the emergence of an exciting
direction in constructing matrix theory for a realistic
cosmology.