Cosmological Singularities from Matrices Sumit R. Das Space-time - - PowerPoint PPT Presentation
Cosmological Singularities from Matrices Sumit R. Das Space-time from Matrices A common slogan in string theory is that space and time are not fundamental, but derived concepts which emerge out of more fundamental structures. In most
¢ A common slogan in string theory is that space
¢ In most cases we do not know much about this
¢ However in some cases we have some hint –
¢ More precisely – the holographic description is
¢ Closed Strings emerge as collective
¢ In these cases the fundamental structures are
2 dimensional strings Matrix Quantum Mechanics M theory/ critical string SUSY Matrix Quantum in light cone gauge Mechanics/ 1+1 YM Strings in 3+1 dimensional N=4 Yang-Mills
5 5
¢ These holographic descriptions have played a
crucial role in our understanding of puzzling aspects of quantum gravitational physics, e.g. Black Holes
¢ Can we use these to address some puzzling
questions in time-dependent space-times – in particular cosmologies where time appears to begin or end – e.g. Big Bangs or Big crunches ?
¢ Can we ask what do we even mean when we say
that time begins or ends ?
¢ In this talk I will discuss some recent attempts in
this direction.
¢ - hermitian matrix
¢ Gauging – states are singlet under SU(N) ¢ Eigenvalues are fermions. Single particle
¢ Density of fermions
2 2
t
¢ To leading order in 1/N dynamics of the scalar
¢ This collective field theory would be in fact the
¢ The fundamental quantum description is in
¢ Collective field theory used to find the emergent
p x
¢ Filled fermi sea ¢ Collective field ¢ Fluctuations
Two scalar fields for the two sides.
m 2
) , 2 ( = ± t m h
¢ The quadratic action for fluctuations ¢ Metric determined up to conformal factor ¢ These two massless scalars are related to the only
two dynamical fields of 2d string theory by a transform which is non-local at the string scale. Both these scalars live in the same space-time
¢ Minkowskian coord. ¢ is independent of
time
¢ Any other conformal
frame will destroy this property
int
H
Penrose Diagram
Diagram should be fuzzy at string scale
S.R.D. and J. Karczmarek, PRD D71 (2005) 086006
¢ An infinite symmetry of
the theory generates time dependent solutions, e.g.
p x
What kind of space-time is perceived by fluctuations around this solution ?
t
finite t Æ
¢ Fluctuations around any classical solution are
¢ To get the causal structure we need to go to
¢ For this solution ¢ As ¢ The space-time
generated has a space like boundary
t t
2
t
¢ This is a geodesically incomplete space-time. The
space-like boundary has regions of strong coupling
¢ Normally one would simply extend the space-time
to complete it
¢ However in this case there is a fundamental
definition of time provided by the matrix model – t
¢ The space-time perceived by closed strings is an
emergent description
¢ There are few other solutions we found which
have more complicated Penrose diagrams but share the feature that the boundary appears space-like.
¢ At the space-like boundary the closed string
theory is strongly coupled except for large values
¢ However the open string theory – i.e. the theory of
fermions is still free, and things are in principle computable
¢ Most likely, we cannot assign a reasonable
space-time description
¢ By a standard chain of dualities, Type IIA string
theory with a compact light cone direction with radius R and with momentum is equivalent to 1+1 dimensional SU(N) Yang- Mills theory with
2 s s YM
s 2
2 2 2 2 2 J I s I s
m ts
When the potential term restricts the scalar fields to be diagonal Action becomes that of Green-Schwarz string in light cone gauge
1 3 2 2 1
N
s
¢ Craps, Sethi and Verlinde : Consider Matrix String
Theory in a background with flat string frame metric, but a dilaton
2 2 2 2 2 J I s Q I Q s
t m ts t
At usual Green-Schwarz string in light cone gauge At non-abelian excitations – no conventional space-time
+
+
¢ Equivalently the YM
theory can be thought to have a constant coupling, but living on the future wedge of the Milne universe
2 2 2 2
t
Q
Constant t
Big Bang Singularity
¢ Motivation : to find a situation where there is
some control of the precise nature of non-abelian excitations
¢ Possible for pp-wave solutions with null linear
dilatons
+
+ +
Qx
123 2 2
2
+
+ Qx
8
+
2 7 2 4 2 2 8 2 3 2 1 2
) ( ... ) ( 6 ) ( ) ( ... ) ( 3 x x x x x + + ˜ ¯ ˆ Á Ë Ê + + + + ˜ ¯ ˆ Á Ë Ê = H m m
2 2 2 2 2
J I s Q I Q s bosonic
t m ts t
2 3 2 1 2
2 7 2 4 2
b a abc Q s Q s
t ts t
8
s 2
Due to RR background flux Due to Metric
t Q s s
3 , 2 , 1 , , = c b a
¢ By rescaling fields and coordinates may write
action in the form
¢ semiclassical limit in which classical
solutions important
¢ In this case the classical solutions are fuzzy
spheres with time-dependent sizes
¢ For large N these are spherical D2 branes in the
2S
s s =
s
a a
c ab c b a
1 2 3
¢ Dynamics governed by the action
¢ In the original IIA theory, these fuzzy spheres are
spherical D2 branes which become smooth for large
above action in this limit.
¢ Equation of motion may be solved analytically for
specific initial conditions, but not for generic ones
2 2 2 2 2
Q s s Q t t
¢ For generic initial
¢ Similarly, very small
¢ We do not have a Sen-Seiberg type
¢ We derived this action by starting with the
¢ The fact that the equations and properties of
¢ Now we have a 2+1
dimensional YM
¢ For small the
Kaluza-Klein modes of the YM in the direction decouple and we have a 1+1 dim YM which becomes usual GS string theory in light cone gauge
B s s
8 8
4 2 2 2 2
s s B YM s s s
s
g
¢ In the presence of a
null linear dilaton the YM theory is weakly coupled at early times and there are fuzzy ellipsoids
¢ But now there is a new
effect : the masses of the KK modes become time dependent
t Q s s n
2 2 2 2 2
Particle Production A state which does not contain any such KK mode at late time is a state with a thermal distribution
In IIB these are D1 branes wrapped around the null compact direction
¢ The Type IIB in pp-wave has two holographic
descriptions The first is the 2+1 dimensional Matrix membrane theory we described.
¢ However this is also dual to the large R charge
sector of a 3+1 dimensional YM theory via AdS/CFT and BMN – some version of quiver gauge theory
¢ We have been able to find a whole class of
cosmological gravity solutions which potentially have CFT duals – but Penrose limit not clear.