Caricatures of Big Bang from Matrices

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Caricatures of Big Bang from Matrices

• Sumit R. Das
• University of Kentucky, Lexington

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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 a few cases we have some hint of what this
  structure could be these are situations where
  the space-time physics has a holographic
  description usually in terms of a field theory
  of matrices.
• These are in fact description of closed string
  dynamics in terms of open strings

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Examples

• Closed String Theory Open String Theory
• 2 dimensional strings Matrix Quantum
  Mechanics
• M theory/ critical string SUSY Matrix Quantum
  
• in light cone gauge Mechanics/ 11 YM
• Strings in 31
  dimensional N4
• 
  Yang-Mills

The Holographic theory usually implies an
underlying non-commutavity
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• 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.

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2d Closed String from Double scaled Matrix
Quantum Mechanics

• - hermitian matrix. This
  is the degree of freedom of open strings joining
  D0 branes
• Gauging states are singlet under SU(N)
• Eigenvalues are fermions. Single particle
  hamiltonian
• Density of fermions

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• To leading order in 1/N, the dynamics of the
  scalar field is given by the action
• This collective field theory would be in fact the
  field theory of closed strings in two dimensions
  the space dimension has emerged out of the
  matrix
• The fundamental quantum description is in terms
  of fermions
• Collective field theory used to find the emergent
  space-time as seen by closed strings

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Ground State and fluctuations

• Filled fermi sea
• Collective field
• Fluctuations
• Two scalar fields for the two sides.

p
x
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• The quadratic action for fluctuations
• Metric determined up to conformal factor
• There are actually two fields for the two
  branches of the fermi surface

• 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.

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• Space time structure is transparent in
  Minkowskian coordinates
• is independent of time
• Any other conformal frame will destroy this
  property

Penrose Diagram
Diagram should be fuzzy at string scale
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A Time-dependent solutionS.R.D. and J.
Karczmarek, PRD D71 (2005) 086006

• An infinite symmetry of the theory generates time
  dependent solutions.
• One example

p
x
Fluctuations are again massless scalars
What kind of space-time is perceived by these
fluctuations ?
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Need to go to Minkowskian coordinates
In units

• The space-time generated has a space like
  boundary
• Similarly a time-reversed solution

The Matrix model time however runs over the full
range
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• 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

It does not make sense to extend the space-time
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Lesson

• The open string time can go over the full range
• The closed string time can be terminated
• The underlying theory of open strings is still
  that of free fermions but there is no clear
  space-time interpretation.

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Space-time in Matrix String Theory for Type IIA

• 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 11 dimensional SU(J)
  Yang-Mills theory with
• on a spatial circle of radius

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Matrices
In flat space (Banks, Seiberg Dijkgraaf,
Verlinde, Verlinde Motl)
This happens in the IR of the gauge theory
When the potential term
restricts the scalar fields to be diagonal
Gauge fields become irrelevant
Action becomes identical to the world-sheet
Green-Schwarz string in light cone gauge
IN THIS LIMIT USUAL SPACE-TIME EMERGES
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Single String
One string of length
Another with length
2 Strings
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Time dependent couplings

• Craps, Sethi and E.Verlinde Matrix String
  Theory in a background with flat string frame
  metric, but a dilaton

At usual Green-Schwarz string
in light cone gauge At
non-abelian excitations
no conventional space-time
However the Yang-Mills theory is weakly coupled
here
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An alternative view
Constant

• Equivalently the YM theory can be thought to have
  a constant coupling, but living on the future
  wedge of the Milne universe

Big Bang Singularity
As in the two dimensional example, the
fundamental time of the Matrix Theory runs over
the full range, but in this Closed string
interpretation there is a beginning of time
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PP Wave Big BangsS.R.D. and J.
Michelson-Phys.Rev.D72(2005)086005,S.R.D, J.
Michelson, to appear

• 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
• For example in IIB theory

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• IIB closed string on a 2-torus with some momentum
  along one direction
• This is dual to a 21 dimensional YM theory on a
  2-torus

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This situation may be generalized to the
pp-wave type of backgrounds with a
time-dependent dilaton
Bosonic part of the action of this 21
dimensional YM
Each has a factor of
Each has a factor of
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Matrix membrane for usual pp-waves

• For time-independent pp-waves (Q0) and for weak
  IIB coupling
• (i) only diagonal Xs survive
• (ii) The gauge field gets dualized
  into a scalar field so we have 8 scalars now
• (iii) The size of the direction is
  small effectively becomes a 11 dimensional
  theory
• (iv) This 11 dimensional theory becomes the
  world-sheet theory of the original IIB string
  moving in this background

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Fuzzy ellipsoids

• semiclassical limit
  in which classical solutions important
• In this case the classical solutions are
  fuzzy ellipsoids formed by Myers effect with
  time-dependent sizes

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For Q0 there is a static solution
These fuzzy ellipsoids encode the non-commutative
nature of the theory. These must be absent in
the phase in which we recover usual perturbative
strings
In the large J limit they become D3 branes
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Time evolution

• For generic initial conditions at the big bang
  the size of the fuzzy ellipsoids vanish at late
  times
• Similarly, very small fuzzy ellipsoids at late
  times grow large at early times

At the big bang a typical state consists of
these fuzzy spheres as well as Strings. At late
times only strings survive
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• The time dependence of is responsible for
  releasing non-abelian degrees of freedom near the
  big bang
• The time dependence in front of means that
  the masses of the Kaluza Klein modes in the
  direction is time dependent
• In terms of the original IIB theory these KK
  modes are D1 branes wrapped around

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Particle depletion

• This results in production or depletion of these
  KK modes with time
• For scalars, the is defined in terms of
  the modes
• The vacuum is defined in terms of the
  modes
• Correspondingly there are creation operators
• and which are not
  equivalent

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• The nontrivial Bogoliubov relations imply that
• If the state at late time does not contain any of
  these modes, the state at early time contains
  lots of pairs of these particles in a squeezed
  state

In this big bang caricature, the universe has
to be in this special squeezed state of the KK
modes at early times to ensure that at late
times we only have perturbative strings
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Big Bangs in AdS/CFT

• The time independent IIB pp-wave has another
  holographic dual this is the large R charge
  sector of a 31 dimensional N4 YM theory
• Can we extend this duality to our null dilaton
  pp-wave ?
• This requires a deformation of
  geometry whose Penrose limit is this pp-wave.
• We havent found this yet
• If we succeed, we can pose interesting
  cosmological questions in this gauge theory.

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Conclusions

• In the toy models considered in this talk, what
  appears to be initial or final singularities from
  the point of view of closed string theory (and
  hence usual gravity) are not really singular in
  the holographic theory
• Rather in these regions the reduction of degrees
  of freedom which leads to an interpretation of
  the space of matrices as space-time is not valid
  in some cases the full noncommutative nature of
  the theory becomes significant
• Mapping these problems into problems in gauge
  theory is likely to yield significant insight