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 cases we do not know much about this fundamental structure. ¢ However in some cases we have some hint – these are situations where the space-time physics has a holographic description – usually in terms of a field theory of matrices.
¢ More precisely – the holographic description is a theory of open strings which live in a lower number of dimensions. These reduces to gauge field theories with gauge groups e.g. SU(N) in some limits. ¢ Closed Strings emerge as collective descriptions of the gauge invariant dynamics. ¢ In these cases the fundamental structures are the matrices of these gauge theories.
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/ 1+1 YM 5 AdS ¥ S Strings in 3+1 dimensional N=4 5 Yang-Mills
¢ 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.
2d Closed String from Double scaled Matrix Quantum Mechanics ¢ - hermitian matrix N ¥ N M ij ( t ) 1 = Ú 2 2 S dt Tr [( D M ) M ] - t 2 ¢ Gauging – states are singlet under SU(N) ¢ Eigenvalues are fermions. Single particle hamiltonian ¢ Density of fermions
¢ To leading order in 1/N dynamics of the scalar field given by the action ¢ This collective field theory would be in fact the field theory of closed strings in two dimensions ¢ The fundamental quantum description is in terms of fermions ¢ Collective field theory used to find the emergent space-time as seen by closed strings
Ground State and fluctuations ¢ Filled fermi sea p ¢ Collective field 2 m ¢ Fluctuations x Two scalar fields for the two sides. ( 2 , t ) 0 h ± m =
¢ 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. Penrose Diagram H ¢ is independent of int time ¢ Any other conformal frame will destroy this property Diagram should be fuzzy at string scale
A Time-dependent solution S.R.D. and J. Karczmarek , PRD D71 (2005) 086006 t Æ -• ¢ An infinite symmetry of p the theory generates time dependent solutions, e.g . x t Æ finite What kind of space-time is perceived by fluctuations around this solution ?
¢ Fluctuations around any classical solution are massless scalars with metric conformal to ¢ To get the causal structure we need to go to Minkowskian coordinates in which the interactions are time independent
¢ For this solution t e e t = 2 t 1 + e t - • £ £ • ¢ As 0 - • £ t £ ¢ The space-time generated has a space like boundary The Matrix model time however runs over the full range
¢ 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
What happens there ? ¢ There are few other solutions we found which have more complicated Penrose diagrams but share the feature that the boundary appears space-like. s ¢ At the space-like boundary the closed string theory is strongly coupled except for large values of ¢ 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
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 J p = - R is equivalent to 1+1 dimensional SU(N) Yang- Mills theory with R g = YM 2 g l s s 2 on a spatial circle of radius l s R = R
X , F : J J Matrices ¥ In flat space 1 = Ú 2 2 I 2 I J 2 S Tr d d { g F ( D X ) [ X , X ] } t s + + s ts 2 m 4 g s g 0 Æ Usual space When the potential term s fi restricts the scalar fields to be diagonal and time Action becomes that of Green-Schwarz string in light cone gauge X ( 2 ) X ( ) s + p = s 1 2 fi X ( 2 ) X ( ) Single String s + p = s 2 3 X ( 2 ) X ( ) s + p = s Smaller cycles = Multiple strings N 1
Time dependent couplings ¢ Craps, Sethi and Verlinde : Consider Matrix String Theory in a background with flat string frame metric, but a dilaton Qx + F = - Q e t 2 2 Q I 2 I J 2 S Tr d d { g e F ( D X ) [ X , X ] } - t = t s + + Ú s ts 2 m 4 g s x + = t t Æ +• At usual Green-Schwarz string in light cone gauge t Æ -• At non-abelian excitations – no conventional space-time However the Yang-Mills theory is weakly coupled here
An alternative view Constant t ¢ Equivalently the YM theory can be thought to have a constant coupling, but living on the future wedge of the Milne universe t = -• 2 2 Q 2 2 ds e ( d d ) t = - t + s Big Bang Singularity As in the two dimensional example, the fundamental time of the Matrix Theory runs over the full range, but in this interpretation there is a beginning of “time”
PP Wave Big Bangs S.R.D. and J. Michelson- hep-th/0508068 ¢ 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 2 2 2 ds 2 dx dx ( x )( dx ) ( d x ) + - + = - H + m + Qx + Qx F e F e Qx + = = m F = - 8 123 + 3 + 2 2 m m Ê ˆ Ê ˆ [ ] [ ] 1 2 3 2 8 2 4 2 7 2 ( x ) ... ( x ) ( x ) ( x ) ... ( x ) H = + + + + + + Á ˜ Á ˜ 3 6 Ë ¯ Ë ¯
Pp-wave Matrix String Action Q g g e - t Æ s s Q e t 2 2 Q I 2 I J 2 S Tr d d { g e F ( D X ) [ X , X ] - t = Ú t s + + s ts bosonic 2 m 4 g s 2 M 1 2 3 2 [( X ) ... ( X ) ] - + + 18 Due to Metric 2 M 4 2 7 2 [( X ) ... ( X ) ] - + + 72 M M Q 8 Q a b c g e X F i e X X X - t t 2 - - e l m s abc ts M s 3 3 g = s R a , b , c 1 , 2 , 3 = Due to RR background flux
¢ By rescaling fields and coordinates may write action in the form 2 S S ( M , g ) ( Mg ) ( 1 , 1 ) s = 1 s Mg 1 ¢ semiclassical limit in which classical >> s solutions important 2 ¢ In this case the classical solutions are fuzzy spheres with time-dependent sizes 3 ¢ For large N these are spherical D2 branes in the original IIA string theory ab a b c a a [ J , J ] i J X ( , ) S ( ) J = e t s = t c
¢ Dynamics governed by the action 2 2 2 2 Q È ˘ J ( J 1 ) dS e 1 t - Ê ˆ Ê ˆ 2 Q S d d S Mg e S - t = s t - - Ú Á ˜ Á ˜ Í ˙ s 2 8 d 3 g t Ë ¯ Ë ¯ Í ˙ Î ˚ s ¢ In the original IIA theory, these fuzzy spheres are spherical D2 branes which become smooth for large J. The DBI+CS action is in precise agreement with above action in this limit. ¢ Equation of motion may be solved analytically for specific initial conditions, but not for generic ones
¢ For generic initial conditions at the big bang the size of the fuzzy spheres vanish at late times ¢ Similarly, very small fuzzy spheres at late times grow large at early times t At the big bang a typical state consists of these fuzzy spheres as well as Strings. At late times only strings survive
¢ We do not have a Sen-Seiberg type derivation of this action ¢ We derived this action by starting with the action of multiple D0 branes in the background – following Myers , Taylor and van Raamsdonk. ¢ The fact that the equations and properties of these fuzzy spheres agree with those of smooth D2 branes in the large-J limit provides a check.
IIB : g , l IIB Big Bangs s s x x 2 R - - ª + p 8 8 x x 2 R ª + p ¢ Now we have a 2+1 B dimensional YM J p = R - g ¢ For small the s Kaluza-Klein modes of SU ( N ) YM : ( , ) - s r r the YM in the direction decouple and 2 l 2 s s ª s + p we have a 1+1 dim YM R which becomes usual 2 l GS string theory in s 2 g r ª r + p light cone gauge s R 2 RR 2 g B = YM 4 g l s s
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