Erik Verlinde University of Amsterdam

A Matrix Big Bang

based on work with Ben Craps and Sav Sethi 1/10 OUTLINE + SUMMARY

Introduction: Space-time singularities

String model: Lightlike linear

Einstein frame: Null singularity Lift to M-theory

M(atrix) theory: (1+1)d Super Yang-Mills on Milne Space

Conclusion: Controllable description of singularity

Apology: No references will be given in this talk. These can be found in our paper: hepth/0506180 2/10 Singularities: • inside black holes • in cosmology: Big Bang/Big Crunch

• space-, time- or light-like in GR: • divergent curvature • geodesic incompleteness

Old hope: “ theory resolves all space-time singularities”

Examples: • : space moddedout by discrete group • : p-cycle shrinks to zero • time-dept orbifolds: minkowski/ lorentztrafo.

Milne space: Null :

≈ + ’ ≈ + ’ ≈ + ’ ≈ + ’ ∆ x ÷ ∆ Λ x ÷ ∆x ÷ ∆ x ÷ ∆ x ÷ → ∆ x ÷ ∆x ÷ →∆ x +vx+ ÷ ∆ − ÷ ∆ −1 − ÷ ∆ ÷ ∆ ÷ « x ◊ « Λ x ◊ − − 1 2 + «x ◊ «x +vx+ 2 v x ◊ Lightlike Linear Dilaton 3/10

2 + − 2 Metric and dilaton: ds = − 2 dx dx + dx i (in string frame) φ = −Qx +

Note that Q Time dependent + string coupling: −Qx + is a redundant g s = e → ∞ for x → −∞ parameter

+ + ½ BPS configuration: δλ = Γ ∂ + φ ε = 0 for Γ ε = 0

Solution to eqnsof motion: R + + = T + + = ∇ + ∂ + φ = 0

+ − 1 2 + Exact CFT on : T B = −∂ X ∂ X + 2 ∂ X i ∂ X i + Q ∂ X

central charge remains unchanged! 4/10 Lightlike linear dilatonas “Big Bang cosmology”

Metric and dilatonin Einstein frame “Expanding Universe”

1 + 2 2 Qx + − 2 + ds E = e (− 2 dx dx + dx i ) φ ( x) = −Qx

1 Qx + In terms of new lightcone coordinates u = e 2 v = 2 x − 2 2 Q ds E = −2dudv + udx i φ(u) = −2logu

Null singularity at u = 0 Space-time is geodessically 1 incomplete Riuiu = → ∞ for u → 0 4u

u > 0 Einstein eqnwith dilatonsource 2 R = T = 1 (∂ φ ) 2 = uu uu 2 u u 2 u = 0 Perturbative string in lightlike linear dilaton(LLD) 5/10 in lightcone gauge X + = p +τ Γ +θ = 0

world sheet action 2 T S w .s . = — d τd σ ((∂X i ) + θ ∂/θ ) exactly like in flat space except for − QX + − Qp +τ string coupling g s = e = e

g-loop n-string amplitudes

∞ + LLD flat − ( 2 g − 2 + n )Qp τ A g ,n = A g ,n × — dτ e τ c τ perturbation theory breaks down for τ c → −∞

signal of a singularity, (but no sign of in this case) lightcone string diagram Lift to M-theory of IIA LLD 6/10

11d metric

4 + 2 + 2 − 3 Qx 2 3 Qx + − 2 ds 11 = e dy + e (− 2 dx dx + dx i )

in new lightcone coordinates 2 Qx + u = e 3 2 −2 2 2 3 − ds 11 = u dy − 2dudv + udx i v = 2 Q x

Null singularity at u = 0

1 2 R = R = − iuiu 4 u yuyu u u > 0

Einstein eqnwithout source u = 0 R uu = R yuyu + ƒ R iuiu = 0 i M(atrix) theory description 7/10

Generalize the BFSS M(atrix) theory

2 T 1 2 T I S BFSS = — dτ tr (X# I + θ θ# − 2 [X I , X J ] − θ ΓI [X ,θ ])

to the curved background

4 + 2 + 2 − 3 Qx 2 3 Qx + − 2 ds11 = e dy + e (− 2dx dx + dxi )

This gives

2 T −2Qτ 2 2 T 1 2Qτ 2 Qτ T i tr(X# i +θ θ# + e Y# −[Y, X j ] −θ [Y,θ ]− 2 e [X i , X j ] + e θ Γ [X i ,θ ])

Next compactifyon S 1 Y ≡ Dσ and tr → — dσ tr

This leads to (1+1)d SYM with time dependent coupling…..

…..or worldsheet! discrete 8/10 Matrix dual of lightlike linear dilatonin DLCQ light-cone quantization Matrix String = (1+1)d super Yang-Mills

2 2 2 T T − 22 2 2− 2 22 −1 T T i i S MS = — dσξdτtrt(r((D(DXXi )i ) ++θθ D/Dθ/ θ++g YgMs F − g YsM [[XXii,,XX jj]] −− ggsYMθθ ΓΓi [i [XX ,θ,θ])])

+ N light-cone momentum p = BounXd−ar≡y cXo−n+di2tiπoRns R break on the worldsheet string coupling 2 2Qτ / R −2 2 α β g s = e = g YM det hαβ ds w.s. = hαβ dξ dξ time dependent worldsheetmetric forward quadrant of Milne space 2 2Qτ / R 2 2 gYM = const. dsw.s. = e (−dτ + dσ )

flat world sheet coordinates ± 1 eQ(τ±σ)/ R ds2 2d +d − ± e±2πQ/ R ± ξ = 2 w.s. = − ξ ξ ξ ≡ ξ Derivation II (a la Seiberg) 9/10 2 + − 2 ds = −2dX dX + dX i Start with 10d LLD background φ = −QX + and define the DLCQ as limit + Shift inX is + − + − not a symmetry ( X , X , X 1 ) ≡ ( X , X , X 1 ) + 2π (0, R , εR ), ε → 0

after Lorentztrafo. one has ≈ X + ’ ≈ ε 0 0 ’≈ x + ’ ∆ ÷ ∆ ÷∆ ÷ ∆ X 1 ÷ = ∆ 1 1 0 ÷∆ x1 ÷ + − + − ∆ − ÷ ∆ −1 −1 −1 ÷∆ − ÷ (x , x , x) ≡ (x , x , x) + 2π (0,0,εR) « X ◊ « ε ε ε ◊« x ◊

Perform T and S-duality The p =N/R sector gets

2 εQx + + − 2 ds = εR e (− 2dx dx + dx i ) mapped on to the low energy

+ φ = −εQx + log εR description of N D1-

+ − ~ + − ~ −1 −1 (x , x , x ) ≡ (x , x , x ) + 2π (0,0,ε R ) stretched in the x~-direction

In the limit ε → 0 string oscillators and decouple 10/10 Conclusion + Generalizations

Lightlike linear dilaton: model for studying singularities in

Matrix theory dual: (1+1)d SYM on Milne with fixed coupling

Near singularity: described by weakly coupled SYM = matrices

Supersymmetry: broken by bc. , potential can be generated!

Does time begin? can one propagate through the Milne singularity?

Generalizations: T- and S-duality maps LLD to solvable backgrounds.

½ BPS solutions exist forarbitrary dilatonprofile:

with Big bang and Big crunch, or bounce….. The End