A matrix big bang Ben Craps Vrije Universiteit Brussel & The International Solvay Institutes High Energy, Cosmology and Strings IHP, Paris, December 13, 2006
Plan • Introduction: D-branes and matrix degrees of freedom • Review of matrix (string) theory • A matrix big bang • Conclusions 1
D-branes: non-perturbative objects in string theory D-branes are extended objects on which open strings can end. The oscillation modes of the open strings are the degrees of freedom of the brane. They include scalar fields X i describing the location/profile of the brane in its transverse dimensions. The tension of a D-brane is proportional to 1/g s , which is very large at weak coupling. X i Polchinski 2
Multiple D-branes have matrix degrees of freedom For two D-branes, one expects fields (X i ) 11 and (X i ) 22 , describing the transverse positions/profiles of the two branes. However, one finds more fields, corresponding to open strings stretching between the two branes: (X i ) 12 and (X i ) 21 . µ ¶ X i X i X i = 11 12 The fields combine in a 2 x 2 matrix X i X i 21 22 V ∼ Tr[ X i , X j ] 2 It turns out that there is a potential This implies that the off-diagonal modes (stretched strings) are very massive when the branes are well-separated. Then only the diagonal modes (brane positions/profiles) are light. When the branes are close to each other, all the matrix degrees of freedom are light! 3
Plan • Introduction: D-branes and matrix degrees of freedom • Review of matrix (string) theory • A matrix big bang • Conclusions 4
Type IIA string theory is M-theory on a circle • perturbative string theory: asymptotic series in g s Type IIA string theory: • what happens for large g s ? Important tool: supersymmetry (BPS states,...) 1 τ D 0 = √ α 0 → D0 becomes light at strong coupling g s There exist BPS bound states of D0-branes with masses N N τ D 0 √ α 0 This matches the spectrum of KK modes for a periodic dimension of radius R 11 = g s Conjecture: 10d type IIA string theory is a circle compactification of an 11d theory, called M-theory The low energy effective field theory for M-theory is 11d supergravity Witten 5
Type IIA string theory is M-theory on a circle (continued) ds 2 11 = G 11 MN ( x µ ) dx M dx N µ ¶ µ 4 ¶ £ dx 10 + C ν ( x µ ) dx ν ¤ 2 − 2 µ ν dx µ dx ν + exp 3 φ ( x µ ) G 10 3 φ ( x µ ) = exp dilaton 10d metric RR 1-form potential Dimensional reduction keeps only modes with . The KK modes with nonzero p 10 = 0 p 10 correspond to D0-branes and their bound states. • We know what it is when compactified on a small circle What is M-theory? • We know its low energy limit • What is its microscopic description? Matrix theory is a non-perturbative description of M-theory in 11d asymptotically Minkowski background (and some compactifications) 6
Matrix theory from DLCQ t x − x + Discrete light-cone quantization (DLCQ) µ x ¶ µ x − R/ ¶ √ x 2 √ ∼ t t + R/ 2 R x − ∼ x − + R i.e. R p + = 2 π N Momentum quantization: R p + Focus on sector with fixed total , i.e. fixed N Define DLCQ as limit of spacelike compactification: Ã q ! µ x ¶ R 2 2 + R 2 x − with R s → 0 s ∼ √ t t + R/ 2 size of spacelike circle Banks, Fischler, Shenker, Susskind; Susskind; Seiberg 7
Matrix theory from DLCQ (continued) Ã q ! µ x ¶ R 2 2 + R 2 x − with R s → 0 s ∼ √ t t + R/ 2 size of spacelike circle Lorentz boost: s µ ¶ µ ¶ µ ¶ x 0 1 + R 2 cosh β sinh β x with = cosh β = t 0 sinh β cosh β t 2 R 2 s µ ¶ µ ¶ x 0 − R s x 0 ∼ Then t 0 t 0 M-theory on spacelike circle M-theory on lightlike circle with radius R s → 0 Banks, Fischler, Shenker, Susskind; Susskind; Seiberg 8
Matrix theory from DLCQ (continued) M-theory on spacelike circle M-theory on lightlike circle with radius R s → 0 q 1 g s = ( R s M p ) 3 / 2 , R s M 3 √ Type IIA string theory with α 0 ≡ M s = But this is p 11d Planck mass In the limit, we get weakly coupled IIA strings ( ), but the string length R s → 0 g s → 0 α 0 → ∞ becomes large ( ), which would seem problematic. p + = N/R However, in M-theory on a lightlike circle, we are interested in states with P − and lightcone energy P 0 = N momentum → N D0 − branes R s After the boost: E 0 = N + ∆ E 0 energy R s 1 1 2 e β ( E 0 − P 0 ) ≈ R P − = ∆ E 0 with √ √ 2( E − P ) = R s √ R s P − ∆ E 0 ≈ R s P − ∆ E 0 So = ⇒ ≈ → 0 RM 3 / 2 R M s p Seiberg 9
Matrix theory from DLCQ (continued) p + = N/R M-theory on lightlike circle with radius R in sector with Type IIA string theory in the presence of N D0-branes with √ α 0 ∆ E 0 → 0 g s → 0 , In this limit, the only non-decoupled degrees of freedom are the NxN matrices X i of the D0-brane worldvolume theory, which reduces to the dimensional reduction of 10d super- Yang-Mills theory: “matrix theory”. Eventually, one wants to decompactify the lightlike circle: p + = N/R R → ∞ , N → ∞ with fixed Taking the matrix theory Lagrangian as a starting point, spacetime emerges from the moduli space of vacua, corresponding to flat directions for diagonal matrix elements. The large N model contains the Fock space of 11d supergravitons in its spectrum. Supersymmetry is essential to protect the flat directions. Banks, Fischler, Shenker, Susskind; Susskind; Seiberg 10
Conclusion: M-theory as matrix quantum mechanics The DLCQ of M-theory in a sector with N units of lightcone momentum is given by the low- energy limit of the worldvolume theory of N D0-branes. This is the dimensional reduction of 9+1 dimensional SYM theory to 0+1 dimensions: matrix quantummechanics. To get uncompactified M-theory, one has to take a large N limit. 11
Matrix description of type IIA strings: matrix string theory compactify x − worldvolume theory of N D0-branes in type Previously: 11d M-theory IIA, with M-theory circle x − along with radius R s compactify x 9 compactify x 9 worldvolume theory of N D0-branes in type 10d IIA string theory IIA compactified on Now: R 9 = M-theory on circle circle with radius , R 9 with radius with M-theory circle compactify x − x − along with radius R s T along x 9 worldvolume theory √ of N D1-branes in type p ) − 1 / 2 À R 9 α 0 = ( R s M 3 But : T-dualize IIB compactified on α 0 /R 9 circle with radius Motl; Banks, Seiberg; Dijkgraaf, Verlinde, Verlinde 12
Matrix description of type IIA strings: matrix string theory (continued) 10d IIA string theory worldvolume theory of with N units of N D1-branes in type IIB lightcone momentum In the previous derivation, the original IIA string theory (with N units of lightcone momentum) was related to an auxiliary IIA string theory (with N units of D0-brane charge) by a 9-11 flip (i.e. viewing two different circles as the M-theory circle). The 9-11 flip is equivalent to a sequence of T-duality, S-duality and T-duality: momentum F1 winding D1 winding D0 charge S T T Thus the original IIA theory is related to the auxiliary IIB theory above by a sequence of T-duality and S-duality. Motl; Banks, Seiberg; Dijkgraaf, Verlinde, Verlinde 13
Matrix string theory: non-perturbative string theory in Minkowski space Matrix string theory is a non-perturbative formulation of type IIA superstring theory in (9+1)-dimensional Minkowski space. It is described by the low-energy effective action of N D1-branes in type IIB string theory, which is super-Yang-Mills theory in 1+1 N = 8 dimensions with gauge group U(N), in a large N limit: µ ¶ Z µ ν − 1 [ X i , X j ] 2 + 1 ( D µ X i ) 2 + θ T D θ T γ i [ X i , θ ] / θ + g 2 s F 2 S = d τ d σ Tr g 2 g s s α are N x N hermitean matrices, transforming in the representations X i , θ α , θ ˙ 8 v , 8 s , 8 c of the SO(8) R-symmetry group of transverse rotations. The worldsheet is an infinite cylinder with coordinates where ( τ , σ ) , σ ∼ σ + 2 π . These are the same fields as in the light-cone Green-Schwarz formulation of the superstring, except that now they are matrix-valued. Relation: eigenvalues of the matrices X i correspond to coordinates of (pieces of) superstring, and similarly for the fermions. Banks, Fischler, Shenker, Susskind; Motl; Banks, Seiberg; Dijkgraaf, Verlinde, Verlinde 14
Matrix strings reduce to perturbative strings at weak string coupling µ ¶ Z µ ν − 1 [ X i , X j ] 2 + 1 ( D µ X i ) 2 + θ T D θ T γ i [ X i , θ ] / θ + g 2 s F 2 S = d τ d σ Tr g 2 g s s √ The YM coupling constant is with g Y M = 1 /g s ` s , α 0 the string length, which we ` s ≡ usually set equal to 1. To compare with perturbative string theory, one takes (This is the infrared limit of g s → 0 . the Yang-Mills theory.) Then the potential is very strong: the matrices are forced to commute and can be simultaneously diagonalized. X i The off-diagonal matrix elements have very large masses and can be integrated out: ab m ∼ || X aa − X bb || g s Supersymmetry ensures that no potential is generated for the diagonal elements! Spacetime then arises dynamically from the moduli space of vacua. Perturbative string interactions have been reproduced from the small g s limit of matrix string theory. Dijkgraaf, Verlinde, Verlinde 15
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