The Phase Diagram of the BMN Matrix Model

Transcription

1 Denjoe O Connor School of Theoretical Physics Dublin Institute for Advanced Studies Dublin, Ireland Workshop on Testing Fundamental Physics Principles Corfu2017, 22-28th September 2017 Background: V. Filev and D.O C. [ and ] Y. Asano, V. Filev, S. Kováčik and D.O C. [ , ].

2 A particle, a string and a membrane

3 The action functional S particle = m dτ proper = m dt 1 v 2. For small (non-relativistic) velocities this gives S particle = m dt + dt mv 2 2

4 Movement in a generic background S particle = m dt g µν dx µ dt dx ν dt this is the Nambu-Goto form of the action. If we rewrite the action using the Lagrange multiplier h as S = 1 dt{h 1 dx µ dx ν g µν hm 2 } 2 dt dt we have the Polyakov form of the action. Eliminating h with its saddle point, h = g/m, recovers the Nambu form. The equations of motion give us the geodesic equation.

5 Coupling to an electromagnetic field S charged particle = m dt g µν dx µ dt dx ν dt dx µ q dt A µdt

6 We can repeat this exercise for a string S NG = 1 2πα dσdτ detg G µν = µ X M ν X N g MN or the Polyakov form, with the Lagrange multiplier metric h µν, S P = 1 4πα dσdτ hh µν G µν Σ The string is very special in that it is a conformally invariant action. Again one can couple the string to e.g. an RR 2-form to get the S NG q µ X M ν X N ɛ µν B MN

7 Quantisation We can quantise the particle or string in either a path integral or Hamiltonian formulation and the results are well appreciated. Both can be generalised to supersymmetric versions with the string leading to string theory and conformal field theory.

8 Membrane propagating in spacetime

9 Membrane Actions Nambu Goto the simplest S NG = M d p+1 x detg G µν = µ X M ν X N g MN Higher form gauge field on the world volume 1 S p form = M (p + 1)! ɛµ 1...µ p+1 C µ1...µ p+1 C µ1...µ p+1 = µ1 X M 1... µp+1 X M p+1 C M1...M p+1 We can add an anti-symmetric part to G µν to get a Dirac-Born-Infeld action. extrinsic curvature terms. Supersymmetric S NG exist only in 4, 5, 7 and 11 dim-spacetime.

10 From Membranes to Matrices (à la Hoppe) The Membrane action, Polyakov form S = T d 3 σ ( ) h h αβ α X µ β X ν η µν Λ 2 M Choose Λ = 1 (rescale X a and T ), and for membrane topology R Σ use the gauge h 0i = 0 and h 00 = 4 ρ det(h ij). The action becomes S = T ρ 4 dt Σ ( d 2 σ Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij)

11 Noting that det( i X a j X b h ab ) = 1 p! {X a 1, X a 2..., X ap }{X b 1, X b 2..., X bp }h a1 b 1 h a2 b 2... h apb p {X a 1, X a 2..., X ap } := ɛ j 1,j 2,...,j p j1 X a 1 j2 X a 2... jp X ap becomes S = T ρ 4 dt Σ ( d 2 σ Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij) S = T ρ 4 dt Σ ( d 2 σ Ẋ µ Ẋ ν η µν 4 ) p!ρ 2 {X a 1, X a 2..., X ap } 2

12 In 2-dim det(h ij ) can be rewritten using {f, g} = ɛ ij i f j g as S = T ρ 4 dt and the constraints become Σ d 2 σ (Ẋ µ Ẋ ν η µν 4ρ ) 2 {X µ, X ν } 2 Ẋ µ i X µ = 0 = {Ẋ µ, X µ } = 0 and Ẋ µ Ẋ µ = 2 ρ 2 {X µ, X ν }{X µ, X ν }. Using lightcone coordinates with X ± = (X 0 ± X D 1 )/ 2 with X + = τ we can solve the constraint for Ẋ and Legendre transform to the Hamiltonian to find G S = T H = With the remaining constraint {P a, X a } = 0. Σ ( 1 ρt Pa P a + T 2ρ {X a, X b } 2 )

13 In this scheme functions are approximated by N N matrices, f F, and Σ f TrF. The Hamiltonian becomes H = d Tr[X i, X j ] 2 i,j=1 and describes a fuzzy relativistic membrane in d + 1 dimensions. Note: Much of the classical topology and geometry are lost in the quantum theory.

14 Once we have the Hamiltonian H we can consider thermal ensembles of membranes whose partition function is given by Z = Tr Phys (e βh ) where the physical constraint means the states are U(N) invariant. The simplest example of a quantum mechanical model with Gauss Law constraint in this class is a family of p gauged Gaussians. Their Euclidean actions are N β D τ X i = τ X i i[a, X i ]. 0 Tr( 1 2 (D τ X i ) m2 (X i ) 2 )

15 Properties of gauge gaussian models The eigenvalues of X i have a Wigner semi-circle distribution. At T = 0, we can gauged A away, while for large T we get a pure matrix model with A one of the matrices. The entry of A as an additional matrix in the dynamics signals a phase transition. In the Gaussian case with p scalars it occurs at T c = m ln p The transition can be observed as centre symmetry breaking in the Polyakov loop. Bosonic matrix membranes are approximately gauge gaussian models V. Filev and D.O C. [ and ]. Note they are the zero volume limit of Yang-Mills compactified on T 3 and on closer inspection they exhibit two phase transitions, very close in temperature.

16 Quantum Gravity At short distances it is expected [Doplicher, Fredenhagen and Roberts, 1995] that spatial co-ordinates, X a should not commute [X a, X b ] 0 in analogy with [x, p] = i in phase space, but [X a, X b ] = iθ ab breaks rotational invariance. We only need the coordinates to commute at low energies.

17 Hand waving à la Polchinski, 2014 (arxiv: ): Take each X a to be an N N matrix and try H 0 = Tr( 1 2 p Ẋ a Ẋ a 1 4 a=1 p [X a, X b ][X a, X b ]) a,b=1 The model describes membranes, Hoppe G S = T H = ( 1 ρt Pa P a + T 2ρ {X a, X b } 2 ) With the remaining constraint {P a, X a } = 0. At low energy, or the bottom of the potential [X a, X b ] = 0.

18 The BFSS model S SMembrane = G C + Fermionic terms The susy version only exists in 4, 5, 7 and 11 spacetime dimensions. BFFS Model The supersymmetric membrane à la Hoppe H = Tr( a=1 Pa P a a,b=1 [X a, X b ][X a, X b ] ΘT γ a [X a, Θ]) The model is claimed to be a non-perturbative 2nd quantised formulation of M-theory. It also describes a system of N interacting D0 branes. Note the flat directions.

19 Finite Temperature Model The partition function and Energy of the model at finite temperature is Z = Tr Phys (e βh ) and E = Tr Phys (He βh ) Z = H

20 The 16 fermionic matrices Θ α = Θ αa t A are quantised as {Θ αa, Θ βb } = 2δ αβ δ AB The Θ αa are 2 8(N2 1) and the Fermionic Hilbert space is H F = H 256 H 256 with H 256 = suggestive of the graviton (44), anti-symmetric tensor (84) and gravitino (128) of 11 d SUGRA. For an attempt to find the ground state see: J. Hoppe et al arxiv:

21 Lagrangian formulation. The BFSS matrix model is also the dimensional reduction of ten dimensional supersymmetric Yang-Mills theory down to one dimension: S M = 1 g 2 { 1 dt Tr 2 (D 0X i ) [X i, X j ] 2 i 2 ΨT C 10 Γ 0 D 0 Ψ + 1 } 2 ΨT C 10 Γ i [X i, Ψ], where Ψ is a thirty two component Majorana Weyl spinor, Γ µ are ten dimensional gamma matrices and C 10 is the charge conjugation matrix satisfying C 10 Γ µ C 1 10 = ΓµT.

22 The BMN or PWMM The supermembrane on the maximally supersymmetric plane wave spacetime ds 2 = 2dx + dx +dx a dx a +dx i dx i dx + dx + (( µ 6 )2 (x i ) 2 +( µ 3 )2 (x a ) 2 ) with dc = µdx 1 dx 2 dx 3 dx + so that F 123+ = µ. This leads to the additional contribution to the Hamiltonian H µ = N ( 2 Tr ( µ 6 )2 (X a ) 2 + ( µ 3 )2 (X i ) 2 + 2µ 3 iɛ ijkx i X j X k + µ ) 4 ΘT γ 123 Θ

23 S µ = 1 β ( 2g 2 dτtr ( µ 0 6 )2 (X a ) 2 + ( µ 3 )2 (X i ) 2 + 2µ 3 iɛ ijkx i X j X k + µ ) 4 ΨT γ 123 Ψ

24 The gravity dual and its geometry Gauge/gravity duality predicts that the strong coupling regime of the theory is described by II A supergravity, which lifts to 11-dimensional supergravity. The bosonic action for eleven-dimensional supergravity is given by S 11D = 1 2κ 2 [ gr F 4 F A 3 F 4 F 4 ] where 2κ 2 11 = 16πG 11 N = (2πlp)9 2π.

25 The relevant solution to eleven dimensional supergravity for the dual geometry to the BFSS model corresponds to N coincident D0 branes in the IIA theory. It is given by ds 2 = H 1 dt 2 + dr 2 + r 2 dω H(dx 10 Cdt) 2 with A 3 = 0 The one-form is given by C = H 1 1 and H = 1 + α 0N r 7 α 0 = (2π) 2 14πg s ls 7. where

26 Including temperature The idea is to include a black hole in the gravitational system. The Hawking termperature provides the temperature of the system. Hawking radiation We expect difficulties at low temperatures, as the system should Hawking radiate. It is argued that this is related to the flat directions and the propensity of the system to leak into these regions.

27 The black hole geometry ds 2 11 = H 1 Fdt 2 + F 1 dr 2 + r 2 dω H(dx 10 Cdt) 2 Set U = r/α and we are interested in α H(U) = 240π5 λ U 7 and the black hole time dilation factor F (U) = 1 U7 0 U 7 with U 0 = 240π 5 α 5 λ. The temperature T λ 1/3 = 1 4πλ 1/3 H 1/2 F (U 0 ) = /2 π 7/2 ( U 0 λ 1/3 ) 5/2. From black hole entropy we obtain the prediction for the Energy S = A ( ) T 9/2 4G N λ 1/3 = E ( ) T 14/5 λn 2 λ 1/3

28 Checks of the predictions We found excellent agreement with this prediction V. Filev and D.O C. [ and ]. The best current results (Berkowitz et al 2016) consistent with gauge gravity give 1 E ( ) 14 ( ) 23 N 2 λ 1/3 = 7.41 T 5 (10.0 ± 0.4) T 5 λ 1/3 λ 1/3 + (5.8 ± 0.5)T T 2 5 +(3.5±2.0)T 11 5 N Using D 4 branes as probes (these adds new fundamental matter). See: M. Berkooz and M. R. Douglas, Five-branes in M(atrix) theory, [hep-th/ ]. In IIA string theory this describes a D0 D4 system.

29 The D4-brane as a probe of the geometry. The dual adds N f D4 probe branes. In the probe approximation N f N c, their dynamics is governed by the Dirac-Born-Infeld action: N f S DBI = (2π) 4 α 5/2 d 4 ξ e det G Φ αβ + (2πα )F αβ, g s where G αβ is the induced metric and F αβ is the U(1) gauge field of the D4-brane. For us F αβ = 0. dω 2 8 = dθ 2 + cos 2 θ dω sin 2 θ dω 2 4 and taking a D4-brane embedding extended along: t, u, Ω 3 with a non-trivial profile θ(u).

30 Embeddings u é sinhql ué coshql ũ sin θ = m + c ũ

31 The condensate and the dual prediction c é T = 0.8 l 1ê mé V. Filev and D. O C. arxiv The data overlaps surprisingly well with the gravity prediction in the region where the D4 brane ends in the black hole.

32 The BMN model The BMN action S BMN = 1 2g 2 { dt Tr (D 0 X i ) 2 ( µ 6 )2 (X a ) 2 ( µ 3 )2 (X i ) 2 iψ T C 10 Γ 0 D 0 Ψ µ 4 ΨT γ 123 Ψ [X i, X j ] 2 2µ 3 iɛ ijkx i X j X k 1 } 2 ΨT C 10 Γ i [X i, Ψ],

33 Large mass expansion For large µ the model becomes the supersymmetric Gaussian model Finite temperature Euclidean Action S BMN = 1 β { 2g 2 dτ Tr (D τ X i ) 2 + ( µ 0 6 )2 (X a ) 2 + ( µ 3 )2 (X i ) 2 Ψ T D τ Ψ + µ } 4 ΨT γ 123 Ψ This model has a phase transition at T c = µ 12 ln 3

34 Perturbative expansion in large µ. Three loop result of Hadizadeh, Ramadanovic, Semenoff and Young [hep-th/ ] T c = µ { λ ln (23 µ } ln ) λ2 µ 6 +

35 Gravity prediction at small µ Costa, Greenspan, Penedones and Santos, [arxiv: ] T SUGRA c lim λ µ 2 µ = (57). The prediction is for low temperatures and small µ the transition temperature approaches zero linearly in µ.

36 Padé approximant prediction of T c with T c = µ { λ 1 + r 1 12 ln 3 µ 3 + r λ 2 } 2 µ 6 + r 1 = ln 3 and r 2 = ( ) Using a Padé Approximant: 1 + r 1 g + r 2 g (r 1 r 2 r1 )g We have T Padé c = µ 12 ln (r 1 r 2 r1 ) λ µ 3 1 r 2 r1 λ µ 3 1 r 2 r1 g

37 Now we can take the small µ limit to obtain a prediction that we can compare with supergravity T Padé c lim λ µ 2 µ = 1 12 ln 3 (1 r 2 1 r 2 ) = This is to be compared with T SUGRA c lim λ µ 2 µ = (57).

38 An initial Phase diagram for the BMN model. T λ 1/3 (μ,t)-phase diagram μ λ 1/3 Orange Large mass expansion Hadizadeh, Ramadanovic, Semenoff, Young, [hep-th/ ]. Brown Gravity prediction of Costa, Greenspan, Penedones, Santos, JHEP03(2015)069 [arxiv: [hep-th]]. Padé approximant: Blue uses only large mass expansion.

39 T λ 1/3 (μ,t)-phase diagram μ λ 1/3 Orange Large mass expansion Hadizadeh, Ramadanovic, Semenoff, Young, [hep-th/ ]. Brown Gravity prediction of Costa, Greenspan, Penedones, Santos, JHEP03(2015)069 [arxiv: [hep-th]]. Padé approximant: Red assumes Costa et al prediction, Blue uses only large mass expansion.

40 Conclusions Bosonic membranes quantised a la Hoppe are well approximated as massive gauged gaussian models. Tests of the BFSS model against non-perturbative studies are in excellent agreement. It is useful to have probes of the geometry. The mass deformed model, i.e. the BMN model is more complicated. Initial phase diagrams indicate agreement with gravity predictions But... More work is needed. A study of non-spherical type IIA black holes would be very useful.

41 Thank you for your attention!