Membrane Matrix models. Workshop on Noncommutative Field Theory and Gravity

Transcription

1 and non-perturbative checks of AdS/CFT Denjoe O Connor Dublin Institute for Advanced Studies Workshop on Noncommutative Field Theory and Gravity Corfu Summer Institute, Corfu, Sept 22nd 2015 Based on work with Veselin Filev [arxiv: ]

2 I will review the formulation of membranes as matrix models and then present some results based on this formulation.

3 From Membranes to Matrices Hoppe in his Ph.D. thesis, with advisor Goldstone, recast the membrane action into a gauge theory of the area-preserving transformations of the membrane surface and then used a matrix regularisation to quantise the model. See [arxiv:hep-th/ ]. The Membrane action S = T d 3 σ( h 2 ( ) h αβ α X µ β X ν η µν Λ Choose Λ = 1 (rescale X a and T ), and for membranes 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 ( Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij)

4 From Membranes to Matrices Hoppe in his Ph.D. thesis, with advisor Goldstone, recast the membrane action into a gauge theory of the area-preserving transformations of the membrane surface and then used a matrix regularisation to quantise the model. See [arxiv:hep-th/ ]. The Membrane action S = T d 3 σ( h 2 ( ) h αβ α X µ β X ν η µν Λ Choose Λ = 1 (rescale X a and T ), and for membranes 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 ( Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij)

5 From Membranes to Matrices Hoppe in his Ph.D. thesis, with advisor Goldstone, recast the membrane action into a gauge theory of the area-preserving transformations of the membrane surface and then used a matrix regularisation to quantise the model. See [arxiv:hep-th/ ]. The Membrane action S = T d 3 σ( h 2 ( ) h αβ α X µ β X ν η µν Λ Choose Λ = 1 (rescale X a and T ), and for membranes 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 ( Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij)

6 From Membranes to Matrices Hoppe in his Ph.D. thesis, with advisor Goldstone, recast the membrane action into a gauge theory of the area-preserving transformations of the membrane surface and then used a matrix regularisation to quantise the model. See [arxiv:hep-th/ ]. The Membrane action S = T d 3 σ( h 2 ( ) h αβ α X µ β X ν η µν Λ Choose Λ = 1 (rescale X a and T ), and for membranes 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 ( Ẋ µ Ẋ ν η µν 4 ) ρ 2 det(h ij)

7 In 2-dim det(h ij ) can be rewritten using {f, g} = ɛ ij i f j g as S = T ρ dt (Ẋ µ Ẋ ν η µν 4ρ ) 4 2 {X µ, X ν } 2 and the constraints become 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. ( 1 ρt Pa P a + T 2ρ {X a, X b } 2 )

8 In 2-dim det(h ij ) can be rewritten using {f, g} = ɛ ij i f j g as S = T ρ dt (Ẋ µ Ẋ ν η µν 4ρ ) 4 2 {X µ, X ν } 2 and the constraints become 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. ( 1 ρt Pa P a + T 2ρ {X a, X b } 2 )

9 Hoppe then introduced matrix regularisation of the membrane and supermembrane, by treating the membrane as a phase space and quantising it. In this scheme functions on the membrane world-volume at fixed time, f (σ 1, σ 2 ) are replaced by N N matrices, f F, with the matrices providing a discrete approximation to the corresponding functions. This is very familiar for those familiar with fuzzy spaces. The Hamiltonian H = p Tr[X i, X j ] 2 i,j=1 describes a quantised fuzzy relativistic membrane in p + 1 dimensions.

10 Hoppe then introduced matrix regularisation of the membrane and supermembrane, by treating the membrane as a phase space and quantising it. In this scheme functions on the membrane world-volume at fixed time, f (σ 1, σ 2 ) are replaced by N N matrices, f F, with the matrices providing a discrete approximation to the corresponding functions. This is very familiar for those familiar with fuzzy spaces. The Hamiltonian H = p Tr[X i, X j ] 2 i,j=1 describes a quantised fuzzy relativistic membrane in p + 1 dimensions.

11 Hoppe then introduced matrix regularisation of the membrane and supermembrane, by treating the membrane as a phase space and quantising it. In this scheme functions on the membrane world-volume at fixed time, f (σ 1, σ 2 ) are replaced by N N matrices, f F, with the matrices providing a discrete approximation to the corresponding functions. This is very familiar for those familiar with fuzzy spaces. The Hamiltonian H = p Tr[X i, X j ] 2 i,j=1 describes a quantised fuzzy relativistic membrane in p + 1 dimensions.

12 Hoppe then introduced matrix regularisation of the membrane and supermembrane, by treating the membrane as a phase space and quantising it. In this scheme functions on the membrane world-volume at fixed time, f (σ 1, σ 2 ) are replaced by N N matrices, f F, with the matrices providing a discrete approximation to the corresponding functions. This is very familiar for those familiar with fuzzy spaces. The Hamiltonian H = p Tr[X i, X j ] 2 i,j=1 describes a quantised fuzzy relativistic membrane in p + 1 dimensions.

13 The Euclidean finite temperature action for the model is S b = 1 g 2 β 0 { 1 dt tr 2 (D tx i ) 2 1 } 4 [X i, X j ] 2 where D t X i = t X i + [A, X i ] and β, the period of the S 1, is the inverse temperature. It is also the high temperature limit of dimensional N = 8 supersymmetric Yang-Mills on R S 1 where β, the period of a spatial S 1 and now not the inverse temperature. The fermions drop out due to their anti-periodic boundary conditions at finite temperature..

14 The Euclidean finite temperature action for the model is S b = 1 g 2 β 0 { 1 dt tr 2 (D tx i ) 2 1 } 4 [X i, X j ] 2 where D t X i = t X i + [A, X i ] and β, the period of the S 1, is the inverse temperature. It is also the high temperature limit of dimensional N = 8 supersymmetric Yang-Mills on R S 1 where β, the period of a spatial S 1 and now not the inverse temperature. The fermions drop out due to their anti-periodic boundary conditions at finite temperature..

15 The Supermembrane A supermembrane action can only be formulated in d = 4, 5, 7 and 11 dimensional space-times was applied Matrix formulation On the Quantum Mechanics of Supermembranes, B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B 305 (1988) 545. Demise The Supermembrane Is Unstable, B. de Wit, M. Luscher and H. Nicolai, Nucl. Phys. B 320 (1989) 135. and then Supermembranes: A Fond Farewell?, B. de Wit and H. Nicolai, DESY Revival M theory as a matrix model: A Conjecture, T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D 55 (1997) 5112 [hep-th/ ].

16 The BFSS model Matrix supermembranes propagating in p + 2 dimensions coincide with p + 1-dim SU(N) Super Yang-Mills theory dimensionally reduced to 1-dim (only time dependence) They can be formulated for p = 2, 3, 5, 9 The BFSS model is the p = 9 case; it also describes a system of N interacting D0 branes.

17 Hamiltonian Formulation The 16 supercharges give the Hamiltonian: Q β = Tr( 1 2 Θ αγ a αβ P a + i 4 Θ αγ ab αβ [X a, X b ]) {Q α, Q β } = δ αβ H + γ a αβ Tr(X a J) H = Tr( 1 2 Pa P a 1 4 [X a, X b ] ΘT γ a [Θ, X a ]) where J is the generator of SU(N) and is zero on physical states J = i[p b, X b ] + Θ α Θ α δ αα N 2 1 2N

18 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, anti-symmetric tensor and gravitino of 11 d SUGRA. For an attempt to find the ground state see: J. Hoppe et al arxiv:

19 Lagrangian formulation. The easiest way to obtain the BFSS matrix model is via dimensional reduction of ten dimensional supersymmetric Yang-Mills theory down to one dimension. The resulting reduced ten dimensional action is given by 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.

20 The AdS/CFT dual geometry Since the model describes the dynamics of D0 branes AdS/CFT gives predictions for the strong regime of the theory. 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. The equations of motion of this system are R MN 1 2 g MNR = 1 2 F 2 MN 1 4 g MN F 4 2 (1) d F F 4 F 4 = 0, df 4 = 0. (2)

21 Then dimensionally reducing an S 1 gives us IIA supergravity. The reduction involves g 11 MN dx M dx N = e 2 3 Φ g 10 mndx m dx n + e 4 3 Φ (dx 10 + C m dx m ) 2 (3) A 10mn dx m dx n = B 2 2πR A lmn dx l dx m dx n = C 3 (4) and where the constant giving the string coupling has been removed from the dilaton. Then with 2κ 2 0 g s 2 = 2κ2 11 2πR where R is the radius of the X 10 circle on which the compactification is done one obtains the IIA supergravity action.

22 The leading α = ls 2 low energy effective field theory on the dual gravity side is given by IIA supergravity the bosonic part of whose action is given in the string frame by S IIA = 1 2κ 2 0 g s 2 d 10 x g{e 2Φ [R+4 dφ H G G 4 2 ]} where H 3 = db 2, G 2 = dc 1, G 4 = dc 3 + H 3 C 1 Eleven dimensional supergravity is the natural strong coupling limit of the IIA superstring. The fields (φ, g mn, B mn ) are from the NS NS sector of the IIA string while the fields (C 1, C 3 ) are from the R R sector.

23 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

24 ds 2 = α ( F H dt 2 + H F du2 + HU 2 dω 8 ) 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 to AdS prediction for the Energy S = A ( ) T 9/2 4G N λ 1/3 = E ( T λn 2 λ 1/3 ) 14/5

25 The observables that we focus on E/N 2 = R 2 = 3 4Nβ 1 Nβ β 0 β < P > = 1 TrU, N U P exp i 0 dt Tr ( [X i, X j ] 2), dt Tr ( X i) 2, β 0 dt A 0 (t).

26 Non-perturbative study via lattice simulations Discretise time to t n = an, (n = 0,..., Λ 1), a = β/λ, and periodic boundary conditions t Λ = Λa = β 0. where D t U n,n+1xn+1 i U n+1,n Xn i a, U n+1,n = U n,n+1 [ ] U n,n+1 = P exp i (n+1)a na dt A(t) parallel transport. Discretised Goldstone Hoppe regulated bosonic membrane action: Λ 1 S b = N tr { 1a X inu n,n+1 X in+1u n,n+1 + 1a (X in) 2 a4 [X in, X jn] } 2, n=0

27 Polyakov loop X»P»\ T

28 EêN T XR 2 \ T Plots of the scaled energy E/N 2 and the extent of space R 2 as functions of the temperature. The dashed curves correspond to the high temperature expansion. One can see that near T 0.9 the plots suggest the existence of a second order phase transition. The energy and temperature in the plots are in units of λ 1/3.

29 The eigenvalue distribution of the holonomy q Plots of the distribution of the holonomy P for temperatures T = (the gapped phase) and T =.9006 (the ungapped phase). The plots are for size N = 16 and lattice spacing a The dashed curves correspond to fits to the Gross-Witten gapped and untapped distributions.

30 Correlation function 6 XTrHX0XtL\ t The correlator Tr ( X 1 (0)X 1 (t) ) for N = 30, β = 10 and lattice spacing a = Fitting to A (e m t + e m(β t) ) = m = E 1 E 0 (1.90 ±.01) λ 1/3

31 The effective dynamics of the Bosonic membrane is given by the action ( 1 S eff N dt Tr 2 1 ) 2Ẋ 2 m2 X 2 One can derive this using a large 1/p expansion which to leading order in large p gives the Euclidean finite temperature action S b = N β 0 dt Tr ( ) 1 2 (D tx ) 2 + p2/3 2 X 2 This model can of course be solved analytically.

32 The gauged Gaussian model has a phase transition: The energy for N = 32 and a = EêN T The gauge field that is responsible for the phase transition. At low temperatures the eigenvalues of A 0 are uniformly distributed but at high temperatures it becomes another matrix whose eigenvalues have a Wigner distribution.

33 A detailed 1/D analysis of the membrane model is given in G. Mandal, M. Mahato and T. Morita, JHEP 1002 (2010) 034 [arxiv: ]. Numerical studies were performed in N. Kawahara, J. Nishimura and S. Takeuchi, JHEP 0710 (2007) 097 [arxiv: ] and refined in T. Azuma, T. Morita and S. Takeuchi, Phys. Rev. Lett. 113 (2014) [arxiv: [hep-th]]

34 Comments The bosonic relativistic membrane has a mass gap and at low temperatures is very well described by a system of oscillators. The result is that the relativistic bosonic membrane has only Planck mass excitations!

35 Adding Fermions Lattice implementation the fermionic part of the action: S f = 1 { } 2g 2 dτ tr ψ α C 9 αβ D τ ψ β ψ α (C 9 γ i ) αβ [X i, ψ β ] We begin by splitting the fermions into two eight component fermions: ψ = (ψ 1, ψ 2 ) and defining the forward and backward derivatives D ± : (D W ) n = (W n U n,n 1 W n 1 U n 1,n )/a,. (5) (D + W ) n = (U n,n+1 W n+1 U n+1,n W n )/a. (6)

36 The discretised kinetic term takes the form: Sf kin = 1 2g 2 dτ tr (ψ α C 9 αβ D τ ψ β) = a Λ 1 } 2g 2 tr {ψ1,n(d T ψ 2 ) n + ψ2,n(d T + ψ 1 ) n n=0 { = 1 g 2 tr Λ 1 n=0 Λ 2 ψ2,nψ T 1,n + n=0 ±ψ T 2,Λ 1 U Λ 1,0ψ 1,0 U 0,Λ 1 } ψ T 2,nU n,n+1 ψ 1,n+1 U n+1,n. The ± gives periodic/anti-periodic boundary conditions. In a static gauge the holonomy is concentrated on a singe link: { Sf kin = 1 g 2 tr Λ 1 n=0 Λ 2 ψ2,nψ T 1,n + n=0 ψ T 2,nψ 1,n+1 ± ψ T 2,Λ 1 D ψ 1,0 D }.

37 Discretised BFSS model: { SBFFS Lattice = N tr 1 Λ 2 X i a nxn+1 i 1 a X Λ 1 i D X 0D i n=0 n=0 Λ 1 ( 1 + a (X n) i 2 a ) 4 [X n, i Xn] j 2 Λ 1 n=0 Λ 1 a Λ 2 ψ2,nψ T 1,n + n=0 } ψ α,n (C 9 γ i ) αβ [Xn, i ψ β,n ] n=0 ψ T 2,nψ 1,n+1 ± ψ T 2,Λ 1 D ψ 1,0 D X i n traceless N N Hermitian matrices, ψ α,n traceless N N Hermitian matrices with Grassmann entries. D = diag{e iθ 1,..., e iθ N },

38 cos Qpf T The pfaffian phase for N = 3 and Λ = 4. The phase remains small for all temperature, but drops at very low temperatures. We believe that this is a lattice effect and is not present in the continuum limit.

39 X»P»\ T The average Polyakov loop P. All dashed curves represent the predictions of the high temperature expansion of N. Kawahara, J. Nishimura and S. Takeuchi, [arxiv: ]

40 XR 2 \ T R 2 = 1 Nβ β dt Tr ( X i) 2 which measures the extent of the 0 eigenvalues of X i.

41 EêN T The internal energy from simulations of 8 N 14 and 8 Λ 16. The data seems to converge on the low temperature prediction of the AdS/CFT correspondence especially when 1/α corrections are included.

42 Our results agrees with other groups: S. Catterall and T. Wiseman, Phys. Rev. D 78 (2008) [arxiv: [hep-th]] K. N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, Phys. Rev. Lett. 100 (2008) [arxiv: [hep-th]]. D. Kadoh and S. Kamata, arxiv: [hep-lat]. V. G. Filev and D. O Connor, arxiv: [hep-th].

43 The Berkooz Douglas model M. Berkooz and M. R. Douglas, [hep-th/ ]. M. Van Raamsdonk, [hep-th/ ]. Describes D0 D4 systems. The IKKT version describes a D( 1) D3 system see M. Van Raamsdonk, [hep-th/ ]. The more general framework involves Dp D(p + 4) systems.

44 Add new bosonic degrees of freedom Φ α as two complex N N f matrices SΦ E = 1 β ( g 2 dτ tr D Φρ τ D τ Φ ρ Φ α (σ A ) α β Jab A [X a, X b ]Φ β 0 1 ) 2 Φ α Φ β Φ β Φ α + Φ α Φ α Φ β Φ β. J A are SU(2) generators SO(4) generators (L ab ) cd = i(δ ad δ bc δ ac δ bd ) J A ab = 1 2 (L A 4) ab εabc (L BC ) ab, K A ab = 1 2 (L A 4) ab εabc (L BC ) ab

45 S χ = 1 g 2 ( tr iχ D 0 χ + χγ a X a χ + 2 i ε αβ χλ α Φ β 2 i ε αβ Φ α λ ) β χ. where λ α = P ι αψ ι. The full model is S BD = S BFSS + S Φ + S χ. The lattice discretisation is again delicate but works and simulations are in progress.

46 Conclusions Bosonic membranes when quantised are massive m p 1/3 l p. Supersymmetric membranes are highly non-trivial with infra-red divergences. Appear to be consistent with AdS/CFT and the interpretation as a non-perturbative formulation of M-theory is promising. Keep tuned!

47 Thank you for your attention!