Monte Carlo Approach to Superstring Theory for Dummies. Masanori Hanada Kyoto/Boulder

Transcription

1 Monte Carlo Approach to Superstring Theory for Dummies Masanori Hanada Kyoto/Boulder

2 Gauge/Gravity Duality (Maldacena1997) String/M Super Yang-Mills definition we want to solve it

3 How? Define the theory. Then: Diagonalize Hamiltonian, or Perform path integral. Let s use numerical methods like lattice QCD simulation.

4 One of the goals: Numerical solution of 3/4 Problem

5 3/4 Problem perturbative YM Energy density in 4d N=4 cn 2 T 4? gravity dual (3/4) cn 2 T 4 t Hooft coupling λ

6 3/4 Problem perturbative YM Energy density in 4d N=4 cn 2 T 4? gravity dual (3/4) cn 2 T 4 λ-expansion α -expansion t Hooft coupling λ

7 3/4 Problem perturbative YM Energy density in 4d N=4 cn 2 T 4 gravity dual derivation from SYM?? (3/4) cn 2 T 4 λ-expansion α -expansion t Hooft coupling λ

8 Gauge/Gravity Duality (Maldacena1997, Itzhaki-Maldacena-Sonnenschein-Yankielowicz 1998) IIA/IIB string around black p-brane definition (p+1)-d U(N)SYM (Dp-branes+strings) p=3 AdS5 S 5 p=0,1,2,3

9 Gauge/Gravity Duality (Maldacena1997, Itzhaki-Maldacena-Sonnenschein-Yankielowicz 1998) IIA/IIB string around black p-brane definition (p+1)-d U(N)SYM (Dp-branes+strings) p=3 AdS5 S 5 p=0,1,2,3 Solve it numerically. For p=0, a precision test is doable.

10 D0-brane quantum mechanics Dimensional reduction of 4d N=4, 10d N=1 Dual to type IIA black zero-brane near t Hooft limit λ = gym 2 N = (mass) 3 λ N 0 T N 0 E N 2 or, equivalently, λ=1 : fix λ -1/3 T N 0 : fix λ -1/3 E N 2

11 String theory prediction λ -1/3 E λ -1/3 T λ=1 BH mass = E = log Z β E/N 2 = 7.41T 14/5 + b T 23/5 + c T 29/5 + + O(1/N 2 )

12 SUGRA (low-t limit) E/N 2 =7.4T 14/5 high-t limit E/N 2 =6T (λ -1/3 E/N 2 ) (λ=1) strong coupling (λ -1/3 T) weak coupling Let s see how they are interpolated.

13 Anagnostopoulos-M.H.-Nishimura-Takeuchi, [hep-th] (E/T)/N 2 An earlier attempt with a mean-field method: Kabat-Lifschytz, 2001 T strong coupling weak coupling Catterall-Wiseman, [hep-th]

14 large-n continuum E/N 2 N= obtained from N=16, 24, 32 Continuum limit from 8,12,16,, 64 lattice points - ( ± ) -( ± ) +( ± ) -( ± ) ± -( ± ) ± finite-n, not continuum strong coupling SUGRA Kadoh-Kamata, 2015 weak coupling M.H. et al, 2008 Monte Carlo String/M-theory Collaboration, 2016

15 Plan How to perform (Euclidean) path integral Markov Chain Monte Carlo (MCMC) How to define super Yang-Mills lattice, supersymmetry and fine tuning problem Application to gauge/gravity duality

16 Before talking about Monte Carlo: Why do we need it?

17 One-way ticket to disaster f(x1,x2,,xp) n bins for p variables n p terms n=100, p=10 n p =10 20 n=100, p=15 n p =10 30 x1,x2,,xp

18 One-way ticket to disaster f(x1,x2,,xp) n bins for p variables n p terms n=100, p=10 n p =10 20 n=100, p=15 n p =10 30 x1,x2,,xp Livermore Lab Sequoia 20PFLOPS ( double-precision floating-point arithmetics/second) operations/5000 seconds operations/634,000 years

19 f(x1,x2,,xp) = exp( S[x1,x2,,xp]) Most of the bins are not important. We should pick up only important bins which contributes to the integral significantly. ( importance sampling ) altitude S[x] (Wikipedia)

20 What is Monte Carlo?

21 Consider field theory on Euclidean spacetime with the action. Generate field configurations with probability e S. ( Important samples are created more.) Then, Such a set of configurations can be generated as long as

22 generate a chain of field configurations with a transition probability Markov chain : transition probability from Ck to Ck+1 does not depend on C0,...,Ck-1 : probability of obtaining C at k-th step Choose so that

23 Nicholas Constantine Metropolis ( ) algorithm = choice of Metropolis simplest Hybrid Monte Carlo (HMC) useful for fermions Rational Hybrid Monte Carlo (RHMC)...etc etc...

24 Basic requirements Markov chain. Irreducibility. Aperiodicity. Detailed balance.

25 Markov Chain C0 C1 C2 Ck Ck+1 P[Ck Ck+1] does not depend on C0, C1,, Ck-1 xk xk+1=xk+δx, -1 Δx 1: uniform random number xk xk+1=xk+δx, xk-1 Δx xk-1 : uniform random number

26 Irreducibility Any two configurations are connected by finite number of steps. L -c Δx c: uniform random number c<l c>l Δx: Gaussian Random

27 Aperiodicity period of C = greatest common divisor of possible number of steps for C C aperiodic : period=1 for any C xk = positive for even k, negative for odd k (period = even) e.g. xk+1=xk*(negative random number)

28 Detailed Balance Condition Why? e S[C] P[C C ] = e S[C ] P[C C] ΣC P[C]P[C C ] = P[C ] for stationary distribution If the detailed balance condition is satisfied, P[C] e S[C] is a stationary distribution, because ΣC e S[C] P[C C ] = ΣC e S[C ] P[C C] = e S[C ] ΣC P[C C] = e S[C ]

29 Gaussian integral

30 Metropolis algorithm (Metropolis-Rosenbluth^2-Teller^2, 1953) (1) vary the field x randomly: (2) accept the new configuration with a probability where Metropolis test

31 Detailed balance ΔS = S[x ] S[x] > 0, without loss of generality. P[x x ] = 0 if x-x > 0.5; e ΔS otherwise P[x x] = 0 if x-x > 0.5; 1 otherwise e S[x] P[x x ] e S[x] e ΔS = e S[x ] e S[x ] P[x x] e S[x ]

32 Initial condition : x=0

33 100 samples 1000 samples , , samples 0.6 samples X X

34 history of x2

35

36 Initial condition : x=10 quickly thermalizes use only these configurations to calculate the expectation value. after the thermalization, configuration with small weight never appears in practice importance sampling

37

38 0 metropolis 1 if exp( ΔS) > metropolis

39 HMC, RHMC, only improve this part.

40 A bad example x x+δx, 1/2 Δx +1 Markov Chain Irreducibility Aperiodicity detailed balance

41 A bad example x x+δx, 1/2 Δx +1 Markov Chain Irreducibility Aperiodicity detailed balance #

42 A bad example x x+δx, 1/2 Δx +1 Markov Chain Irreducibility Aperiodicity detailed balance #

43 improving the efficiency

44 Initial condition : x=10 x changes little by little there is a correlation

45 How to adjust step size x x+δx, c Δx +c c too small 100% acceptance, almost random walk; inefficient Δx c, acceptance = 100% c too large rarely accepted; inefficient Δx 1, acceptance 1/c

46 How to adjust step size x n xk xk+n 1 independent samples n=1 for large c c n 1 for small c when accepted

47 How to adjust step size x n xk xk+n 1 independent samples n=1 for large c c n 1 for small c when accepted (acceptance rate) O(c) steps O(1/c 2 ) steps for 1 independent sample

48 How to adjust step size c 4.0 looks good

49 We learned how to calculate VEV s <x>, <x2 > etc How can we calculate the partition function? Z= dx exp( S[x])

50 We want to calculate Z= dx exp( S[x]). Suppose Z0= dx exp( S0[x]) is known; e.g. Gaussian integral. Then, we can use Z=(Z/Z0) Z0, and:

51 However Overlap Problem

52 S[x;c]=(x c) 2 /2 S=S[x;100], S0=[x;0] almost every time we sample here weight exp(s0 S) exp( 5000) sampled only every exp(+5000) times weight exp(s0 S) exp(+5000) Importance sampling is not working; huge error appears

53 Sk=S[x;k]=(x k) 2 /2, k=0,1,2,,100 exp( Sk) and exp( Sk+1) have enough pverlap Zk+1/Zk can be calculated reliably

54 ABJM matrix model Partition function of ABJM theory on S 3 via supersymmetric localization Beautiful analytic results Kapustin, Willett, Yaakov Drukker, Marino, Putrov, Fuji, Hirano, Moriyama,

55

56 Sign Problem What if the action is complex? S=SR+iSI The weight is now complex not probability (ABJM matrix model with different variables)

57 Sign Problem Reweighting can still work in principle In practice it is very hard: almost 0/0

58 Fermion Fermions appear in a bilinear form. (if not.. make them bilinear by introducing auxiliary fields!) can be integrated out by hand. So, simply use the effective action, not always real

59 (no-)sign problem in SYM Almost no sign at high-t Even at low-t, we can justify phase quenching at least numerically. (will be explained later)

60 Plan How to perform (Euclidean) path integral Markov Chain Monte Carlo (MCMC) How to define super Yang-Mills lattice, supersymmetry and fine tuning problem Application to gauge/gravity duality

61 warm-up example : Pure Yang-Mills (bosonic)

62 Wilson s lattice gauge theory Unitary link variable : lattice spacing ν μ x

63 Exact symmetries Gauge symmetry 90 degree rotation discrete translation Charge conjugation, parity These symmetries exist at discretized level.

64 Continuum limit respects exact symmetries at discretized level. Exact symmetries at discretized level gauge invariance, translational invariance, rotationally invariant,... in the continuum limit. What happens if the gauge symmetry is explicitly (not spontaneously) broken, (e.g. the sharp momentum cutoff prescription)?

65 We are interested in low-energy, long-distance physics (compared to the lattice spacing ). So let us integrate out high frequency modes. Then... gauge symmetry breaking radiative corrections can appear. To kill them, one has to add counterterms to lattice action, whose coefficients must be fine-tuned! fine tuning problem This is the reason why we must preserve symmetries exactly.

66 Super Yang-Mills

67 No-Go for lattice SYM SUSY algebra contains infinitesimal translation. Infinitesimal translation is broken on lattice by construction. So it is impossible to keep all supercharges exactly on lattice. Then SUSY breaking radiative corrections appear in general. Still it is possible to preserve a part of supercharges. (subalgebra which does not contain )

68 Basic ideas (Kaplan-Katz-Unsal 2002) Keep a few supercharges exact on lattice. Use it (and other discrete symmetries) to forbid SUSY breaking radiative corrections. Only extended SUSY can be realized for a technical reason. In (0+1)-d and (1+1)-d, no fine tuning to all order in perturbation.

69 (0+1)-d SYM Matrix quantum mechanics is UV finite. No fine tuning! (4d N=4 is also UV finite, but that relies on cancellations of the divergences...) We don t have to use lattice. It is sufficient just to fix the gauge & introduce momentum cutoff. (M.H.-Nishimura-Takeuchi, 2007) Lattice can also work, of course. (parallelization is easier)

70 (2+1)-d maximal SYM (Maldacena-Sheikh Jabbari-van Raamsdonk, 2003) Start with the Berenstein-Maldacena-Nastase Matrix model, which can be formulated without fine tuning. I,J=1,...,9; a,b,c=1,2,3; i=4,...,9 +(fermions) BMN model has (modified) 16 SUSY

71 Fuzzy sphere preserves 16 SUSY. Around it one obtains (2+1)-d SYM on noncommutative space. Fuzzy D2 brane out of N D0 branes (R. C. Myers 1999) Lattice is embedded in matrices. Large-N = continuum limit With maximal SUSY, commutative limit of the noncommutative space is smooth. (no UV/IR mixing) (Matusis-Susskind-Toumbas 00)

72 s spin-s representation (2s+1) (2s+1) matrix s spherical lattice consisting of (2s+1) 2 points ( fuzzy sphere ) D2-brane made of D0-branes k-copies of spin-s fuzzy sphere = k-coincident D2 N = (2s+1)k

73 (3+1)-d SYM 4d N=1 pure SYM : lattice chiral fermion assures SUSY (Kaplan 1984, Curci-Veneziano 1986) 4d N=4 : Again Hybrid formulation: Lattice + fuzzy sphere (M.H.-Matsuura-Sugino 2010, M.H. 2010) Large-N Eguchi-Kawai reduction(ishii-ishiki-shimasaki-tsuchiya, 2008) Fine tuning with 4d lattice (Catterall, Damgaard, Degrand, Joseph, Giedt, Schaich,., 2012 )

74 Application to Superstring Theory

75 Gauge/Gravity Duality (Maldacena1997, Itzhaki-Maldacena-Sonnenschein-Yankielowicz 1998) IIA/IIB string around black p-brane definition (p+1)-d U(N)SYM (Dp-branes+strings) p=3 AdS5 S 5 p=0,1,2,3 Solve it numerically.

76 Fine tuning problem 2d 3d 4d 2d, simulation 1d, simulation Cohen, Kaplan, Katz, Unsal Sugino; Catterall Maldacena, Sheikh-Jabbari, van Raamsdonk 2003 M.H., Matsuura, Sugino 2010, 2011 M.H., Kanamori 2009, 2010; Gigure, Kadoh 2015 M.H., Nishimura, Takeuchi 2007; Catterall, Wiseman 2007 Sign problem Anagnostopoulos, M.H., Nishimura, Takeuchi 2007 M.H., Kanamori 2010 M.H., Nishimura, Sekino, Yoneya 2011 Monte Carlo String/M-theory Collaboration, 2016 No problem for many important theories.

77 Gauge/Gravity Duality (Maldacena1997, Itzhaki-Maldacena-Sonnenschein-Yankielowicz 1998) IIA/IIB string around black p-brane definition (p+1)-d U(N)SYM (Dp-branes+strings) p=3 AdS5 S 5 p=0,1,2,3 Solve it numerically. For p=0, a precision test is doable.

78 Precision p=0

79 D0-brane quantum mechanics Dimensional reduction of 4d N=4, 10d N=1 Dual to type IIA black zero-brane near t Hooft limit λ = gym 2 N = (mass) 3 λ N 0 T N 0 E N 2 or, equivalently, λ=1 : fix λ -1/3 T N 0 : fix λ -1/3 E N 2

80 String theory prediction λ -1/3 E λ -1/3 T λ=1 BH mass = E = log Z β E/N 2 = 7.41T 14/5 + b T 23/5 + c T 29/5 + + O(1/N 2 ) (α /RBH 2 ) 3 (α /RBH 2 ) 5

81 SUGRA (low-t limit) E/N 2 =7.4T 14/5 high-t limit E/N 2 =6T (λ -1/3 E/N 2 ) (λ=1) strong coupling (λ -1/3 T) weak coupling Let s see how they are interpolated.

82 flat direction

83 problem with flat direction There is a flat direction even at quantum level.

84 eigenvalues = position of D0-branes bound state of eigenvalues = black hole flat direction gas of D0-branes One has to restrict the path integral in order to extract the black hole.

85 How to tame the flat direction (1) In string theory, this BH is stable at gs=0. In the gauge theory, bound state should become more stable as N becomes larger N N 1 N N 1 phase space suppression exp( N)

86 How to tame the flat direction (2) In string theory, this BH is stable at gs=0. In the gauge theory, bound state should become more stable as N becomes larger N=4 N=8 dttrxi 2 /N N=16 Monte Carlo time

87 Σ M TrXM 2 Monte Carlo time

88 Σ M TrXM 2 dot-com bubble Lehman shock S&P 500 asset price bubble Nikkei Stock Average Monte Carlo time

89 SUGRA (low-t limit) E/N 2 =7.4T 14/5 high-t limit E/N 2 =6T (λ -1/3 E/N 2 ) (λ=1) strong coupling (λ -1/3 T) weak coupling Let s see how they are interpolated.

90 Anagnostopoulos-M.H.-Nishimura-Takeuchi, [hep-th] Some history (E/T)/N 2 An earlier attempt with a mean-field method: Kabat-Lifschytz, 2001 T strong coupling weak coupling Catterall-Wiseman, [hep-th] Qualitatively Good! but Not good for any fit Not large-n, not continuum consistent with SUGRA due to cutoff effect & large error bar.

91 Anagnostopoulos-M.H.-Nishimura-Takeuchi, [hep-th] Some history (E/T)/N 2 flat direction An earlier attempt with a mean-field method: Kabat-Lifschytz, 2001 T strong coupling weak coupling Catterall-Wiseman, [hep-th] Qualitatively Good! but Not good for any fit Not large-n, not continuum consistent with SUGRA due to cutoff effect & large error bar.

92 Some history (cont d) M.H.-Hyakutake-Nishimura-Takeuchi, [hep-th] Kadoh-Kamata, [hep-lat] (See also Filev-O connor 2015)

93 M.H.-Hyakutake-Nishimura-Takeuchi, [hep-th] Kadoh-Kamata, [hep-lat] strong coupling weak coupling We need large-n and continuum result!

94 Use lattice regularization. MPI parallelization. ( Burn electricity N N temporal S 1 (# of sites = L =Nt) matrices

95 Monte Carlo String/M-theory Collaboration (MCSMC) K-supercomputer (RIKEN, Kobe, Japan) Vulcan (LLNL, Livermore, USA) Enrico Rinaldi Evan Berkowitz

96 Naive or unimproved regularization (# of sites = L =Nt)

97 improved regularization (# of sites = L =Nt)

98 fermion sign problem PfD is not real positive sign problem We use phase quench, i.e. Pf D instead of PfD It can be justified numerical. (will be explained later)

99 16 cores / node typically 2 64 nodes / job for this project

100 continuum limit (L ) with fixed N different regularizations lead to the same continuum limit. N=16, T=1.0

101 simultaneous L & N limit T=0.5 (Note that 1/N 2 -expansion works at regularized level)

102 String theory prediction λ -1/3 E λ -1/3 T λ=1 BH mass = E = log Z β E/N 2 = 7.41T 14/5 + b T 23/5 + c T 29/5 + + O(1/N 2 ) Took N, and did 3-parameter fit by E/N 2 = at 14/5 + b T 23/5 + c T 29/5

103 New large-n continuum E/N 2 N= obtained from N=16, 24, 32 Continuum limit from 8,12,16,, 64 lattice points - ( ± ) -( ± ) +( ± ) -( ± ) ± -( ± ) ± finite-n, not continuum strong coupling SUGRA Kadoh-Kamata, 2015 weak coupling M.H. et al, 2008 MCSMC, PRD 2016

104 SUGRA = finite-t E/N 2 = at 14/5 + b T 23/5 + c T 29/5 a = 7.4 +/ parameter fit (4-parameter is too much) [hep-th], [hep-lat] E/N 2 = 7.41T 14/5 + b T 23/5 + c T 29/5 + + O(1/N 2 )

105 STRING = finite-t E/N 2 = 7.41T 14/5 + b T p + c T p+6/5 3-parameter fit (4-parameter is too much) p = 4.6 +/ [hep-th], [hep-lat] E/N 2 = 7.41T 14/5 + b T 23/5 + c T 29/5 + + O(1/N 2 )

106 Preliminary new result trying

107 1/N vs Quantum String

108 gs correction in the gravity side (Y. Hyakutake, PTEP 2013) E/N 2 = 7.41T T T (1/N 2 )( 5.77T 0.4 +at ) +(1/N 4 )(bt 2.6 +ct ) +... Does SYM describe Quantum Gravity?

109 N=16, 24, N=3,4,5 - e 10 HHIN T 0.4 String Theory -5.77T 0.4 +b 1 T 2.2 b 0 T 0.4 +b 1 T 2.2 -

110 N=16, 24, e 10 HHIN String Theory -5.77T 0.4 +b 1 T 2.2 b 0 T 0.4 +b 1 T 2.2 e10 - (MCSMC, 2016) L : number of lattice points L = 8, 12, 16, 24, 32, 48, 64 N = 16, 24, 32

111 N=3, 4, e 10 HHIN String Theory -5.77T 0.4 +b 1 T 2.2 b 0 T 0.4 +b 1 T 2.2 (If you want to know the technical detail about how we controlled the flat direction at such small N, I can explain it later.) dual gravity prediction 5.77T coefficient of 1/N 2 c T M.H.-Hyakutake-Ishiki-Nishimura, Science 2014

112 Sign problem & its cure M.H., Nishimura, Sekino, Yoneya 2010 Buchoff, M.H., Matsuura 2012 MCSMC, 2016

113 Sign problem (1) Fermions appear in a bilinear form. (if not, make them bilinear by introducing auxiliary fields.) can be integrated out by hand. Monte Carlo cannot be used if it is not real positive (In maximal SYM, Pfaffian appears instead of determinant)

114 Sign problem (2) In Monte Carlo simulations, configurations are generated with probability Then by collecting many configuration one can approximate the expectation value as Crucial assumption:

115 Sign problem (3) In maximal SYM, detd is complex. The phase-quenched weight can still be studied. Phase can be taken into account by the phase reweighting in principle, but usually it s hopelessly hard. 0/0

116 No Sign problem (4) histogram of an observable Numerically observed in (0+1)-d and (1+1)-d.

117 No Sign problem (5) Even at large N/volume, where phase cannot be calculated. No correlation Polyakov loop is strongly correlated with the phase (MCSMC, 2016)

118 un-gauged matrix model Maldacena, Milekhin, to appear Berkowitz, M.H., Rinaldi, Vranas, to appear

119 At and gauge singlet constraint EOM of A t = Gauss s law ΣM ( txm i[at,xm]) 2 ΣM[ txm, XM] = 0 In operator language, ΣM[ txm, XM] is the generator of gauge transformation Gauss s law = Gauge singlet constraint remove At by hand remove gauge singlet constraint

120 gauge singlet = closed string gauge non-singlet = open string

121 gauged matrix model = BH made of closed strings ungauged matrix model = BH made of open strings open string ending at boundary

122 gauged matrix model = BH made of closed strings ungauged matrix model = BH made of open strings open string ending at boundary closed string (Maldacena-Milekhin, to appear)

123 ungauged matrix model = gauged matrix model + short open strings at boundary? all representations which can be made of adj degeneracy of adj states (must be integer) energy of adj state (Maldacena-Milekhin, to appear)

124 λ -1/3 ΔE (log scale) Preliminary λ -1/3 β d a =2, ca 1?

125 F 2 = (1/βN) Tr[XI,XJ] 2 λ -2/3 ΔF 2 (log scale) Preliminary λ -1/3 β

126 bosonic theory

127 Gauge symmetry may not be essential for the duality. Other dimensions? Generic QFT? ( may solve various technical problems associated with lattice regularization, sign problem, Hamiltonian formulation,.)

128 Wilson loop Correlation function Theories less SUSY M.H., Miwa, Nishimura, Takeuchi 2008 M.H., Nishimura, Sekino, Yoneya 2011 Gasbarro, M.H., Rinaldi, Vranas, in progress M.H., Matsuura, Nishimura, Robles-Llana 2010 Probing geometry by D-brane MCSMC, to appear this week or next week Better to re-run the simulations for better precision

129 Many more things to do GPU parallelization More efficient algorithm ( n-th root trick looks promising) 11d SUGRA from BFSS, M2/M5 from BMN More nontrivial dynamics 4d SYM Real time It has just begun!