Putting M theory on a computer

0.1 MB·

by  Jun Nishimura

Putting M theory on a computer
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2 Jun Nishimura∗
HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japan,
n
a andGraduateUniversityforAdvancedStudies(SOKENDAI),Tsukuba305-0801,Japan
J E-mail: [email protected]

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2 KonstantinosN. Anagnostopoulos

NationalTechnicalUniversityofAthens,ZografouCampus,GR-15780Athens,Greece
]
t E-mail: [email protected]
a
l
- MasanoriHanada
p
e TheoreticalPhysicsLaboratory,RIKENNishinaCenter,
h
2-1Hirosawa,Wako,Saitama351-0198,Japan
[
E-mail: [email protected]

1
v ShingoTakeuchi
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GraduateUniversityforAdvancedStudies(SOKENDAI),Tsukuba305-0801,Japan
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2 E-mail: [email protected]
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.
1 We proposeanon-latticesimulationforstudyingsupersymmetricmatrixquantummechanicsin
0
a non-perturbativemanner. Inparticular,ourmethodenablesustoputM theoryona computer
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0 based on its matrix formulation proposed by Banks, Fischler, Shenker and Susskind. Here we
:
v presentMonteCarloresultsofthesamematrixmodelbutinadifferentparameterregion,which
i
X correspondsto the ’t Hooftlarge-N limit at finite temperature. In the strong couplinglimit the
r modelhasadualdescriptionintermsoftheND0-branesolutionin10dtypeIIAsupergravity.Our
a
results providehighlynontrivialevidencesfor the conjecturedduality. Inparticular, the energy
(andhencetheentropy)ofthenon-extremalblackholehasbeenreproducedbysolvingdirectly
thestronglycoupleddynamicsoftheD0-braneeffectivetheory.
TheXXVInternationalSymposiumonLatticeFieldTheory
July30-August42007
Regensburg,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
PuttingMtheoryonacomputer JunNishimura
1. Introduction
Large-N gauge theories are playing more and more important roles in theoretical particle
physics. In particular, they are considered to beuseful informulating superstring/M theories non-
perturbatively extending the idea of matrix models, which was successful for string theories in
sub-critical dimensions. Forinstance, ithasbeenconjectured thatcritical string/M theories canbe
formulated in terms of matrix models, which can be formally obtained by dimensionally reduc-
ing U(N) super Yang-Mills theory in ten dimensions to D=0,1,2 dimensions. The D=1 case
corresponds totheMatrixtheory[1],whichisconjectured todescribe MTheorymicroscopically.
Another important conjecture, which has been studied intensively over the decade, concerns
the duality between the strongly coupled large-N gauge theory and the weakly coupled super-
gravity. The best understood example is the AdS/CFT correspondence, but there are numerous
extensions tonon-conformal fieldtheories aswell. Inparticular, large-N gauge theories inlowdi-
mensionshavebeenstudiedintensivelyatfinitetemperature,whichrevealedintriguingconnections
totheblack-hole thermodynamics [2,3,4,5].
MonteCarlosimulationoflarge-N gaugetheoriesisexpectedtobeveryusefulinordertocon-
firm these conjectures or to make use of them. Indeed, the totally reduced models [6] (the gauge
theory reduced toD=0dimension) havebeenstudied inrefs.[7,8]. IntheD≥1case, somesort
of “discretization” is needed in order to put the theory on a computer. However, lattice simula-
tion of supersymmetric gauge theories is not straightforward. In some cases the lack of manifest
supersymmetry just necessitates cumbersome fine-tuning, but in the worse cases the chiral and/or
Majorana nature offermions makesitdifficult toevenformulate anappropriate lattice theory. We
propose tocircumventalltheseproblemsinherentinthelatticeapproachbyadopting anon-lattice
approach [9] for one-dimensional supersymmetric gauge theories. This approach, in particular,
enablesustoputMtheoryonacomputerusingtheMatrixtheory[1],whichtakestheformofa1d
U(N)gaugetheorywith16supercharges.
Here we demonstrate our approach by studying the same model but in a different parameter
region, which corresponds to the ’t Hooft large-N limit at finite temperature [10]. In the strong
coupling limit the model has a dual description [2] in terms of the N D0-brane solution in type
IIA supergravity. Our results provide highly nontrivial evidences for the conjectured duality. In
particular, theenergy (andhencetheentropy) ofthenon-extremal blackholehasbeenreproduced
bysolving directlythestrongly coupleddynamicsoftheD0-braneeffectivetheory.
2. SUSY matrixquantum mechanics with16 supercharges
Themodelcanbeobtainedformallybydimensionally reducing10dN =1superYang-Mills
theoryto1d. Theactionisgivenby
1 b 1 1 1 1
S= g2 dttr 2(DtXi)2−4[Xi,Xj]2+2y a Dty a −2y a (gi)ab [Xi,y b ] , (2.1)
0
Z (cid:26) (cid:27)
where D =¶ −i[A(t), · ] represents the covariant derivative with the gauge field A(t) being an
t t
N×N Hermitianmatrix. Thismodelcanbeviewedasaone-dimensional U(N)gaugetheory with
adjoint matters. The bosonic matrices X(t) (i= 1,···,9) come from spatial components of the
i
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PuttingMtheoryonacomputer JunNishimura
10d gauge field, while the fermionic matrices y a (t) (a =1,···,16) come from aMajorana-Weyl
spinor in 10d. The 16×16 matrices g in (2.1) act on spinor indices and satisfy the Euclidean
i
Clifford algebra {g ,g }=2d . Weimpose periodic and anti-periodic boundary conditions onthe
i j ij
bosons and fermions, respectively. The extent b in the Euclidean time direction then corresponds
to the inverse temperature b ≡1/T. The’t Hooft coupling constant isgiven by l ≡g2N,and the
dimensionless effectivecouplingconstantisgivenbyl˜ =l /T3. Withoutlossofgeneralityweset
l =1,hencelow(high)T corresponds tostrong(weak)coupling strength, respectively.
3. Non-latticesimulationforSUSY matrix quantum mechanics
Wefixthegaugebythestaticdiagonal gauge
1
A(t)= diag(a ,···a ), (3.1)
b 1 N
wherea canbechosentosatisfytheconstraint max (a )−min (a )≤2p usingthelargegauge
a a a a a
transformation. We have to add to the action a term S =−(cid:229) 2ln sina a−a b , which appears
FP a<b 2
fromtheFaddeev-Popovprocedure.
(cid:12) (cid:12)
WemakeaFourierexpansion (cid:12) (cid:12)
L L ′
Xiab(t)= (cid:229) X˜ianbeiw nt ; y aab(t)= (cid:229) y˜aabreiw rt , (3.2)
n=−L r=−L ′
wherew = 2p and L ′≡L −1/2. Theindices nandr takeinteger andhalf-integer values, respec-
b
tively,corresponding totheimposedboundary conditions. Introducing ashorthand notation
f(1)···f(p) ≡ (cid:229) f(1)···f(p) , (3.3)
n k1 kp
(cid:16) (cid:17) k1+···+kp=n
wecanwritetheaction(2.1)asS=S +S ,where
b f
S = Nb 1 (cid:229)L nw −a a−a b 2X˜ba X˜ab−1tr [X˜,X˜ ]2
b "2n=−L (cid:18) b (cid:19) i,−n in 4 i j 0#
(cid:16) (cid:17)
Sf = 12Nb r=(cid:229)L−′L ′"i(cid:18)rw −a a−b a b(cid:19)y˜aba,−ry˜aabr−(gi)ab tr y˜a ,−r [X˜i,y˜b ] r #. (3.4)
n (cid:16) (cid:17) o
Itisimportantthatwehaveintroduced thecutoffL afterfixingthegaugenon-perturbatively. This
ispossibleonlyin1d. Inhigherdimensions,themomentumcutoffregularization inevitablybreaks
the gauge invariance. In the bosonic case, we have checked explicitly [9] that the results of the
non-lattice simulation agreewiththeresults ofthelatticesimulationinthecontinuum limit.
Note that our action is nothing but the gauge-fixed action in the continuum except for having
a Fourier mode cutoff. This leads to various advantages over the lattice approach proposed in ref.
[11]. Supersymmetry, whichismildlybroken bythecutoff, isshown(in1dWess-Zumino model)
toberestoredmuchfasterthanthecontinuum limitisachieved. Infact,thecontinuum limitisalso
approachedfasterthanonewouldnaivelyexpectfromthenumberofdegreesoffreedom. Theseare
understandablefromthefactthatthemodesabovethecutoffarenaturallysuppressedbythekinetic
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PuttingMtheoryonacomputer JunNishimura
term. A further (albeit technical) advantage of our formulation is that the Fourier acceleration,
which eliminates the critical slowing down completely [12], can be implemented without extra
cost since we are dealing with Fourier modes directly. We consider that all these merits of the
present approach compensate the superficial increase in the computational effort by the factor of
O(L )comparedtothelatticeapproach [11]withthesamenumberofdegreesoffreedom.
The fermionic action Sf may be written in the form Sf = 12MAa r;Bb sy˜aAry˜bBs, where we have
expandedy˜a r=(cid:229) NA=21y˜aArtAintermsofU(N)generatorstA. Integratingoutthefermions,weobtain
thePfaffianPfM,whichiscomplexforgenericconfigurations oftheremainingbosonicvariables.
However,itturnsouttoberealpositivewithhighaccuracyinthetemperatureregionstudiedinthe
present work. Hence wecan replace it by |PfM|=det(D1/4), where D =M†M. One can then
applytheRationalHybridMonteCarloalgorithm [13]tostudythesystem inanefficientway.
4. Results
30
11..22
25
11..00
20
2N 15 00..88
E/ 00..4455 00..5500
10
N=12, L =4
5 N=14, L =4
black hole
HTE
0
0.0 1.0 2.0 3.0 4.0 5.0
T
Figure1:Theenergy(normalizedbyN2)isplottedagainstT.Thedashedlinerepresentstheresultobtained
by HTEup to thenextleadingorderfor N =12 [14]. Thesolid line representstheasymptoticpower-law
behavioratsmallT predictedbythegauge/gravityduality.Theupperleftpanelzoomsuptheregion,where
thepower-lawbehaviorsetsin.
In fig. 1 we plot the internal energy defined by E = ¶ (b F), where F is the free energy
¶b
ofthesystem. Ourresults interpolate nicely theweakcoupling behavior —calculated bythehigh
temperatureexpansion(HTE)uptothenextleadingorder[14]—andthestrongcouplingbehavior
E =7.4·T2.8 predicted by the gauge/gravity duality [2] from the dual black-hole geometry [15].
N2
Thepower-lawbehaviorsetsinatT ≃0.5,whichisreasonablesincetheeffectivecouplingconstant
isgivenbyl˜ =1/T3 inourconvention.
In ref. [4] the Gaussian expansion method was applied to the present model, and the energy
obtainedattheleadingorderwasfittednicelytothepowerlawE/N2=3.4·T2.7within0.25.T .
1. ThisresultisinreasonableagreementwithourdataatT ∼1,butdisagreesatlowertemperature.
In fig. 2 we plot the absolute value of the Polyakov line P= 1 (cid:229) N eia a, which is the order
N a=1
parameter fortheSSBoftheU(1)symmetry. Itchanges smoothly fortherange ofT investigated,
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PuttingMtheoryonacomputer JunNishimura
1.00
0.90
æP|
|
Æ
0.80
N=12, L =4
N=14, L =4
HTE
0.70
0.0 1.0 2.0 3.0 4.0 5.0
T
Figure 2: The Polyakovline is plotted againstT. The dashed line representsthe result of HTE up to the
nextleadingorderforN=12[14]. Thedottedlinerepresentsafittoeq.(4.1)witha=0.15andb=0.072.
1.0
0.8
0.6
æP|
|
Æ 0.4
0.2 N=16
N=32
HTE
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
T
Figure3: ThePolyakovlineforthebosonicmodel[16]. ThedashedlinerepresentstheresultofHTEupto
thenextleadingorderforN=16[14].
whichimpliestheabsenceofaphasetransition aspredicted bythegauge/gravity duality[3,5]. At
lowT itcanbefittednicelytotheasymptoticbehavior characteristic toadeconfined theory:
a
h|P|i=exp − +b . (4.1)
T
(cid:16) (cid:17)
Thisisinstrikingcontrast tothebosonic case[16]showninfig.3forcomparison1.
1Inref.[16]itwasfoundthatthereareactuallythreephasesinthebosonicmodel. Theintermediatephaseappears
inaverynarrowrangeoftemperatureTc2<T <Tc1,whereTc1=0.905(2)andTc2=0.8761(3),anditischaracterized
bythenon-uniformeigenvaluedistributionoftheholonomymatrix. Theorderofphasetransitionsaresecondorderat
T =Tc1,andthirdorderatT =Tc2.
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PuttingMtheoryonacomputer JunNishimura
5. Summary andfuture prospects
We have presented the first Monte Carlo results for the maximally supersymmetric matrix
quantum mechanics. Thenon-lattice simulation enabled ustostudythelowtemperature behavior,
whichwasnotaccessiblebyHTE.Thisprovidedhighlynon-trivialevidencesforthegauge/gravity
duality. Inparticular, weobserved thattheinternalenergyasymptotes nicelyatlowtemperatureto
theresultobtained fromthedualblack-hole geometry.
Ourresultssuggestthatnotonlythepowerbutalsothecoefficientofthepower-lawbehavioris
reproducedcorrectlybythegaugetheoryintheN→¥ andl˜ →¥ limits. Thisimpliesthatwewere
able to identify the microscopic degrees of freedom, which accounts for the Bekenstein-Hawking
entropy forthe10dnon-extremal blackhole. Theyarenothing buttheopenstrings attached tothe
D0-branes,whicharedescribedbythegaugetheory. Thisshouldbecomparedwithref.[17],which
studied extremalblackholesandreliedonthesupersymmetric non-renormalization theorem.
Assumingthedualitytoholdinthestrongersense,onemaygoonandinvestigatethequantum
and stringy corrections to the black-hole thermodynamics from the gauge theory side as finite-N
andfinite-l˜ effects. Inparticular itwouldbeinteresting tounderstand thephysicalmeaningofthe
infrared instability observed inoursimulation [10]fromthatperspective.
When we simulate M theory, we should impose periodic boundary conditions on fermions,
and then the system withfiniteN corresponds toasector ofM theory compactified on alight-like
circle[18]. However,thePfaffianwillnotbeclosetorealpositiveduetothefermioniczeromodes
unlike the situation in the present work. That may cause a technical problem known as the sign
problem when one tries to investigate the large-N behavior. The same problem occurs in Monte
Carlo studies of the totally reduced models, in which the phase of the Pfaffian is speculated [19]
to induce the spontaneous breaking of SO(10) symmetry down to SO(4). This pattern of SSB
is indeed suggested by the Gaussian expansion method [20], and it provides a natural scenario
for the dynamical generation of 4d space-time [21]. In ref. [22] it was conjectured, based on the
Eguchi-Kawai equivalence, that a similar phenomenon occurs also in the supersymmetric matrix
quantummechanics. Wehopetoaddresssuchanissuefromfirstprinciplesbyusingthenon-lattice
simulation methodtogetherwiththeideaproposed inref.[8]toovercomethesignproblem.
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