A new approach to the complex-action problem and its application to a nonperturbative study of superstring theory

7 Pages·0.21 MB·English

by  K. N. Anagnostopoulos

A new approach to the complex-action problem
and its application to a nonperturbative study of superstring theory
K.N. Anagnostopoulos∗
Department of Physics, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Greece
J. Nishimura†
The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark, and
Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
(Dated: February 1, 2008)
Monte Carlo simulations of a system whose action has an imaginary part are considered to be
extremely difficult. We propose a new approach to this ‘complex-action problem’, which utilizes
a factorization property of distribution functions. The basic idea is quite general, and it removes
theso-called overlap problem completely. Hereweapply themethodtoanonperturbativestudyof
3
superstringtheoryusingitsmatrixformulation. Inthisparticularexample,thedistributionfunction
0
turnsouttobepositivedefinite,whichallowsustoreducetheproblemevenfurther. Ournumerical
0
results suggest an intuitiveexplanation for thedynamical generation of 4d space-time.
2

n PACSnumbers: 05.10.Ln,11.25.-w,11.25.Sq
a
J

0 I. INTRODUCTION creasingly difficult as the system size increases. In this
3 sense, our approach does not solve the complex action
problem completely. This should be contrasted to the
It occurs in many interesting systems ranging from
2 meron-cluster algorithm [1], with which one can study a
condensed matter physics to high-energy physics that
v
special class of complex-action systems by computer ef-
1 their action has an imaginary part. Some examples for
fortsincreasingatmostbysomepowerofthesystemsize.
4 instance in high-energy physics are the finite density
0 QCD, Chern-Simons theories, systems with topological The factorization method eliminates the overlap prob-
8 terms (like the θ-term in QCD), and systems with chi- lem,whichcomposessomeportionofthecomplexaction
0 problem, but not the whole. However, the resolution of
ral fermions. While this is not a conceptual problem, it
1 the overlapproblemisinfactasubstantialprogress. For
poses a technical problem when one attempts to study
0
instance, Refs. [2] developed a new method to weaken
these systems by Monte Carlo simulations, which would
/
h otherwise provide a powerful tool to understand their the same problem in finite density QCD, and the criti-
t calpointwassuccessfullyidentified. Thereforeweexpect
- properties from first principles (see Refs. [1, 2, 3] for re-
p that the complete resolution of the overlap problem al-
cent works).
e lows us to address various interesting questions related
h In this Letter we propose a new approach to this
tocomplex-actionsystemswiththepresentcomputerre-
: ‘complex-action problem’. Suppose we want to obtain
v sources. Since our method is based on the general prop-
an expectation value of some observable. Then, as a
i erty of distribution functions, it can be applied to any
X more fundamental object, we consider the distribution
complex-action systems.
r function associated with that observable. In general the
a distribution function has a factorization property, which Inthisarticleweareconcernedwithanonperturbative
study of superstring theory using its matrix formulation
relatesittothe distributionfunctionassociatedwiththe
[4]. Eventually we would like to examine the possibility
same observable but calculated omitting the imaginary
that our 4-dimensional space time appears dynamically
part of the action. The effect of the imaginary part is
in 10-dimensionalstring theory [5, 6, 7, 8, 9, 10]. Monte
representedbyacorrectionfactorwhichcanbe obtained
Carlo simulation of the matrix model suffers from the
by a constrained Monte Carlo simulation. One of the
complex action problem, and there are evidences that
virtues of this method is that it removes the so-called
the imaginary part of the action plays a crucial role in
overlap problem completely. This problem comes from
thedynamicalreductionofthespace-timedimensionality
the fact that the two distribution functions — one for
[7]. We will discuss how we can study such an issue by
the full model and the other for the model omitting the
Monte Carlo simulation using the new approach.
imaginary part — have little overlap in general. The
method avoids this problem by ‘forcing’ the simulation
to sample the important region for the full model.
II. THE SUPERSTRING MATRIX MODEL
Thedeterminationofthecorrectionfactorbecomesin-
AsanonperturbativedefinitionoftypeIIBsuperstring
theoryin10dimensions,Ishibashi,Kawai,Kitazawaand
∗Electronicaddress: [email protected] Tsuchiya [4] proposed a matrix model, which can be
†Electronicaddress: [email protected] formally obtained by the zero-volume limit of D = 10,
2
= 1, pure super Yang-Mills theory. The partition where the symbol denotes a VEV with respect to
0
N h · i
function of the type IIB matrix model (and its obvious the partition function
generalizations to D =4,6) can be written as
Z = dA e−S0 . (6)
0
Z = dA e−Sb Z [A] , (1) Z
f
Z
TheVEV canbeevaluatedbystandardMonteCarlo
whehremreitAiaµn(mµa=tri1c,e·s·,·a,nDd)Sare=D bo1soTnric([NA×,AN]2tr)aicseltehses simulationhs·.iH0owever, eiΓ 0 is nothing but the ratio of
b −4g2 µ ν the two partition funct(cid:10)ions(cid:11) Z0/Z, and therefore it be-
bosonic part of the action. The factor Zf[A] represents havesase−N2∆F atlargeN,where∆F >0 isthe differ-
the quantity obtained by integration over the fermionic
ence of the free energy density of the corresponding two
matrices, and its explicit form is given for example in
systems. Thisenormouscancellation(notethat eiΓ =1
Refs. [7, 11]. The convergence of the integral (1) for | |
for each configuration) is caused by the fluctuation of
arbitrary N 2 was first conjectured [12] and proved
≥ the phase Γ, which grows linearly with the number of
recently [13]. The only parameter g in the model can be
fermionic degrees of freedom, which is of O(N2). As a
absorbedbyrescalingAµ 7→√gAµ,whichmeansthatgis resultthenumberofconfigurationsrequiredtoobtainthe
merelyascaleparameterratherthanacouplingconstant.
VEV eiΓ with sufficient accuracy grows as econst.N2.
Therefore, one can determine the g dependence of any 0
quantities ondimensionalgrounds[16]. Throughoutthis Thesa(cid:10)me(cid:11)istrueforthenumerator λieiΓ 0 in(5). This
paper, we make our statements in such a way that they is the notorious ‘complex action pro(cid:10)blem’(cid:11)(or rather the
do not depend on the choice of g. ‘sign problem’, as we see below), which occurs also in
In this model space-time is represented by A , and many other interesting systems.
µ
hence treateddynamically[5]. It is Euclideanas a result In fact we may simplify the expression (5) by using a
of the Wick rotation, which is always necessary in path symmetry. We note that under parity transformation :
integral formalisms. Its dimensionality is dynamically
determined and can be probed by the moment of inertia AP = A
1 − 1 (7)
tensor defined by [6] (cid:26)APi =Ai for 2≤i≤D ,
1
Tµν = Tr(AµAν) . (2) the fermion integral Zf[A] becomes complex conjugate
N [7], while the bosonic action S is invariant. Since the
b
Since Tµν is a D D real symmetric matrix, it has D observable λi is also invariant, we can rewrite (5) as
×
real eigenvalues corresponding to the principal moments
of inertia, which we denote as λi (i=1,···,D) with the λi = hλi cosΓi0 . (8)
ordering h i cosΓ
h i0
λ1 >λ2 > >λD >0 . (3) Note,however,thattheproblemstillremains,sincecosΓ
···
flips its sign violently as a function of A .
µ
Let us define the VEV with respect to the partition
hOi
function (1). If we find that λ with i = 1, ,d is
i
h i ···
much larger than the others, we may conclude that the
IV. THE NEW METHOD
dimensionality of the dynamical space-time is d .
A. The factorization property of distributions
III. THE COMPLEX ACTION PROBLEM
The model(6)omitting the phase Γ wasstudiedupto
The fermion integral Z [A] in the partition function N = 768 and N = 512 for D = 6 and D = 10 respec-
f
(1) is complex in general for D = 10, N 4 and for tively using the low-energy effective theory [5]. There
D =6,N 3[7]. Letusrestrictourselvesto≥these cases it was found that λ /(gN1/2) approaches a universal
i 0
≥ h i
in what follows. Parameterizing the fermion integral as constant independent of i as N increases. This means
Z [A]=exp(Γ +iΓ), the partition function (1) may be that the dynamical space-time becomes isotropic in D-
f R
written as dimensions at N = , and hence the absence of SSB of
∞
SO(D) symmetry, if one omits the phase Γ.
Z = dA e−S0 eiΓ , (4) We normalize the principal moments of inertia λ as
i
Z
λ
where S0 = Sb ΓR is real. According to the standard λ˜ d=ef i . (9)
− i
reweighting method, one evaluates the VEV hλii as hλii0
hλii= (cid:10)λeiieΓiΓ(cid:11)0 , (5) Tthheepnhtahsee.deTvhiaetrieolnevoafnhtλ˜qiiufersotmion1irsewphreestehnetrstthheedeeffveicattsioonf
h i0
3
depends on i at large N. In order to obtain the expec- The position of the peak x is given by the solution to
p
tationvalue λ˜ , we considerthe distributionassociated
h ii ∂
with the observable λ˜ : 0= lnρ (x)=f(0)(x) λ V′( λ x) , (18)
i ∂x i,V i −h ii0 h ii0
ρ (x)d=ef δ(x λ˜ ) . (10) where we have defined
i i
h − i
∂
As an important property of the distribution ρ (x), it f(0)(x)d=ef lnρ(0)(x) . (19)
i i ∂x i
factorizes as
This implies that λ V′( λ x ) gives the value of
ρi(x)= C1 ρ(i0)(x)wi(x) , (11) fi(0)(x) at x = xp.h Siiin0ce wheiit0akpe γ sufficiently large,
the distribution ρ (x) has a sharp peak, and we can
i,V
where C is a normalization constant given by safelyreplacethepositionofthepeakx bytheexpecta-
p
tionvalue λ˜ . Onceweobtainf(0)(x),wecanobtain
C d=ef heiΓi0 =hcosΓi0 . (12) ρ(i0)(x) byh iii,V i
The real positive function ρ(0)(x) is defined by x
i ρ(0)(x)=exp dzf(0)(z)+const. , (20)
i (cid:20)Z i (cid:21)
0
ρ(0)(x)d=ef δ(x λ˜ ) , (13)
i h − i i0 wherethe integrationconstantcanbe determinedbythe
(0)
which is nothing but the distribution of λ˜i in the model normalization of ρi (x).
(6) without Γ. The function ρ(0)(x) is peaked at x = 1
i
due to the chosennormalization(9). The function w (x)
i C. Resolution of the overlap problem
in (11) can be regarded as the correction factor repre-
senting the effect of Γ, and it is given explicitly as
From ρ(0)(x) and w (x), we may obtain the VEV λ˜
i i h ii
w (x)d=ef eiΓ = cosΓ , (14) by
i i,x i,x
h i h i
∞ ∞dxxρ(0)(x)w (x)
where the symbol i,x denotes a VEV with respect to λ˜ = dxxρ (x)= 0 i i . (21)
a yet another partihti·oin function h ii Z0 i R 0∞dxρ(i0)(x)wi(x)
R
Actually this simply amounts to using the reweighting
Z = dAe−S0δ(x λ˜ ) . (15)
i,x Z − i formula (8) but calculating the VEVs on the r.h.s. by
∞
Given all these definitions, it is straightforwardto prove λ˜ cosΓ = dxxρ(0)(x)w (x) (22)
the relation (11). h i i0 Z i i
0
∞
cosΓ = dxρ(0)(x)w (x) . (23)
h i0 Z i i
B. Monte Carlo evaluation of ρ(i0)(x) and wi(x) 0
This reveals one of the virtues of our approach as com-
pared with the standard reweighting method using the
In order to obtain the function w (x), we have to sim-
i formula (8) directly. Suppose we are to obtain the l.h.s.
ulate (15). In practice we simulate instead the system
of (22) and (23) by simulating the system (6). Then
for most of the time, λ˜ takes the value at the peak of
Zi,V =Z dA e−S0 e−V(λi) , (16) ρ(0)(x). However,inordiertoobtaintheVEVsaccurately
i
we have to sample configurations whose λ˜ takes a value
i
where V(λi) is some potential introduced only for the where ρ(0)(x)w (x) becomes large. In general the over-
i-th principal moment of inertia. The explicit form of | i i |
lapofthetwofunctionsbecomesexponentiallysmallwith
the potential we used in the study is V(z) = 1γ(z
2 − the system size, and this makes the important sampling
ξ)2, where γ and ξ are real parameters. The results are
ineffective. Therefore, this ‘overlap problem’ composes
insensitivetoγasfarasitissufficientlylargeandwetook
some portion of the complex-action problem. The new
γ = 1000.0. Let us denote the VEV associated with the
approach eliminates this problem by ‘forcing’ the simu-
partition function (16) as . Then the expectation
hOii,V lation to sample the important region.
value cosΓ providesthevalueofw (x)atx= λ .
i,V i i i,V
h i h i
The function ρ(0)(x) can be obtained from the same
i
simulation (16). Note that the distribution function for D. Further improvement in the case wi(x)>0
λ˜ in the system (16) is given by
i
So far, we have been discussing the general properties
ρ (x)d=ef δ(x λ˜ ) ρ(0)(x)e−V(hλii0x) . (17) of the new method. In the case at hand, we can actually
i,V h − i ii,V ∝ i
4
0
-0.002
-0.004
-0.006
-0.008 N= 4
N= 8
N=12
-0.01
N=16
N=18
-0.012 N=20
Φ (x)
4
Φ (x)
5
-0.014
0 1 2 3 4 5 6
FIG. 1: The function 1 lnw (x) is plotted for N =12,16,18,20. For x < 1 we also plot data for N = 4,8 to clarify the
N2 4
convergence. We extract the scaling function Φ (x) by fitting the data to some analytic function, which is represented by the
4
solid line. Thedashed line represents Φ (x),which is obtained similarly from the scaling behavior of 1 lnw (x).
5 N2 5
further reduce the problem by using the fact that the correctionfactorforlargerN byw (x)=eN2Φi(x),where
i
correctionfactorsw (x)areactuallypositivedefinite,and the multiplication by N2 and the exponentiation would
i
so is the full distribution function ρ (x). (Note that this magnify the errors in Φ (x) considerably. This does not
i i
is not guaranteed in general.) This allows us to obtain occur when we obtain the VEV λ˜ by solving (24).
i
the VEV λ˜ by minimizing the ‘free energy density’ h i
i
F (x) = h1ilogρ (x), instead of using (21). For that
wiesimply−nNee2dtosiolveF′(x)=0,whichisequivalentto
i V. RESULTS
1 d 1
N2fi(0)(x)=−dx(cid:20)N2 lnwi(x)(cid:21) . (24) Monte Carlo simulation of (16) can be performed by
using the algorithm developed for the model (6) in Ref.
[11]. The required computational effort is O(N6). In
The function in the bracket [ ] is expected to approach
· this work, we use instead the low-energyeffective theory
a well-defined function as N increases :
proposed in Ref. [5] and further developed in Ref. [8].
1 The required computational effort becomes O(N3). For
lnw (x) Φ (x) . (25)
N2 i → i the definition of the low-energy effective theory as well
as all the technical details including parameters used in
Let us note that wi(x) is nothing but the expectation the simulations, we follow Ref. [8]. The validity of the
value of eiΓ in the system (15). According to the ar- low-energy effective theory in studying the extent of the
gument below (6), wi(x) for fixed x decreases as e−αN2 dynamical space time is discussed in Ref. [11]. We also
at large N. The constant α may depend on x, hence notethatthecomplex-actionproblemsurvivesinpassing
the assertion. Indeed our numerical results in Fig. 1 (al- from the full theory to the low-energy effective theory,
thoughtheachievedvaluesofN arenotverylarge)seem and hence we expect that the effects of the phase on the
to support this argument. Once we extract the scaling reduction of space-time dimensionality should be visible
function Φi(x), we may use it instead of N12 lnwi(x) in alsointhelow-energyeffectivetheory,ifitisthereatall.
(24) for larger N. Thus we are able to obtain the VEV Here we study the D =6 case (instead of D =10,which
λ˜ for much larger N than those allowing the direct corresponds to the type IIB matrix model) to decrease
i
h i
Monte Carlo evaluation of the correction factor w (x). the computational efforts further.
i
Thepositivedefinitenessofw (x)iscrucialforsuchan In Fig. 1, we plot 1 lnw (x). The correction factor
i N2 4
extrapolation technique to work. If we were to calculate w (x) has a minimum at x 1 andit becomes largerfor
4
the VEV λ˜ by (21), we would need to calculate the bothx<1andx>1. This∼canbeunderstoodasfollows.
i
h i
5
N= 64
N=128
0.015 - Φ’ (x)
4
0.01
0.005
0
-0.005
-0.01
0 2 4 6 8 10 12
FIG. 2: The function 1 f(0)(x) is plotted for N = 64,128. The solid line represents −Φ′(x), which we calculate from the
N2 4 4
scaling function Φ (x) extracted in Fig. 1.
4
Let us recallagainthat w (x) is the expectation value of the fact that ρ(0)(x) is peaked at x = 1. The solid
i i
eiΓ inthesystem(15),whereλ˜ isconstrainedtoagiven line represents Φ′(x). The intersections of the two
i − 4
valueofx. Atx=1,thesystem(15)isalmostequivalent curves provide the solutions to (24). At N = 128, we
to the system (6), because λ˜i would be close to 1 even find that the distribution ρ4(x) has two peaks; one at
without the constraint. (From this, it also follows that x = xs < 1 and the other at x = xl > 1. The ratio
wi(1) takes almost the same value for all i.) Therefore, of the peak height R = ρ4(xs)/ρ4(xl) can be written as
the dominant configurations of the system (15) at x = R = exp N2( s l) , where s and l are the area
{ A −A } A A
1 is isotropic at large N [8]. On the other hand, the of the regions surrounded by the two curves. We obtain
dominant configurations of the system (15) at small x s 5.0 10−4 and l 4.5 10−3, from which we
A ∼ × A ∼ ×
are (i 1)-dimensional, since the constraint forces λ˜ to conclude that the peak at x > 1 is dominant. In Fig. 3
i
besma−ll,andduetotheordering(3),alltheλ˜ withj we show the results of a similar analysis for ρ5(x). We
j
≥ find that the distribution ρ (x) at N =128 also has two
i become small. Similarly the dominant configurations 5
peaks; one at x < 1 and the other at x > 1. However,
of the system (15) at large x are almost i-dimensional,
here we obtain 2.0 10−3 and 3.8 10−3,
since the constraint forces λ˜i to be large,and due to the which are compaArasb∼le. × Al ∼ ×
ordering(3), alltheλ˜ with j i become large. Now let
j
≤
usrecallthatthephaseΓvanisheswhentheconfiguration
A has the dimensionality d d , where d = 4,6 for
cr cr
D =6,10,respectively[7]. A≤s aconsequence, w (x) gets VI. SUMMARY AND DISCUSSIONS
4
larger in both x<1 and x>1 regimes.
As mentioned already, Fig. 1 supports the scaling be- We have proposed a new method to study complex-
havior (25) with increasing N. The scaling function action systems by Monte Carlo simulations. In partic-
Φ (x) can be extracted by fitting the data to some ana- ular we discussed how we can use the method to inves-
4
lytic function. We find that Φ (x) approaches 0 linearly tigate the possibility that four-dimensional space time
4
asx 0,anditapproachessomenegativeconstantexpo- is dynamically generated in the type IIB matrix model.
→
nentially as x . We observea similar scalingbehav- Thespace-timedimensionalityisprobedbytheeigenval-
ior for 1 lnw→(x∞). The corresponding scaling function ues λ of the moment of inertia tensor and we study the
N2 5 i
Φ5(x) is plotted in Fig. 1 for comparison. distribution of each eigenvalue. The distribution ρ(0)(x)
i
Fig. 2 represents a graphical solution of (24) for i = obtained without the phase Γ has a single peak, which
4. The open and closed circles describe the function is located at x = 1. The effect of the phase Γ on the
1 f(0)(x) for N = 64,128 respectively. It is positive distributionfunctionisrepresentedbythemultiplication
N2 4
at x < 1 and turns negative at x > 1, which reflects of the correction factor w (x) as stated in (11).
i
6
N= 64
N=128
0.015 - Φ’ (x)
5
0.01
0.005
0
-0.005
-0.01
0 2 4 6 8 10 12
FIG. 3: The function 1 f(0)(x) is plotted for N = 64,128. The solid line represents −Φ′(x), which we calculate from the
N2 5 5
scaling function Φ (x) shown in Fig. 1.
5
Our results for the 4-th and 5-theigenvalues (i=4,5) which implies that such a system is not easy to collapse
in the D = 6 case show that the correction factor w (x) into a configuration with dimensions 3. The conse-
i
≤
strongly suppresses the peak of ρ(0)(x) at x = 1 and quencewouldbethatρ(0)(x)ismuchmoresuppressedin
i 4
favours both smaller x and larger x. As a result, we ob- the small x regime than ρ(0)(x) at large N. We consider
5
serve that the distribution ρi(x) including the effects of thatthispreventsthepeakatx<1fromdominatingfor
the phase, in fact, has a double peak structure. More- ρ (x), andasaresultweobtain4dspace-time. Sincethe
4
over, the two peaks tend to move away from x=1 as N above argument is based only on the scaling behaviors
is increased. It is important to determine which of the and the branched polymer description, it is expected to
two peaks becomes dominant in the large N limit. At be valid also in the D = 10 case. (While this paper was
N =128,we observethat the peak at x>1 is dominant being revised, an analytic evidence for the dominance of
for both ρ4(x) and ρ5(x). We note, however, that it is 4d space-time was also reported [14].)
much more dominant for ρ4(x) than for ρ5(x). Our new approachto complex-actionsystems is based
From Figs. 2 and 3, we observe that the function on the factorization property (11) of distribution func-
1 f(0)(x) changes drastically as we go from N = 64 tions, which is quite general. As we discussed in Sec-
N2 i
to N =128. In fact we find that 1f(0)(x) scales (notice tion IVC, it resolves the overlapproblem completely. In
N i a separate paper we will report on a test of the new
thenormalizationfactor 1),asshowninFig.4fori=4.
N method in a Random Matrix Theory for finite density
The scaling region extends from x 1, where 1f(0)(x) QCD, where exact results in the thermodynamic limit
∼ N i
crosses zero, namely the place where ρ(0)(x) has a peak. are successfully obtained [15]. We hope that the ‘factor-
i
A similar scaling behavior is observed for i = 5. This ization method’ allows us to study interesting complex-
scaling behavior is understandable if we recall that the action systems in various branches of physics.
long-distance property of the system is controlled by a
branched-polymer like system [5], which is essentially a
system with N degrees of freedom. If we naively extrap- Acknowledgments
olate this scaling behavior of 1f(0)(x) to larger N, the
N i
l.h.s. of (24) becomes negligible. It follows that the peak We would like to thank J. Ambjørn, H. Aoki, W. Bi-
at x < 1 eventually dominates for both i = 4,5, consid- etenholz, Z. Burda, S. Iso, H. Kawai, E. Kiritsis, P. Or-
ering the asymptotic behaviors of Φ (x) as x 0 and land,M.Oshikawa,B.PeterssonandG.Vernizzifor dis-
i
→
x . This means that the space-time dimensional- cussions. The computation has been done partly on Fu-
→ ∞
ity becomes d 3. However, it is well-known that the jitsuVPP700EatTheInstituteofPhysicalandChemical
≤
Hausdorff dimension of a branched polymer is d = 4, Research(RIKEN),andNECSX5atResearchCenterfor
H
7
5
N= 16
N= 32
4 N= 64
N=128
3
2
1
0
-1
0 2 4 6 8 10 12
FIG. 4: The function 1f(0)(x)is plotted for N =16,32,64,128. A clear scaling behavior is observed.
N 4
Nuclear Physics (RCNP) of Osaka University. K.N.A.’s work of J.N. was supported in part by Grant-in-Aid for
researchwas partially supported by RTN grants HPRN- Scientific Research (No. 14740163) from the Ministry of
CT-2000-00122, HPRN-CT-2000-00131 and HPRN-CT- Education, Culture, Sports, Science and Technology.
1999-00161 and the INTAS contract N 99 0590. The
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