Dynamical Compactification of Extra Dimensions in the Euclidean IKKT Matrix Model via Spontaneous Symmetry Breaking
Konstantinos N. Anagnostopoulos, Takehiro Azuma, Yuta Ito, Jun Nishimura, Toshiyuki Okubo, Stratos Kovalkov Papadoudis
aa r X i v : . [ h e p - l a t ] M a y Dynamical Compactification of Extra Dimensions inthe Euclidean IKKT Matrix Model via SpontaneousSymmetry Breaking
Konstantinos N. Anagnostopoulos
Physics Department, National Technical University of Athens, Zografou Campus, GR-15780Zografou, GreeceE-mail: [email protected]
Takehiro Azuma
Institute for Fundamental Sciences, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa,Osaka, 572-8508, JapanE-mail: [email protected]
Yuta Ito
National Institute of Technology, Tokuyama College, Gakuendai, Shunan, Yamaguchi 745-8585,Japan and KEK Theory Center, High Energy Accelerator Research Organization, 1-1 Oho,Tsukuba, Ibaraki 305-0801, JapanE-mail: [email protected]
Jun Nishimura
KEK Theory Center, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba,Ibaraki 305-0801, Japan and Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho,Tsukuba, Ibaraki 305-0801, JapanE-mail: [email protected]
Toshiyuki Okubo
Faculty of Science and Technology, Meijo University, Nagoya, 468-8502, JapanE-mail: [email protected]
Stratos Kovalkov Papadoudis ∗ Physics Department, National Technical University of Athens, Zografou Campus, GR-15780Zografou, GreeceE-mail: [email protected] c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ he IKKT matrix model has been conjectured to provide a promising nonperturbative formula-tion of superstring theory. In this model, spacetime emerges dynamically from the microscopicmatrix degrees of freedom in the large- N limit, and Monte Carlo simulations of the Lorentzianversion provide evidence of an emergent (3+1)-dimensional expanding space-time. In this talk,we discuss the Euclidean version of the IKKT matrix model and provide evidence of dynamicalcompactification of the extra dimensions via the spontaneous symmetry breaking (SSB) of the10D rotational symmetry. We perform numerical simulations of a system with a severe com-plex action problem by using the complex Langevin method (CLM). The CLM suffers from thesingular-drift problem and we deform the model in order to avoid it. We study the SSB pattern aswe vary the deformation parameter and we conclude that the original model has an SO(3) sym-metric vacuum, in agreement with previous calculations using the Gaussian expansion method(GEM). We employ the GEM to the deformed model and we obtain results consistent with theones obtained by CLM. Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity"(CORFU2019)31 August - 25 September 2019Corfu, Greece ∗ Speaker. ynamical Compactification of Extra Dimensions
Stratos Kovalkov Papadoudis
1. Introduction
The type IIB matrix model [1], also known as the IKKT model, has been proposed as a nonper-turbative formulation of superstring theory. Spacetime emerges dynamically from the eigenvaluesof the bosonic matrices in the large- N limit [2] and it is possible that the extra dimensions are com-pactified dynamically via a non perturbative mechanism. Furthermore, it is possible that a uniquevacuum exists in the theory, thereby solving the so-called landscape problem.The action of the model can be formally viewed as the dimensional reduction of the 10 D , N =
1, SU( N ) super Yang-Mills (SYM) theory to zero dimensions. Numerical simulations [3, 4,5, 6, 7, 8, 9, 10, 11] suggest that continuum time emerges dynamically and 3 dimensional spaceundergoes rapid expansion after a critical time t c , while the remaining 6 dimensions do not expand.The cosmological time is defined by the eigenvalues of the temporal matrix A , and their infiniteand homogeneous distribution in the large N limit is a nontrivial dynamic result, allowing one todefine a continuum and infinitely extending time. Furthermore, the spatial matrices A i have a banddiagonal structure in the SU( N ) basis used to diagonalize A that makes possible to define space ata given time. Then one can study the structure of space, and especially its size, which for t > t c , itis found to be expanding in the three dominant dimensions. This expansion is exponential at earlytimes [5] and becomes a power law at later times [8], making possible the emergence of a realisticcosmology from the dynamics of the microscopic degrees of freedom.Simulations of the model using Monte Carlo techniques is hard because of the complex actionproblem. This originates from the e iS b factor in the path integral, where S b is the bosonic part ofthe action. This problem can be avoided by integrating the scale factor of the bosonic matrices[3, 4, 5, 6, 7, 8, 9]. One obtains a sharply peaked function of S b near the origin, which can then beapproximated by a Gaussian. The complex action problem vanishes, but the approximation favorssingular spatial configurations [10]. In [11] the 6D model at late times was studied without this ap-proximation, and the Complex Langevin Method (CLM) [12, 13] was used in order to confront thecomplex action problem. A two parameter deformation of the model was used for making the sim-ulations possible, and provided evidence that when these parameters become small, configurationswith non trivial spatial structure dominate the path integral.In this work we study the Euclidean version of the IKKT matrix model. This model, as well asrelated ones, have been studied extensively [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27],because they are more tractable numerically and are finite [15, 16], making the introduction ofinfrared cutoffs unnecessary. There is strong evidence that dynamical compactification of extradimensions in these models is realized via spontaneous symmetry breaking (SSB) of the SO(10)rotational symmetry of the model. The effective action of the model S eff = S R + i Γ , obtained afterintegrating out the fermionic degrees of freedom, is complex. The fluctuations of the phase e i Γ fora generic SO( d ), d > d > i Γ ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis plays a critical role in suppressing large dimensional configurations.The Monte Carlo simulations of the Euclidean IIB matrix model confront a very strong com-plex action problem, which needs to be addressed by using special methods. Straightforwardreweighting is not possible, since it makes the computational effort to increase exponentially withthe matrix size N , which needs to be extrapolated to infinity. A density of states based method wasused in [20, 34, 35, 22, 23, 24, 25], allowing one to study relatively large systems and providingevidence that SSB occurs from first principles. But it turned out to be hard to determine the patternof SSB, and it was not until recently [26, 36, 27] that this question could be addressed by using theComplex Langevin Method (CLM)[12, 13, 37]. The CLM is applied by complexifying the degreesof freedom and defining a stochastic process where the expectation values with respect to this pro-cess are equal to the expectation values defined in the original system. This method fails in manycases, and it was not until recently that with the help of new techniques [38, 39, 40, 41, 42] it waspossible to meet the conditions for the equivalence of the stochastic process defined by the CLMand the original system, and obtain correct results for an extended range of parameters [43, 44, 45,21, 46, 47, 48]. The CLM has been applied successfully to many systems in lattice quantum fieldtheory [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 43, 60, 61, 62, 63, 64, 65, 48, 66, 67, 68, 69, 70],and matrix models [71, 72, 21, 73, 26, 11, 48, 74, 75, 36, 27].In this talk we review the application of the CLM to the Euclidean type IIB matrix modeldiscussed in [26, 36, 27]. The successful application of the method requires the deformation ofthe model by two parameters m f and ε , which deform the Dirac operator and the Bosonic part ofthe action with mass like terms. This way, one can avoid the singular drift problem [21], whichcan lead to erroneous results in systems where singularities of the drift dominate in the stochasticprocess of the CLM [40]. In the IIB matrix model the singular drift problem occurs when theeigenvalues of the Pfaffian, when put into a canonical Youla’s form, accumulate near zero. Thismethod was applied successfully on the 6D IIB model [26], obtaining SSB to SO(3), consistentwith GEM calculations in [32]. The 10D model was studied in [27]. In that case, the simulationswere harder because the fermionic degrees of freedom increase by a factor of four and the finitesize effects are more severe due to the increase of the dimensionality of the target space. Theresults are consistent with GEM calculations [33], showing SSB to SO(3). A careful extrapolationof the results, first to large N , then to small ε and finally to small m f is necessary in order to obtainthe correct large N limit of the original model. This makes the calculation tricky, and the MonteCarlo simulations were done in parallel with GEM calculations of the m f –deformed model. Bycomputing the free energy of vacua of different dimensionality using the GEM, it was possibleto find physical solutions for the SO( d ), d = , , m f decreases, lower dimensional vacua turn out to have lower free energies and the pattern of SSB issimilar to the one obtained by the CLM. 2 ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis
2. The IKKT matrix model
The IKKT matrix model [1] is defined by the action: S = S b + S f , where (2.1) S b = − N tr [ A µ , A ν ][ A µ , A ν ] , (2.2) S f = − N tr (cid:0) ψ α ( C Γ µ ) αβ [ A µ , ψ β ] (cid:1) . (2.3)For D =
10, the matrices A µ ( µ = , , , . . . , D −
1) are N × N traceless Hermitian matrices whichtransform like vectors, and ψ α ( α = , , . . . , D / − ) are N × N traceless matrices with Grassmannentries which transform like Majorana-Weyl spinors. The 2 D / − × D / − matrices Γ µ and C arethe gamma matrices after Weyl projection and the charge conjugation matrix, respectively, in tendimensions.In order to obtain the Euclidean version of the IKKT matrix model, we perform the Wickrotation A = iA D , Γ = − i Γ D . (2.4)The metric now is δ µν ( µ , ν = , . . . , D ) and the partition function becomes Z = Z dAd ψ e − S = Z dA e − S b Pf M , (2.5)where the 2 D / − ( N − ) × D / − ( N − ) anti-symmetric matrix M is defined by its action ψ α → ( M ψ ) α = ( C Γ µ ) αβ [ A µ , ψ β ] (2.6)on the linear space of traceless complex N × N matrices. The model has an SO( D ) rotationalsymmetry acting on A µ and ψ α .Dynamical compactification of extra dimensions can be realized via the SSB of the SO( D )symmetry to SO( d ), with d < D . The order parameters of the SSB can be taken to be [21, 26] λ µ = N tr ( A µ ) , µ = , . . . , D , (2.7)where no sum over µ is taken. Then, we break the SO( D ) symmetry explicitly by deforming themodel by ∆ S b = N ε D ∑ µ = m µ tr (cid:0) A µ (cid:1) (2.8)using the parameters ε and 0 < m ≤ . . . ≤ m D . The SSB pattern will arise in the ε → after taking the large- N limit. For finite N , the choice 0 < m ≤ . . . ≤ m D yields h λ i ≥ . . . ≥ h λ D i . Ifthere is no SSB, all h λ µ i will be equal in the N → ∞ and ε → D ) rotational symmetry has been studied by using the GEM. This methoduses a systematic expansion around a Gaussian action S , which contains many parameters. Inorder to make the calculation tractable, the number of parameters is reduced by considering SO( d )–symmetric ansatzes for d < D . The free energy and the expectation values of the observables is3 ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis calculated in the expansion around S as a function of these parameters, and the physical solutionsare computed by finding a region in the parameter space where the results are independent of theseparameters. In this way, one obtains nonperturbative information about the model [76]. Whenapplied to the IKKT matrix model [29, 30, 77, 78, 79, 80, 81, 31, 82, 32, 33], the GEM yields thefree energy and the average extent of spacetime in each direction. The D = R ( d ) for µ = , . . . , d , whereas the shrunken directions to have andextent r = h λ µ i ≈ . µ = d + , . . . , D , which is independent of d . The values of R ( d ) aresuch that R d ( d ) r D − d ≈ l D , (2.9)where v ≡ l D is the spacetime volume. It turns out that l ≈ .
627 and R ( ) ≈ .
76. The D = d )ansatzes with 2 ≤ d ≤
7. The SO(3) ansatz turns out to have the lowest free energy, also showingSSB to SO(3). In that case R ( ) ≈ . , r ≈ . , l ≈ . . (2.10)These results are consistent with the ones obtained by Monte Carlo simulations [25]. They arefinite, in contrast to the case of the Lorentzian model, where R ( d ) expands indefinitely.
3. The Complex Langevin Method
In this section we review how to apply the CLM to the Euclidean IKKT matrix model. Thepartition function can be written as Z = Z dA e − S eff , (3.1)where the effective action S eff = S b − log Pf M is complex. In the CLM the matrices A µ becomegeneral complex traceless matrices, which amounts to the complexification of the degrees of free-dom in the CLM. The time evolution in the fictitious Langevin time is given by d (cid:0) A µ ( t ) (cid:1) i j dt = − ∂ S eff [ A µ ( t )] ∂ (cid:0) A µ (cid:1) ji + (cid:0) η µ (cid:1) i j ( t ) . (3.2)The noise η µ ( t ) is made from traceless Hermitian matrices whose elements are random variablesobeying the Gaussian distribution ∝ exp (cid:18) − R tr (cid:8) η µ ( t ) (cid:9) dt (cid:19) . The first term on the right-handside is the drift term ∂ S eff ∂ (cid:0) A µ (cid:1) ji = ∂ S b ∂ (cid:0) A µ (cid:1) ji −
12 Tr M − ∂ M ∂ (cid:0) A µ (cid:1) ji ! , (3.3)where Tr represents the trace of a 16 ( N − ) × ( N − ) matrix. The expectation value of anobservable O [ A µ ] can be calculated from h O [ A µ ] i = T Z t + Tt O [ A µ ( t )] dt , (3.4)4 ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis where A µ ( t ) is a general complex matrix solution of (3.2), t is the thermalization time, and T islarge enough in order to obtain good statistics. Upon complexification of the matrices A µ ( t ) , theobservable O [ A µ ( t )] depends on general complex matrices. The analyticity of the function O [ A µ ] plays a crucial role in the proof of the validity of (3.4)[38, 39, 41]. The discretization of (3.2) isgiven by (cid:0) A µ (cid:1) i j ( t + ∆ t ) = (cid:0) A µ (cid:1) i j ( t ) − ∆ t ∂ S [ A µ ( t )] ∂ (cid:0) A µ (cid:1) ji + √ ∆ t (cid:0) η µ (cid:1) i j ( t ) , (3.5)which we have used in order to solve (3.2) numerically.The solutions of the stochastic equation (3.2) are random variables with distribution P ( A ( R ) µ , A ( I ) µ ; t ) ,where A ( R ) µ ( t ) = ( A µ ( t ) + A † µ ( t )) / A ( I ) µ ( t ) = ( A µ ( t ) − A † µ ( t )) / i , and in order that in the t → ∞ limit an observable O [ A µ ] has an expectation value in this distribution equal to h O [ A µ ] i , given bythe path integral (2.5), certain conditions need to be met. In [41] it was shown that if the magnitudeof the drift falls off exponentially or faster, then these conditions are met. For this, the A µ ( t ) shouldnot make long trips in the anti-Hermitian direction. Gauge cooling has been applied in order torestrict those trips and meet this condition[26, 27]. Adaptive stepsize techniques have also beenapplied in order to keep the stability of the time evolution. Furthermore, one has to avoid the singu-lar drift problem. This problem occurs in the term with M − in Eq. (3.3), when the eigenvalues of M accumulate densely near zero. In order to avoid this problem, we deform the fermionic actionby adding a term ∆ S f = − im f N (cid:16) ψ α ( C Γ Γ †9 Γ ) αβ ψ β (cid:17) , (3.6)where m f is the deformation parameter. This term shifts the eigenvalue distribution of M awayfrom the origin [21], but it also breaks the SO(10) symmetry down to SO ( ) × SO ( ) explicitly.Therefore, we examine if the remaining SO(7) symmetry breaks down to smaller subgroups as m f is varied and discuss what occurs at m f =
0. As m f → ∞ , the fermionic degrees of freedom decoupleand we obtained the so–called bosonic model. This model is known to be SO(10) symmetric andno SSB occurs [25].The severeness of the singular drift problem depends on the parameters m f and ε . For largeenough m f the problem disappears, but as m f is lowered the problem reappears for small enough ε . In our simulations we monitor the fall off of the drift and make sure that it falls of faster thanexponentially. The singular drift problem is expected to vanish for large enough N [27].
4. Results
In our numerical investigation we consider whether the remaining SO(7) symmetry is brokendown to a smaller group. We study the model deformed by (2.8) and (3.6). We use m f = . .
4, 1 .
0, 0 .
9, 0 .
7, in order to extrapolate to the undeformed IKKT model at m f =
0. The m µ in Eq. (2.8) are chosen so that this term does not break SO(10) completely, because otherwisethe spectrum of m µ becomes too wide to make the ε → m f = . m µ = ( . , . , . , , , , , , , ) , which enables us to distinguish SO( d ) vacua with d = , , , ,
7. For smaller values of m f , we choose m µ = ( . , . , , , , , , , , ) , whichenables us to distinguish SO( d ) vacua with d = , , ,
7. In particular, we may confirm that theSO(3) symmetry remains unbroken by seeing that h λ i = h λ i and h λ i agree in the N → ∞ and5 ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis ε → m µ has a drawback that λ and λ are mixed upbecause of m = m , and hence one cannot distinguish SO(5) and SO(6) vacua. This does not causeany harm, however, as far as we find that h λ i and h λ i do not agree in the N → ∞ and ε → (cid:1) µ ( m f = . , (cid:2) ) (cid:0) ρ µ ( m f = . , ε ) ε ρ µ ( m f = . , ε ) ε ρ µ ( m f = . , ε ) ε ρ µ ( m f (cid:3)(cid:4)(cid:5)(cid:6) ε ) ε Figure 1:
The ρ µ ( m f , ε ) in Eq. (4.2) are plotted against ε for m f = . m f = . m f = . m f = . m f = . m µ =( . , . , . , , , , , , , ) for m f = . m µ = ( . , . , , , , , , , , ) for the other values of m f .The continuous lines are polynomial fits in ε . For m f = . m f the fits are quadratic in ε . In the m f = . ( ρ + ρ + ρ ) / ρ , ρ , ρ , ρ and ( ρ + ρ + ρ ) /
3. For the other plots, the curves from top to bottomare ( ρ + ρ ) / ρ , ρ , ρ , ( ρ + ρ ) / ( ρ + ρ + ρ ) / In order to probe the SSB, one has to take the N → ∞ limit first and then the ε → m µ in Eq. (2.8), the large- N limit is obtained by first computing the ratio ρ µ ( m f , ε , N ) = h λ µ i m f , ε , N ∑ ν = h λ ν i m f , ε , N , (4.1)and then by making a large- N extrapolation ρ µ ( m f , ε ) = lim N → ∞ ρ µ ( m f , ε , N ) . (4.2)6 ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis
The large- N extrapolation is performed by plotting ρ µ ( m f , ε , N ) against 1 / N and making a quadraticfit with respect to 1 / N . Then we make the ε → ρ µ ( m f ) = lim ε → ρ µ ( m f , ε ) (4.3)by fitting ρ µ ( m f , ε ) to a polynomial in ε . In Fig. 1 we plot the large- N extrapolated values ρ µ ( m f , ε ) as a function of ε for m f = .
0, 1.4, 1.0, 0.9 and 0 . ε → m f ≤ . ε →
0. This crossover is expected to vanish in thelarge– N limit.From the extrapolated values ρ µ ( m f ) , we find that the SO(7) symmetry of the deformed modelis not spontaneously broken at m f = .
0, but it is actually broken to SO(4) for m f = . m f = .
0, 0 .
9, 0 .
7. Thus as m f is decreased, the SO(7) symmetry seems to be spontaneouslybroken to smaller subgroups gradually in the same way as it was observed in the D = m f . This is based on the fact that the Pfaffian vanishes identically for strictly 2D configurations[28, 83], which implies that the mechanism of SSB due to the phase of the Pfaffian no longer worksthere . Hence our results are consistent with the results obtained by the GEM for the undeformedmodel, which show that the SO(3) vacuum has the smallest free energy. -7-6-5-4-3-2-1 0 1 2 1 1.5 2 2.5 3 3.5 4 4.5 5 f r ee ene r g y m f SO(7)SO(6)
Figure 2:
The free energy calculated up to three loops for the solutions found with the SO(7) and SO(6)ansatzes are plotted against the fermion mass m f . We observe a clear tendency that the SO(6) symmetric vac-uum is more favored as m f is decreased, whereas the free energy for the two ansatzes tends to be degenerateas m f is increased. The above mentioned results have also been checked against a calculation using the GEM onthe deformed model (3.6). The GEM has been applied to the Euclidean IKKT matrix model in[33], where SO(10) was found to be broken down to SO(3). The GEM has the advantage that the This is also reflected in the GEM results [32, 33] for the free energy of the SO( d ) vacuum, which becomes muchlarger for d = d ≥ ynamical Compactification of Extra Dimensions Stratos Kovalkov Papadoudis large- N limit is easily taken by considering only planar graphs and that the small- ε extrapolationis not necessary. The systematic errors of the GEM are due to the truncation of the expansion andthe errors in determining the parameters that give the physical solutions. As such, the two methodscan be considered to be completely independent. In [27], we performed a three-loop calculationusing SO( d ) symmetric ansatzes for d = ,
7, and calculated the free energy. We observed that bydecreasing m f , the free energy of the SO(6) vacuum becomes smaller than the free energy of theSO(7) vacuum. In Fig. 2 we plot the free energy calculated up to three loops for the solutions foundwith the SO(7) and SO(6) ansatzes against the fermion mass m f . We observe a clear tendency thatthe SO(6) symmetric vacuum is more favored as m f is decreased. However, the free energy forthe two ansatzes tends to become degenerate as m f is increased. In this situation it is difficult toidentify the critical point, given the accuracy of the GEM results.At m f = .
0, the extent of space was found to agree very well between the two methods. In[27], it is found that ρ = · · · = ρ = . , ρ = ρ = ρ = . , (4.4)whereas the CLM results of Fig. 1 give ρ = · · · = ρ = . , ( ρ + ρ + ρ ) / = . . (4.5)Therefore, for the first time, we have a first principle study of the Euclidean IKKT matrixmodel that produced clear results on the question of dynamical compactification of extra dimen-sions via SSB of the SO(10) rotational symmetry of the model. The SO(10) rotational symmetrybreaking of the Euclidean IKKT matrix model down to SO(3) due to the phase of the Pfaffian isinteresting, but it makes the model somewhat difficult to interpret. Given the promising propertiesof the Lorentzian model [3, 4, 5, 6, 7, 8, 9, 10, 11], we consider that the naive Wick rotation tothe Euclidean model is not the right direction to pursue. On the other hand, the fact that the CLMenabled us to obtain a clear SSB pattern for the deformed model, which suffers from a severe signproblem, is encouraging. We hope that the CLM is equally useful in investigating the LorentzianIKKT model, in particular in the presence of fermionic matrices, which are not included yet inRef. [11]. Acknowledgements
The authors would like to thank S. Iso, H. Kawai, H. Steinacker and A. Tsuchiya for valuablediscussions. T. A. was supported in part by Grant-in-Aid for Scientific Research (No. 17K05425)from Japan Society for the Promotion of Science. Computations were carried out using computa-tional facilities at KEKCC and the NTUA het cluster. This work was also supported by computa-tional time granted by the Greek Research & Technology Network (GRNET) in the National HPCfacility - ARIS - under project ID IKKT10D.
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