Complex Langevin studies of the dynamical compactification of extra dimensions in the Euclidean IKKT matrix model
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 ] S e p Complex Langevin studies of the dynamical compactification of extra dimensions inthe Euclidean IKKT matrix model
Konstantinos N.
Anagnostopoulos , ∗ Takehiro
Azuma , † Yuta
Ito , ‡ Jun
Nishimura ,
4, 5, § Toshiyuki
Okubo , ¶ and Stratos Kovalkov Papadoudis ∗∗ National Technical University of Athens, Zografou Campus, GR-15780 Athens, Greece Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka, 572-8508, Japan National Institute of Technology, Tokuyama College,Gakuendai, Shunan, Yamaguchi 745-8585, Japan High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan Faculty of Science and Technology, Meijo University, Nagoya, 468-8502, Japan
The type IIB matrix model, also known as the IKKT matrix model, is a promising candidate fora nonperturbative formulation of superstring theory. In this talk we study the Euclidean version ofthe IKKT matrix model, which has a “sign problem” due to the Pfaffian coming from integratingout the fermionic degrees of freedom. To study the spontaneous breaking of the SO(10) rotationalsymmetry, we apply the Complex Langevin Method (CLM) to the Euclidean IKKT matrix model.We conclude that the SO(10) symmetry is broken to SO(3), in agreement with the previous studiesby the Gaussian Expansion Method (GEM). We also apply the GEM to the deformed model and findconsistency with the CLM result. These are proceedings of Takehiro Azuma’s talk at Asia-PacificSymposium for Lattice Field Theory (APLAT 2020) on August 4-7, 2020, based on the paper [1].
I. INTRODUCTION
Large- N reduced models have been proposed as thenon-perturbative definition of superstring theory. In par-ticular, the type IIB matrix model, also known as theIKKT matrix model [2], is regarded as one of the mostpromising approaches. The theory is formally defined bythe dimensional reduction of the ten-dimensional N = 1super-Yang-Mills theory to zero dimensions. We inter-pret the eigenvalues of the bosonic matrices as the space-time coordinate, and the spacetime is dynamically gener-ated from the matrices’ degrees of freedom. Superstringtheory is well-defined in the ten-dimensional spacetime,and it is an important question how our four-dimensionalspacetime emerges dynamically.The Euclidean version of the IKKT matrix model is ob-tained after a Wick rotation of the temporal direction. Ithas a manifest SO(10) rotational symmetry, whose spon-taneous breaking implies the dynamical generation of thelower-dimensional spacetime. In the Euclidean version,it has been known that the spontaneous symmetry break-ing (SSB) of SO(10) is not realized in the phase-quenchedmodel, and that the complex phase of the Pfaffian thatcomes from integrating out the fermionic degrees of free-dom plays an important role in the SSB of SO(10) [3–8].On the other hand, it is difficult to numerically study thesystems with complex phase, due to the so-called “com-plex action problem”, or “sign system”. If we are to ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] study the SSB of SO(10), we need to overcome the “signproblem”.Various approaches to the “sign problem” have beenso far proposed. One of the promising approaches tothe “sign problem” is the “Complex Langevin Method”(CLM) [9, 10], which attempts to define a stochastic pro-cess for the complexified variables. Recently, the CLMhas gained enormous attention because the condition forthe equivalence to the original path integral has beenclarified [11–17]. The CLM has been previously appliedto the toy models to capture the rotational symmetrybreaking in the Euclidean IKKT matrix model [18, 19].In [19], five of the authors (K.N.A., T.A., Y.I., J.N.and S.K.P.) have investigated the six-dimensional ver-sion of the Euclidean IKKT matrix model. Similarly tothe IKKT matrix model, the six-dimensional version alsohas a “sign problem” from the determinant which is ob-tained by integrating out the fermion. Using the CLM,it has been shown that the SO(6) rotational symmetryis broken to SO(3), as suggested by Gaussian ExpansionMethod (GEM) [20, 21].In this talk, we apply the CLM to the Euclidean IKKTmatrix model, extending the study of the six-dimensionalversion [19]. This is numerically much more involvedthan the six-dimensional version, since the size of thegamma matrices after Weyl projection increases from 4to 16. Also, the finite- N effects become severer, whichrequires simulations at large N to make sensible large- N extrapolations. This makes the extrapolation withrespect to the deformation parameters, which were usedfor the CLM of the toy models [18, 19], more difficult.This leads us to study the model with a mass deformationof the fermion using the GEM, which is compared withthe CLM result.These proceedings are organized as follows. In Sec. II,we introduce the Euclidean version of the IKKT matrixmodel, which we focus on. In Sec. III, we apply the CLMto the IKKT matrix model. In Sec. IV, we present theresult of the GEM applied to the deformed model, whichis well compared with the CLM result. Sec. V is devotedto a summary and discussions. II. EUCLIDEAN VERSION OF THE IKKTMATRIX MODEL
The action S of the IKKT matrix model [2] is given by S = S b + S f , where (1) S b = − N A µ , A ν ][ A µ , A ν ] , (2) S f = − N (cid:0) ¯ ψ α (Γ µ ) αβ [ A µ , ψ β ] (cid:1) . (3)The bosons A µ ( µ = 1 , , · · · ,
10) are N × N tracelessHermitian matrices, and the Majorana-Weyl spinors ψ α ( α = 1 , , · · · ,
16) are N × N traceless matrices withGrassmann entries. Γ µ are the 16 ×
16 gamma matri-ces after Weyl projection. ¯ ψ = ψ C , with C being the16 ×
16 charge conjugation matrix. In the Euclidean ver-sion, the indices are contracted by δ µν = (1 , , , · · · , Z = Z dAdψe − S = Z dAe − S b Pf M = Z dAe − S eff . (4)The effective action S eff is defined as S eff = S b − log Pf M . (5) M is a 16( N − × N −
1) anti-symmetric matrix,which represents a linear transformation ψ α → ( M ψ ) α = ( C Γ µ ) αβ [ A µ , ψ β ] , (6)acting on the linear space of traceless complex N × N ma-trices ψ α . In the Euclidean version, the partition functionis finite without any cutoff [22, 23]. However, the Pfaf-fian Pf M is complex in general and we face a severe signproblem. We define its phase Γ as Pf M = | Pf M| e i Γ .It has been shown that in the phase-quenched model,in which Pf M is replaced with | Pf M| , the SSB of theSO(10) rotational symmetry does not occur [4, 6–8].The SSB of the SO(10) symmetry has been studied viathe GEM, and it has turned out that SO(10) is sponta-neously broken to SO(3) [20, 21]. We consider the 10 × T µν = 1 N tr( A µ A ν ) . (7)We define its 10 eigenvalues as λ µ with the ordering λ > λ > · · · > λ . In the SO( d ) vacuum, the V.E.V.’s h λ i , · · · , h λ d i grow and the V.E.V.’s h λ d +1 i , · · · , h λ i shrink in the large- N limit. The results of the studies ofthe SO( d ) symmetric vacua for 2 ≤ d ≤ r = lim N →∞ √ λ n ( n = d + 1 , · · · ,
10) is r ≃ . d (universal compactification scale).2. The ten-dimensional volume of the Euclidean space-time does not depend on d , except for d = 2 (constantvolume property). For the extent of the extended direc-tions R = lim N →∞ √ λ n ( n = 1 , , · · · , d ), the volume is V = R d r − d = l , with l ≃ . d =3, which suggests the dynamical emergence of three -dimensional spacetime.The Euclidean version has been studied by Monte Carlosimulation using the factorization method [7, 8]. Theseworks provided strong numerical evidence for the real-ization of SSB of the SO(10) rotational symmetry, butthey were not able to determine the precise SSB patternaccurately enough. III. COMPLEX LANGEVIN STUDIES OF THEIKKT MATRIX MODEL
In this section, we apply the CLM to the effective ac-tion of the IKKT matrix model S eff defined by (5). Inthe CLM, we extend the degrees of freedom of A µ fromthe Hermitian traceless matrices to the general complextraceless matrices. The CLM consists of solving the fol-lowing Langevin equation dA µ ( t ) ij dt = − ∂S eff ∂A µ ( t ) ji + η µ ( t ) ij , where (8) ∂S eff ∂A µ ( t ) ji = ∂S b ∂A µ ( t ) ji −
12 Tr (cid:18) ∂ M ∂A µ ( t ) ji M − (cid:19) . (9)Tr is the trace with respect to the 16( N − × N − t is a fictitious time,which should not be confused with the real time. η µ ( t ) ij is the Hermitian white noise that follows the probabilitydistribution ∝ exp (cid:0) − R tr η ( t ) dt (cid:1) . This is independentfor different times t, t ′ . The white noise is rendered trace-less as η ( t ) ii → η ( t ) ii − N tr η ( t ). The expectation valueof an observable is evaluated as hO [ A µ ] i = 1 T Z t + Tt O [ A µ ( t )] dt. (10) t is the thermalization time, and T is sufficiently large toobtain good statistics. The holomorphy of the observable O is important for the proof of the validity of (10) [11,12, 17]. The Langevin equation (8) is put on a computerby the discretization ( A µ )( t + ∆ t ) ij = ( A µ )( t ) ij − (∆ t ) ∂S eff ∂A µ ( t ) ji + p (∆ t ) η µ ( t ) ij . (11) The term p (∆ t ) stems from the normalization of η µ ( t ), so that it follows the probability distribution ∝ exp (cid:0) − P t tr η ( t ) (cid:1) .We cannot extract a reliable result equivalent to thepath integral from the CLM when we encounter the fol-lowing two problems: One is the “excursion problem”,which occurs when A µ is too far from Hermitian. Theother is the “singular drift problem”, which occurs whenthe drift term (9) becomes large due to the accumulationof some of the eigenvalues of M close to zero. In orderto justify the CLM, the probability distribution of the“drift norm” u = vuut N X µ =1 N X i,j =1 (cid:12)(cid:12)(cid:12)(cid:12) ∂S eff ∂ ( A µ ) ji (cid:12)(cid:12)(cid:12)(cid:12) , (12)which is measured during the complex Langevin simula-tion, should fall off exponentially or faster. If we look atthe “drift term”, we get the drift of the CLM [17].In order to avoid the “excursion problem”, we usethe technique named “gauge cooling” [13, 16], to keep A µ closer to Hermitian matrices. This consists ofminimizing the “Hermiticity norm” defined by N H = − N P µ =1 tr { ( A µ − A † µ ) } at each step in solving thediscretized Langevin equation (11).In applying the CLM to the IKKT matrix model, weadd the following two mass terms to the action S as de-fined in (1) [18, 19]:∆ S b = N ε X µ =1 m µ tr( A µ ) , (13)∆ S f = − im f N ψ α ( C Γ Γ † Γ ) αβ ψ β ) . (14)In order to probe the SSB, we break the SO(10) sym-metry explicitly by adding the bosonic mass term (13).Here, m µ satisfies 0 < m ≤ · · · ≤ m . We consider thefollowing order parameter for the SSB of SO(10): λ µ = 1 N tr( A µ ) , ( µ = 1 , , · · · , . (15)Here, there is no summation of µ . We consider (15) in-stead of the eigenvalues of T µν defined by (7), to avoidthe subtleties in the holomorphy of the observables. Wetake the ε → after taking the large- N limit.The mass term (14) is added to avoid the singular driftproblem coming from the near-zero eigenvalues of M , byshifting the eigenvalue distribution of M on the com-plex plane away from the origin. The mass term (14)breaks the SO(10) symmetry to SO(7) × SO(3). Westudy whether SO(7) is broken to smaller subgroups aswe vary m f , and discuss what happens in the m f → m f → + ∞ limit is the bosonic model, sincethe fermionic degrees of freedom decouple. It is knownthat there is no SSB of SO(10) in the bosonic model [3].In our simulations, we choose the range of the parameters( m f , ε ) for each N , so that the probability distribution ofthe “drift norm” (12) falls off exponentially or faster.We apply the CLM to the action, with the originalmodel (1) deformed by (13) and (14): S ′ = S + ∆ S b + ∆ S f . (16)The effective action (5) is modified accordingly. To makethe ε → m µ carefully. Here, we take m µ as m µ = (0 . , . , . , , , , , , ,
8) ( m f = 3 . , (17) m µ = (0 . , . , , , , , , , ,
8) ( m f ≤ . , (18)so that we can distinguish the SO( d ) vacua with d =3 , , , , m f = 3 .
0, and d = 2 , , , m f ≤ . N : ρ µ ( m f , ε, N ) = h λ µ i m f ,ε,N P ν =1 h λ ν i m f ,ε,N , (19)where h λ µ i m f ,ε,N is the V.E.V. of the observable (15) withrespect to the deformed action (16) at finite N . Then,we make a large- N extrapolation ρ µ ( m f , ε ) = lim N → + ∞ ρ µ ( m f , ε, N ) . (20)An example of this procedure is shown in FIG. 1. TheEuclidean IKKT matrix model suffers so severe finite- N effects that it requires a quadratic fit with respect to N .After taking the large- N limit, we plot ρ µ ( m f , ε )against ε in FIG. 1. Some of the small- ε points are ex-cluded from the fitting, due to the difficulty in overcom-ing the finite- N effects. As we read off the ε → m f = 3 . m f , we observethe SSB of SO(7) to SO(4) at m f = 1 .
4, and to SO(3)at m f = 0 . , . , .
0, respectively. This is consistent withthe GEM result for the undeformed model (1), in whichthe SO(3) vacuum has the smallest free energy [21].
IV. COMPARISON WITH THE GEM RESULT
In this section, we present the result of the GEM forthe action S ′′ = S + ∆ S f , with the original model (1)deformed only by (14). The basic idea of the GEM is torewrite the action S ′′ as S ′′ = S + ( S ′′ − S ) , where (21) S = N X µ =1 M µ tr( A µ ) + N X α,β =1 A αβ tr( ψ α ψ β ) , (22) A αβ = − im f ( C Γ Γ † Γ ) αβ + X µ,ν,ρ =1 i m µνρ ( C Γ µ Γ † ν Γ ρ ) αβ .S and ( S ′′ − S ) are regarded as the “classical ac-tion” and the “one-loop counter term”, respectively. Wechoose the ordering of the parameter M µ as 0 < M ≤· · · ≤ M . The other parameter m µνρ is a totally anti-symmetric 3-form. We focus on the two cases, in whichwe impose SO( d ) × Z with d = 6 and d = 7, where Z represents the group of cyclic permutation of the 8th, 9thand 10th directions. In the following, we call this “theSO( d ) ansatz”.In FIG. 2 (Left) we plot the free energy calculated upto the three loops for the solutions found with the SO(7)and SO(6) ansatze against m f . As we decrease m f , we (cid:1) µ ( m f , (cid:2) , N ) (cid:0) + (cid:3) )/2 (cid:4) (cid:5) (cid:6) ( (cid:7) + (cid:8) )/2( (cid:9) + (cid:10) + (cid:11) )/3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.1 0.2 0.3 0.4 0.5 ρ µ ( m f = . , ε ) ε ρ µ ( m f = . , ε ) ε ρ µ ( m f = . , ε ) ε ρ µ ( m f = . , ε ) ε ρ µ ( m f (cid:12)(cid:13)(cid:14)(cid:15) ε ) ε FIG. 1. (Top Left) The large- N extrapolation of ρ µ ( m f , ε, N ) for m f = 1 . , ε = 0 .
2, with m µ given by (18). The ρ µ ( m f , ε, N )are averaged for µ = 1 , µ = 6 , µ = 8 , ,
10 in order to increase statistics. We make a quadratic fit with respect to N .(The Rest) ρ µ ( m f , ε ) after taking the large- N limit are plotted against ε for m f = 3 . m f = 1 . m f = 1 . m f = 0 . m f = 0 . m f = 3 . m f ≤ .
4, respectively. The curves from top to bottom are ( ρ + ρ + ρ ) , ρ , ρ , ρ , ρ , ( ρ + ρ + ρ ) for m f = 3 .
0, and ( ρ + ρ ) , ρ , ρ , ρ , ( ρ + ρ ) , ( ρ + ρ + ρ ) for m f ≤ .
4, respectively. see a clear tendency that the SO(6) symmetric vacuumis favored. And the free energy for SO(7) and SO(6) isdegenerate for large m f . Also in FIG. 2 (Right) we plotthe extent of space λ µ against m f for the SO(7) ansatz.At m f = 3 .
0, we have λ = · · · λ = 0 . , λ = λ = λ = 0 . ρ = · · · = ρ = 0 . , ρ = ρ = ρ = 0 . , where ρ µ = λ µ P ν =1 λ ν . This is in impressive agreementwith the CLM result(CLM): ρ = · · · = ρ = 0 . , ρ + ρ + ρ . , which is read off from FIG. 1 (Top-Right). V. CONCLUSION AND OUTLOOK
In this talk, we have studied the Euclidean versionof the IKKT matrix model via the CLM, to elucidate how the spacetime is dynamically generated in super-string theory. Similarly to the previous work on the six-dimensional version [19], we have used the deformationtechnique to overcome the singular-drift problem. Wehave found that as we decrease m f , the lower-dimensionalspacetime is chosen. At m f ≤ .
0, the SO(10) symme-try is spontaneously broken to SO(3). Although it isdifficult to make a quantitatively sensible m f → ρ µ , we conclude that the SO(3)vacuum is chosen in the undeformed model ( m f = 0).Also, we have studied the mass deformed model usingthe GEM, and compared the free energy for the SO(7)and SO(6) ansatze. Here again, as we decrease m f , theansatze for lower-dimensional spacetime are energeticallyfavored. At m f = 3 .
0, where the SO(7) vacuum is real-ized, we have seen a quantitative agreement between theCLM and GEM results of the spacetime extent.The CLM can be applied to many interesting systemswith the sign problem. In [24], an attempt has been made -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) e x t en t o f s pa c e λ i m f SO(7) solution λ λ FIG. 2. (Left) The free energy calculated up to three loops for the solutions for SO(7) and SO(6) ansatzes are plotted against m f . (Right) The extent of space (15) for the SO(7) ansatz is plotted against m f . Here, we plot λ = · · · = λ and λ = λ = λ . to apply the CLM to the Lorentzian version of the IKKTmatrix model, where the complex phase comes from e iS b .It was found that for a certain deformed bosonic modelthe emergent three-dimensional expanding space has aclear departure from the fuzzy-sphere-like Pauli-matrixstructure. We hope to pursue this direction and eluci-date the structure of the expanding space as suggestedby superstring theory. Also, to study the nature of theexpansion of the universe, we need to simulate the modelat larger N . The impact on the fermionic degrees of freedom should be studied. We hope to report on moreanalysis in future publications [25]. Acknowledgement.—
The works of T.A. were sup-ported in part by Grant-in-Aid for Scientific Research(No. 17K05425) from Japan Society for the Promotionof Science. Computations were carried out with KEKCCand NTUA Het Cluster. This work was also supportedby computational time granted by the Greek Researchand Technology Network (GRNET) in the National HPCfacility — ARIS — under project ID “IKKT10D”. [1] K. N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura,T. Okubo and S. Kovalkov Papadoudis, JHEP , 069(2020) [arXiv:2002.07410].[2] N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya,Nucl. Phys. B , 467 (1997) [arXiv:hep-th/9612115].[3] T. Hotta, J. Nishimura and A. Tsuchiya, Nucl. Phys. B , 543-575 (1999) [arXiv:hep-th/9811220].[4] J. Ambjorn, K. N. Anagnostopoulos, W. Bietenholz,T. Hotta and J. Nishimura, JHEP , 013 (2000)[arXiv:hep-th/0003208].[5] J. Nishimura and G. Vernizzi, JHEP , 015 (2000)[arXiv:hep-th/0003223].[6] J. Ambjorn, K. N. Anagnostopoulos, W. Bietenholz,T. Hotta and J. Nishimura, JHEP , 011 (2000)[arXiv:hep-th/0005147].[7] K. N. Anagnostopoulos, T. Azuma and J. Nishimura,JHEP , 009 (2013) [arXiv:1306.6135].[8] K. N. Anagnostopoulos, T. Azuma and J. Nishimura,PoS LATTICE2015 , 307 (2016) [arXiv:1509.05079].[9] G. Parisi, Phys. Lett. B , 393-395 (1983)[10] J. R. Klauder, Phys. Rev. A , 2036-2047 (1984)[11] G. Aarts, E. Seiler and I. O. Stamatescu, Phys. Rev. D , 054508 (2010) [arXiv:0912.3360].[12] G. Aarts, F. A. James, E. Seiler and I. O. Stamatescu,Eur. Phys. J. C , 1756 (2011) [arXiv:1101.3270]. [13] E. Seiler, D. Sexty and I. O. Stamatescu, Phys. Lett. B , 213-216 (2013) [arXiv:1211.3709].[14] J. Nishimura and S. Shimasaki, Phys. Rev. D , no.1,011501 (2015) [arXiv:1504.08359].[15] K. Nagata, J. Nishimura and S. Shimasaki, PTEP ,no.1, 013B01 (2016) [arXiv:1508.02377].[16] K. Nagata, J. Nishimura and S. Shimasaki, JHEP ,073 (2016) [arXiv:1604.07717].[17] K. Nagata, J. Nishimura and S. Shimasaki, Phys. Rev.D , no.11, 114515 (2016) [arXiv:1606.07627].[18] Y. Ito and J. Nishimura, JHEP , 009 (2016)[arXiv:1609.04501].[19] K. N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimuraand S. K. Papadoudis, JHEP , 151 (2018)[arXiv:1712.07562].[20] T. Aoyama, J. Nishimura and T. Okubo, Prog. Theor.Phys. , 537-563 (2011) [arXiv:1007.0883].[21] J. Nishimura, T. Okubo and F. Sugino, JHEP , 135(2011) [arXiv:1108.1293].[22] W. Krauth, H. Nicolai and M. Staudacher, Phys. Lett. B , 31-41 (1998) [arXiv:hep-th/9803117].[23] P. Austing and J. F. Wheater, JHEP , 019 (2001)[arXiv:hep-th/0103159].[24] J. Nishimura and A. Tsuchiya, JHEP06