Monte Carlo studies of dynamical compactification of extra dimensions in a model of nonperturbative string theory Lattice 2015, Jul 15th 2015 (Wed) 18:30-21:00.

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Presentation transcript:

Monte Carlo studies of dynamical compactification of extra dimensions in a model of nonperturbative string theory Lattice 2015, Jul 15th 2015 (Wed) 18:30-21:00 [Board 24] Konstantinos N. Anagnostopoulos, Takehiro Azuma and Jun Nishimura 1. Introduction Result of Gaussian Expansion Method (GEM) The phase wn(x) scales at large N as Difficulties in putting complex partition functions on computers. Order parameter of the SSB of SO(10). ・Around x≅1: fn(0)(x)/N scales at large N: e.g. lattice QCD, matrix models for string theory Extended d-dim. and shrunken (10-d) dim. at N→∞ ⇒　SSB SO(10)→SO(d) ・Around x<0.4: fn(0)(x)/N2 scales at large N 　→ existence of the hardcore potential. 1. Sign problem: The reweighting requires 　configs. exp[O(N2)] Main Results of GEM [J. Nishimura, T. Okubo and F. Sugino, arXiv:1108.1293] Preliminary Monte Carlo results: x3,4,5 are close to the GEM result ≅0.40. <*>0 = (V.E.V. for phase-quenched Z0) ・Universal compactification scale r2 ≅0.15 for SO(d) ansatz (d=2,3,…7). ・Constant volume property except d=2 V=Rd×r10-d=l10, l2≅0.38 ・SSB SO(10)→SO(3). 2. Overlap problem: Discrepancy of important configs. between Z0 and Z. 2. Factorization method Method to sample important configs. for Z. n=3 SO(2) [J. Nishimura and K.N. Anagnostopoulos, hep-th/0108041 K.N. Anagnostopoulos, T.A. and J. Nishimura, arXiv:1009.4504] Mechanism of SSB in Euclidean IKKT model Constrain the observables Partition function of the model: correlated with the phase Γ. Normalization n=4 SO(3) Factorization of the distribution function ρ. The Pfaffian PfM is complex in the Euclidean case. Complex phase Γ is crucial for the SSB of SO(10). [J. Nishimura and G. Vernizzi hep-th/0003223] No SSB in the phase-quenched partition function. n=5 SO(4) <*>0=V.E.V. for Z0 Comparison of the free energy Simulation of Zx with a proper choice of the set Σ ⇒sample the important region for Z. Free energy for the SO(d) vacuum: 4. Result of Monte Carlo simulation Evaluation of the observables It turns out sufficient to constrain only one eigenvalue λd+1 →0 at large N Peak of ρ at V=(system size)→∞. = Minimum of the free energy SO(4) ⇒　SO(d) vacuum SO(3) The SO(2) vacuum is disfavored. ⇒Solve the saddle-point equation Applicable to general systems with sign problem. ⇒(r/l)2[≃0.15/0.38=0.40 (GEM)] SO(2) 3. The IKKT model We study the SO(d) symmetric vacua (d=2,3,4) 　 x1=…=xd >1>xd+1,…,x10 The large eigenvalues λ1,…λd do not affect much the fluctuation of the phase. Promising candidate for nonperturbative string theory [N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, hep-th/9612115] 5. Summary Solve the saddle-point equation for n=d+1. Simulation by Rational Hybrid Monte Carlo (RHMC) algorithm. We have studied the dynamical compactification of the spacetime in the Euclidean IKKT model. ・Euclidean case after the Wick rotation 　A0→iA 10, Γ0→-iΓ 10. ・Aμ, Ψα ⇒N×N Hemitian matrices　　(μ=1,2,…,d=10, α,β=1,2,…,16) ・Eigenvalues of Aμ ⇒spacetime coordinate ・Spontaneous Symmetry Breaking (SSB) of SO(10) ⇒dynamical emergence of spacetime. Monte Carlo simulation via factorization method　⇒We have obtained the results 　　　　　　　consistent with GEM: ・Universal compactification scale for SO(2,3,4) vacuum. ・SO(2) vacuum is disfavored. where Solution ⇒ in the SO(d) vacuum.