Papers
MONTE CARLO STUDIES OF SUPERSYMMETRIC MATRIX QUANTUM MECHANICS WITH SIXTEEN SUPERCHARGES AT FINITE TEMPERATURE
Phys. Rev. Lett. 100, 021601 (2008)
We present the first Monte Carlo results for supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature. The recently proposed non-lattice simulation enables us to include the effects of fermionic matrices in a transparent and reliable manner. The internal energy nicely interpolates the weak coupling behavior obtained by the high temperature expansion, and the strong coupling behavior predicted from the dual black hole geometry. The Polyakov line takes large values even at low temperature suggesting the absence of a phase transition in sharp contrast to the bosonic case. These results provide highly non-trivial evidences for the gauge/gravity duality.
Like (3)NEW APPROACH TO THE COMPLEX-ACTION PROBLEM AND ITS APPLICATION TO A NONPERTURBATIVE STUDY OF SUPERSTRING THEORY
Phys. Rev. D 66, 106008 (2002)
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 the so-called overlap problem completely. Here we apply the method to a nonperturbative study of superstring theory using its matrix formulation. In this particular example, the distribution function turns out to be positive definite, which allows us to reduce the problem even further. Our numerical results suggest an intuitive explanation for the dynamical generation of 4d space-time.
LARGE N DYNAMICS OF DIMENSIONALLY REDUCED 4D SU(N) SUPER YANG-MILLS THEORY
J.High Energy Phys. 07:013,2000
We perform Monte Carlo simulations of a supersymmetric matrix model, which is obtained by dimensional reduction of 4D SU(N) super Yang-Mills theory. The model can be considered as a four-dimensional counterpart of the IIB matrix model. We extract the space-time structure represented by the eigenvalues of bosonic matrices. In particular we compare the large N behavior of the space-time extent with the result obtained from a low energy effective theory. We measure various Wilson loop correlators which represent string amplitudes and we observe a nontrivial universal scaling in N. We also observe that the Eguchi-Kawai equivalence to ordinary gauge theory does hold at least within a finite range of scale. Comparison with the results for the bosonic case clarifies the role of supersymmetry in the large N dynamics. It does affect the multi-point correlators qualitatively, but the Eguchi-Kawai equivalence is observed even in the bosonic case.
CROSSING THE c=1 BARRIER IN 2d LORENTZIAN QUANTUM GRAVITY
Phys.Rev.D61:044010,2000
In an extension of earlier work we investigate the behaviour of two-dimensional Lorentzian quantum gravity under coupling to a conformal field theory with c>1. This is done by analyzing numerically a system of eight Ising models (corresponding to c=4) coupled to dynamically triangulated Lorentzian geometries. It is known that a single Ising model couples weakly to Lorentzian quantum gravity, in the sense that the Hausdorff dimension of the ensemble of two-geometries is two (as in pure Lorentzian quantum gravity) and the matter behaviour is governed by the Onsager exponents. By increasing the amount of matter to 8 Ising models, we find that the geometry of the combined system has undergone a phase transition. The new phase is characterized by an anomalous scaling of spatial length relative to proper time at large distances, and as a consequence the Hausdorff dimension is now three. In spite of this qualitative change in the geometric sector, and a very strong interaction between matter and geometry, the critical exponents of the Ising model retain their Onsager values. This provides evidence for the conjecture that the KPZ values of the critical exponents in 2d Euclidean quantum gravity are entirely due to the presence of baby universes. Lastly, we summarize the lessons learned so far from 2d Lorentzian quantum gravity.
QUANTUM GEOMETRY OF 2d GRAVITY COUPLED TO UNITARY MATTER
Nucl.Phys.B497:445-475,1997.
We show that there exists a divergent correlation length in 2d quantum gravity for the matter fields close to the critical point provided one uses the invariant geodesic distance as the measure of distance. The corresponding reparameterization invariant two-point functions satisfy all scaling relations known from the ordinary theory of critical phenomena and the KPZ exponents are determined by the power-like fall off of these two-point functions. The only difference compared to flat space is the appearance of a dynamically generated fractal dimension d_h in the scaling relations. We analyze numerically the fractal properties of space-time for Ising and three-states Potts model coupled to 2d dimensional quantum gravity using finite size scaling as well as small distance scaling of invariant correlation functions. Our data are consistent with d_h=4, but we cannot rule out completely the conjecture d_H = -2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing exponent of a spin-less primary field of conformal weight (n+1,n+1). We compute the moments <L^n> and the loop-length distribution function and show that the fractal properties associated with these observables are identical, with good accuracy, to the pure gravity case.
THE FLAT PHASE OF CRYSTALLINE MEMBRANES.
J.Phys. I France 6: 1321-1345,1996.
We present the results of a high-statistics Monte Carlo simulation of a phantom crystalline (fixed-connectivity) membrane with free boundary. We verify the existence of a flat phase by examining lattices of size up to $128^2$. The Hamiltonian of the model is the sum of a simple spring pair potential, with no hard-core repulsion, and bending energy. The only free parameter is the the bending rigidity $\kappa$. In-plane elastic constants are not explicitly introduced. We obtain the remarkable result that this simple model dynamically generates the elastic constants required to stabilise the flat phase. We present measurements of the size (Flory) exponent $\nu$ and the roughness exponent $\zeta$. We also determine the critical exponents $\eta$ and $\eta_u$ describing the scale dependence of the bending rigidity ($\kappa(q) \sim q^{-\eta}$) and the induced elastic constants ($\lambda(q) \sim \mu(q) \sim q^{\eta_u}$). At bending rigidity $\kappa = 1.1$, we find $\nu = 0.95(5)$ (Hausdorff dimension $d_H = 2/\nu = 2.1(1)$), $\zeta = 0.64(2)$ and $\eta_u = 0.50(1)$. These results are consistent with the scaling relation $\zeta = (2+\eta_u)/4$. The additional scaling relation $\eta = 2(1-\zeta)$ implies $\eta = 0.72(4)$. A direct measurement of $\eta$ from the power-law decay of the normal-normal correlation function yields $\eta \approx 0.6$ on the $128^2$ lattice.
THE SOLUTION SPACE OF THE UNITARY MATRIX MODEL STRING EQUATION AND THE SATO GRASSMANNIAN.
Commun.Math.Phys.148:469-486,1992.
The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell $\Gr$ of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form $\lb \cp ,\cq_- \rb =\hbox{\rm 1}$, with $\cp$ and $\cq_-$ $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints $\L_n\,(n\geq 0)$, where $\L_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model.
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