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Statistical Physics Approach to M-Theory Integrals

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Statistical Physics Approach to M-Theory Integrals Nonperturbative Quantum Effects 2000 PROCEEDINGS Statistical Physics Approach to M-theory Integrals∗ Werner Krauth and Matthias Staudacher CNRS-Laboratoire de Physique Statistique, Ecole Normale Sup´erieure 24, rue Lhomond F-75231 Paris Cedex 05 E-mail: [email protected] Albert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik Am M¨uhlenberg 1 D-14476 Golm E-mail: [email protected] Abstract: We explain the concepts of computational statistical physics which have proven very help- ful in the study of Yang-Mills integrals, an ubiquitous new class of matrix models. Issues treated are: Absolute convergence versus Monte Carlo computability of near-singular integrals, singularity detec- tion by Markov-chain methods, applications to asymptotic eigenvalue distributions and to numerical evaluations of multiple bosonic and supersymmetric integrals. In many cases already, it has been possible to resolve controversies between conflicting analytical results using the methods presented here. Keywords: Monte Carlo Methods, M-theory, Matrix Models, Yang-Mills Theory. 1. Introduction theories. Furthermore, they appear in proposed formulations of string theory (the IKKT model) Recent work in field theory has revealed the exis- and M-theory. It remains to be elucidated whether tence of an important new class of gauge-invariant they contain further non-perturbative informa- matrix models. At the difference of the clas- tion on gauge theories via the Eguchi-Kawai mech- sic Wigner-type models, interest now focusses anism. on integrals of D non-linearly coupled matrices For the sake of brevity (cf. [1] for complete Xµ,µ = 1,...D. The Xµ are constructed from definitions), we write down (even for supersym- A the generators T of the fundamental representa- metric Yang-Mills theories) only the effective bo- A tion of a given Lie algebra Lie(G): Xµ = Xµ TA, sonic integral, which is obtained after integrat- with A = 1,..., dim(G). The group G may be ing out the (= number of supersymmetries) N SU(N), but the orthogonal, symplectic and ex- Grassmann-valued fermionic matrices ceptional groups have also come under close scrutiny A recently. N dXµ D,G = Z √2π × These ordinary, multiple Riemann integrals Z A,µ 1 Y stem from a dimensional reduction of D-dimen- Tr [Xµ,Xν ][Xµ,Xν ] 4g2 A e ( Xµ ). (1.1) arXiv:cond-mat/0010127v1 [cond-mat.stat-mech] 9 Oct 2000 sional Euclidean continuum Yang-Mills theory to P zero dimensions. They have important impli- In this equation, ( X ) is the Pfaffian of a cer- P { } cations, as the integrals yield the bulk part of tain matrix , which can be constructed from the Witten index of supersymmetric quantum M the adjoint representation of the Xµ. mechanical gauge theories, and appear in multi- During the last few years, intense effort has instanton calculations of large N susy Yang-Mills been brought to bear on these integrals, ranging ∗Talk presented by W. Krauth from the rigorous exact solution for SU(2) [2],[3] Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacher to ultra-sophisticated analytical calculations [4] cial for Yang-Mills integrals that we present them which lent support to earlier conjectures [5]. in the next section in the simplified context of a We have initiated a project with the aim 1 dimensional integral. − to obtain direct non-perturbative information on these integrals by numerical Monte Carlo calcu- 2. Existence & Computability lation. In several cases already, this approach has allowed to clarify analytic properties of the inte- Consider the integral grals, both in the supersymmetric and the purely 1 −α bosonic case (where the Pfaffian in eq.(1.1) is I(α)= dxµ(x)x (2.1) 0 simply omitted). We have also obtained very pre- Z with a constant weight function µ(x) = 1, which cise values (statistical estimates) of for several Z we introduce for later convenience. In this toy low-ranked groups, estimates which were suffi- problem, the singularity at x = 0 plays the role of ciently precise to decide between differing ana- a valley, as discussed before, in the more complex lytical conjectures. The basic strength of the nu- Yang-Mills integral. merical approach is however to allow the com- putation of a wide range of observables (Wilson partial sum for Integral 9 loops, eigenvalue distributions), and much work α =0.9 remains to be done. 8 The integrals in eq.(1.1) resemble partition 7 functions in statistical physics. Our initial hope 6 was to reduce eq.(1.1) to a standard form, most 5 Partial sum simply by the transformation 4 3 2 A XA A N dXµ − µ ( Xµ ) 2 2 2 0 5000 10000 = e σ F 2 , D,G XA MC-time Z √2π µ µ,A − 2 Z Y e 2σ (1.2) Figure 1: Partial sums St (cf. eq.(2.2)) for α = 0.9 P where ( XA ) is the integrand in the second vs Monte Carlo time t. The numerical estimate for F µ line of eq.(1.1). In this form, the integral can this integral seems to converge to the wrong result. (in principle) be computed directly using Monte Carlo methods. To do so, it would suffice to We may compute the integral eq.(2.1) by the generate Gaussian distributed random numbers Monte Carlo method in the following way: as the A weight function is constant (µ(x) = 1), we pick t Xµ , and to average the term [ ] in eq.(1.2) over uniformly distributed points xt with 0 xt 1 this distribution. The straightforward approach ≤ ≤ is thwarted by the fact that the integrals eq.(1.2) and compute - and, equivalently, eq.(1.1) - only barely con- t −α verge, if at all. I St =1/t x , (2.2) ∼ i The reason for this bad convergence lies in i=1 X the existence of “valleys” in the action in eq.(1.1). where t = 1, 2 ... is the Monte Carlo time. A For example, any configuration of mutually com- typical outcome for the partial sums St during a muting Xµ ([Xµ,Xν ]=0 µ,ν) gives rise to a Monte Carlo calculation for α = 0.9 is shown ∀ subspace of matrices with vanishing action, which in figure 1. The calculation is seemingly cor- stretches out to infinity, and leads to a large con- rect, as standard error analysis gives a result tribution to . There is presently no mathemat- I(α = 0.9) = 6.13 0.46, without emitting any Z ± ical proof that these singularities are integrable warnings! Carrying on the simulation for much (cf. [9] [10] for perturbative results). longer times, we would every so often generate Very importantly, integrals may exist, with- an extremely small xt, which in one step would out being computable by straightforward Monte hike up the partial sum, and change the error es- Carlo methods. This distinction between exis- timate. Repeatedly, we would get tricked into ac- tence and (Monte Carlo) computability is so cru- cepting “stabilized values” of the integral, which 2 Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacher would probably still not correspond to the true Markov chain sampling of [0,1] 0.7 value I(α =0.9) = 10! α =0.9 0.6 Clearly, there is a problem with the com- 0.5 putability of the integral, which can be traced 0.4 back to its infinite variance. Calling = x−α, 0.3 O Position the variance is given by 0.2 0.1 2 2 V ar = dxµ(x) (x) [ dxµ(x) (x)] . 0 O − O 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Z Z (2.3) MC-time The error in the Monte Carlo evaluation eq.(2.2) Figure 2: Position xt vs Monte Carlo time t for the ′ behaves like V ar/t, and, for α 0.5, is infinite. Markov chain with stationary distribution µ (x) = ≥ −0.9 This situation is virtually impossible to diagnose x . The time evolution of xt does not get stuck, p 1 −0.9 from within the simulation itself. because the integral 0 dxx exists. We have developed a highly efficient tool to R numerically check for (absolute) convergence of integrals. The idea (translated to the case of the Markov chain sampling of [0,1] 0.7 present toy problem) is to perform Markov chain α =1.8 0.6 random walk simulation with a stationary distri- 0.5 bution µ′(x) = µ(x)x−α and µ′′(x) = µ(x)x−2α. 0.4 to check for existence of the integral and finite- 0.3 ness of the variance, respectively. In an effort Position 0.2 to be completely explicit, this means to choose 0.1 a small displacement interval δt, uniformly dis- 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 tributed between +ǫ and ǫ, and to go from xt − MC-time to xt+1 according to the following probability ta- ′′ −1.8 ble Figure 3: Same as figure 2, but for µ (x) = x . xt gets stuck at x ∼ 0 very quickly, signalling that the variance of the integral eq.(2.1) is infinite. xt + δt w/ probability x = min(1,µ′(x )/µ(x )) . t+1 t+1 t x else t The method can be easily adapted to mul- (2.4) tidimensional integrals by monitoring an auto- (cf. [14]). During these simulations (which are correlation function rather than the position x , neither used nor useful to compute the integral t eq.(2.1) itself), we are exclusively interested in In our applications, the method has been finding out whether the Markov chain eq.(2.4) successful much beyond our initial expectations.
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