We explain the concepts of computational statistical physics which have
proven very helpful 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 detection 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.Comment: 6 pages, talk presented by WK at conference 'Non- perturbative
  Quantum Effects 2000', Paris, Sept 200

Krauth, Werner

Staudacher, Matthias

English

arXiv

We explain the concepts of computational statistical physics which have proven very helpful 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 detection 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

Krauth, W.

Staudacher, M.

MPG.PuRe

Statistical Physics Approach to M-theory Integrals

arXiv:cond-mat/0010127 v1   9 Oct 2000Nonperturbative Quantum Effects 2000PROCEEDINGSStatistical Physics Approach to M-theory Integrals∗Werner Krauth and Matthias StaudacherCNRS-Laboratoire de Physique Statistique, Ecole Normale Supe´rieure24, rue LhomondF-75231 Paris Cedex 05E-mail: krauth@lps.ens.frAlbert-Einstein-Institut, Max-Planck-Institut fu¨r GravitationsphysikAm Mu¨hlenberg 1D-14476 GolmE-mail: matthias@aei-potsdam.mpg.deAbstract: 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 numericalevaluations of multiple bosonic and supersymmetric integrals. In many cases already, it has beenpossible to resolve controversies between conflicting analytical results using the methods presentedhere.Keywords: Monte Carlo Methods, M-theory, Matrix Models, Yang-Mills Theory.1. IntroductionRecent work in field theory has revealed the exis-tence of an important new class of gauge-invariantmatrix models. At the difference of the clas-sic Wigner-type models, interest now focusseson integrals of D non-linearly coupled matricesXµ, µ = 1, . . . D. The Xµ are constructed fromthe generators TA of the fundamental representa-tion of a given Lie algebra Lie(G): Xµ = XAµ TA,with A = 1, . . . ,dim(G). The group G may beSU(N), but the orthogonal, symplectic and ex-ceptional groups have also come under close scrutinyrecently.These ordinary, multiple Riemann integralsstem from a dimensional reduction of D-dimen-sional Euclidean continuum Yang-Mills theory tozero dimensions. They have important impli-cations, as the integrals yield the bulk part ofthe Witten index of supersymmetric quantummechanical gauge theories, and appear in multi-instanton calculations of largeN susy Yang-Mills∗Talk presented by W. Krauththeories. Furthermore, they appear in proposedformulations of string theory (the IKKT model)and M-theory. It remains to be elucidated whetherthey contain further non-perturbative informa-tion on gauge theories via the Eguchi-Kawaimech-anism.For the sake of brevity (cf. [1] for completedefinitions), we write down (even for supersym-metric Yang-Mills theories) only the effective bo-sonic integral, which is obtained after integrat-ing out the N (= number of supersymmetries)Grassmann-valued fermionic matricesZND,G =∫ ∏A,µdXAµ√2π×e14g2Tr [Xµ,Xν ][Xµ,Xν ](P {XAµ }). (1.1)In this equation, P({X}) is the Pfaffian of a cer-tain matrix M, which can be constructed fromthe adjoint representation of the Xµ.During the last few years, intense effort hasbeen brought to bear on these integrals, rangingfrom the rigorous exact solution for SU(2) [2],[3]Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacherto ultra-sophisticated analytical calculations [4]which lent support to earlier conjectures [5].We have initiated a project with the aimto obtain direct non-perturbative information onthese integrals by numerical Monte Carlo calcu-lation. In several cases already, this approach hasallowed to clarify analytic properties of the inte-grals, both in the supersymmetric and the purelybosonic case (where the Pfaffian in eq.(1.1) issimply omitted). We have also obtained very pre-cise values (statistical estimates) of Z for severallow-ranked groups, estimates which were suffi-ciently precise to decide between differing ana-lytical conjectures. The basic strength of the nu-merical approach is however to allow the com-putation of a wide range of observables (Wilsonloops, eigenvalue distributions), and much workremains to be done.The integrals in eq.(1.1) resemble partitionfunctions in statistical physics. Our initial hopewas to reduce eq.(1.1) to a standard form, mostsimply by the transformationZND,G =∫ ∏µ,AdXAµ√2πe−XA2µ2σ2F({XAµ })e−∑ XA2µ2σ2 ,(1.2)where F({XAµ }) is the integrand in the secondline of eq.(1.1). In this form, the integral can(in principle) be computed directly using MonteCarlo methods. To do so, it would suffice togenerate Gaussian distributed random numbersXAµ , and to average the term [ ] in eq.(1.2) overthis distribution. The straightforward approachis thwarted by the fact that the integrals eq.(1.2)- and, equivalently, eq.(1.1) - only barely con-verge, if at all.The reason for this bad convergence lies inthe existence of “valleys” in the action in eq.(1.1).For example, any configuration of mutually com-muting Xµ ([Xµ, Xν ] = 0 ∀µ, ν) gives rise to asubspace of matrices with vanishing action, whichstretches out to infinity, and leads to a large con-tribution to Z. There is presently no mathemat-ical proof that these singularities are integrable(cf. [9] [10] for perturbative results).Very importantly, integrals may exist, with-out being computable by straightforward MonteCarlo methods. This distinction between exis-tence and (Monte Carlo) computability is so cru-cial for Yang-Mills integrals that we present themin the next section in the simplified context of a1−dimensional integral.2. Existence & ComputabilityConsider the integralI(α) =∫ 10dxµ(x)x−α (2.1)with a constant weight function µ(x) = 1, whichwe introduce for later convenience. In this toyproblem, the singularity at x = 0 plays the role ofa valley, as discussed before, in the more complexYang-Mills integral.234567890 5000 10000Partial sumMC-timepartial sum for Integralα =0.9Figure 1: Partial sums St (cf. eq.(2.2)) for α = 0.9vs Monte Carlo time t. The numerical estimate forthis integral seems to converge to the wrong result.We may compute the integral eq.(2.1) by theMonte Carlo method in the following way: as theweight function is constant (µ(x) = 1), we pick tuniformly distributed points xt with 0 ≤ xt ≤ 1and computeI ∼ St = 1/tt∑i=1x−αi , (2.2)where t = 1, 2 . . . is the Monte Carlo time. Atypical outcome for the partial sums St during aMonte Carlo calculation for α = 0.9 is shownin figure 1. The calculation is seemingly cor-rect, as standard error analysis gives a resultI(α = 0.9) = 6.13 ± 0.46, without emitting anywarnings! Carrying on the simulation for muchlonger times, we would every so often generatean extremely small xt, which in one step wouldhike up the partial sum, and change the error es-timate. Repeatedly, we would get tricked into ac-cepting “stabilized values” of the integral, which2Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacherwould probably still not correspond to the truevalue I(α = 0.9) = 10!Clearly, there is a problem with the com-putability of the integral, which can be tracedback to its infinite variance. Calling O = x−α,the variance is given byV ar =∫dxµ(x)O(x)2 − [∫dxµ(x)O(x)]2 .(2.3)The error in the Monte Carlo evaluation eq.(2.2)behaves like√V ar/t, and, for α ≥ 0.5, is infinite.This situation is virtually impossible to diagnosefrom within the simulation itself.We have developed a highly efficient tool tonumerically check for (absolute) convergence ofintegrals. The idea (translated to the case of thepresent toy problem) is to perform Markov chainrandom walk simulation with a stationary distri-bution µ′(x) = µ(x)x−α and µ′′(x) = µ(x)x−2α.to check for existence of the integral and finite-ness of the variance, respectively. In an effortto be completely explicit, this means to choosea small displacement interval δt, uniformly dis-tributed between +ǫ and −ǫ, and to go from xtto xt+1 according to the following probability ta-blext+1 =xt + δt w/ probabilitymin(1, µ′(xt+1)/µ(xt))xt else.(2.4)(cf. [14]). During these simulations (which areneither used nor useful to compute the integraleq.(2.1) itself), we are exclusively interested infinding out whether the Markov chain eq.(2.4)gets stuck. If so, it has become attracted by apoint x0 with ∫ x0+ǫx0dxµ′(x) =∞ (2.5)i. e. a non-integrable singularity. In figures 2 and3 , we show xt for Markov chains with stationarydistributions µ′(x) and µ′′(x), respectively. Fig-ure 3, in particular, implies that the variance ofthe integral eq.(2.1) is infinite so that the resultof fig. 1 cannot be trusted, while figure 2 assuresus that the integral exists.00.10.20.30.40.50.60.70 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000PositionMC-timeMarkov chain sampling of [0,1]α =0.9Figure 2: Position xt vs Monte Carlo time t for theMarkov chain with stationary distribution µ′(x) =x−0.9. The time evolution of xt does not get stuck,because the integral∫10dxx−0.9 exists.00.10.20.30.40.50.60.70 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000PositionMC-timeMarkov chain sampling of [0,1]α =1.8Figure 3: Same as figure 2, but for µ′′(x) = x−1.8.xt gets stuck at x ∼ 0 very quickly, signalling thatthe variance of the integral eq.(2.1) is infinite.The method can be easily adapted to mul-tidimensional integrals by monitoring an auto-correlation function rather than the position xt,In our applications, the method has beensuccessful much beyond our initial expectations.Besides its “consulting” role within the MonteCarlo framework (as explained in the caption offigures 2 and 3), we have used it extensively to es-tablish the existence conditions for bosonic andsusy Yang-Mills integrals, which have not beenobtained analytically beyond the 1-loop level. Wehave also adopted the method to obtain impor-tant information on the asymptotic behavior ofintegrals.We conclude the discussion of our toy prob-lem by showing how, after all, the integral eq.(2.1)can be computed by Monte Carlo methods. Con-3Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudachersider firstQ(α2, α1) =∫ 10x−α2dx∫ 10x−α1dx=∫ 10measure︷ ︸︸ ︷x−α1operator︷ ︸︸ ︷xα1−α2 dx∫ 10x−α1︸ ︷︷ ︸measuredx(2.6)According to the discussion in section 3, Q(α2, α1)can be computed from random numbers distributedas µ(x) = x−α1 as long as2α2 − α1 < 1. (2.7)α1 α2 Q(α2, α1) error (in %)0.0 0.15 1.179 0.2 %0.15 0.55 1.881 1 %0.55 0.75 1.799 1 %0.75 0.85 1.656 0.7 %0.85 0.9 1.508 0.6 %Table 1: Values of α1, α2, for which the quantityQ(α2, α1) is computed according to eq.(2.6). Theintegral eq.(2.1) is well approximated by∏Q.It is easy to see that all of the pairs (α1, α2)in table 1 satisfy the bound of eq.(2.7), and theMonte Carlo data forQ(α2, α1) can thus be trusted,just as the final result∫ 10dx x−0.9 =∏Q = 9.98± 0.16. (2.8)3. “Measurement” = “Comparison”After these preliminary steps, we finally confrontthe Monte Carlo measurement of the Yang-Millsintegrals. In this context, we recall from our ba-sic physics training the heading of this section.Translated to the context of a Monte Carlo calcu-lation, the measure/compare equivalence meansthat the integral eq.(1.1) has always to be writtenasZND,G =∫ ∏A,µdXAµ√2πµ({XAµ }){F({XAµ })µ({XAµ })}.(3.1)In the { } in eq.(3.1), we compare F to themeasure, which we are free to choose (but whichwe have to be able to integrate analytically). Asmentioned before, the Gaussians of eq.(1.2) aretoo different from F to work. A straightforwardgeneralization of the approach eq.(2.6) was foundto be wanting: the mismatch between F and µcould only be smoothed with a very large numberof steps (α1, α2) in eq.(2.6).A much better approach has come from theobservation that F can be compactified onto thesurface of a hypersphere, because both the actionand the Pfaffian are homogeneous functions ofthe radius R =√∑XAµ , which can thereforebe integrated out. Introducing polar coordinatesR,Ω and noting F˜ = ∫∞0dRRd−1F(Ω, R), wearrive at the ultimate formulation of the integralZND,G =∫dΩF˜(Ω). (3.2)This means that the integrand F˜ is compared tothe constant function on the surface of a hyper-sphere in dimension d = dim(G)N .The integral eq.(3.2) can still not be eval-uated directly, so that the strategy of eq.(2.6)has to be used. Here, we simply compute a fewratios of the integrals∫dΩ[F˜(Ω)]αfor differ-ent values 0 < α < 1. In this case, of course,pairs (α2, α1) are tested by the qualitative MonteCarlo algorithm, as analytical convergence con-ditions in the spirit of eq.(2.7) are lacking. Af-ter having expended an extraordinary amount ofrigor on these very difficult integrals, we never-theless obtain well-controlled predictions, to besurveyed below.4. SynopsisThe methods presented in the previous sectionswere used to compute a number of results whichare fully discussed in [1], [6], [7], [8]. For comple-mentary Monte Carlo studies, using somewhatdifferent techniques, see [9], [11]. To give an in-dication of the scope and the quality of the data,we present here our recent calculations for gaugegroups other than SU(N), as well as an intrigu-ing qualitative result concerning the asymptoticsof the eigenvalue distributions.The first example concerns the evaluation ofthe integrals for the gauge groups SO(N), Sp(2N)and G2. These calculations can be connected toother theoretical work essentially by dividing Zby the volume FG of the group G. In this way,4Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacherwe arrive at a numerical value for the bulk contri-bution to the quantum-mechanical Witten index,which is given byindD0 (G) =1FGZND,G. (4.1)In table 3 we list our Monte Carlo resultsfor this bulk index, obtained by the methods ex-plained above, for groups up to rank three. Wefurthermore compare these data to analytic pre-dictions from the generalization of the deforma-tion method of Moore et al. [4] to these groups[8]. Note the excellent precision (2% statisticalerror) for groups up to SO(7), where the inte-gral eq.(1.1) lies in 84 dimensions. Intriguingly,both our numerical and analytical results are atvariance with a previous conjecture [12] for non-unitary groups. In the special case of SO(7),e.g., ref. [12] obtains the fraction 15/128 whichis incompatible with our data.Group Monte Carlo ExactG indD=40 (G)SO(3) 0.2503 ± 0.0006 1/4SO(4) 0.0627 ± 0.0013 1/16SO(5) 0.1406 ± 0.001 9/64SO(6) 0.0620 ± 0.001 1/16SO(7) 0.0966 ± 0.0017 25/256Sp(2) 0.2500 ± 0.0002 1/4Sp(4) 0.139 ± 0.0015 9/64Sp(6) 0.0973 ± 0.003 51/512G2 0.173 ± 0.003 151/864Table 2: Monte Carlo results versus proposed(BRST deformation method) exact values for theD = 4 bulk index.Let us mention that at present the calcula-tions for D = 4 and D = 6 are considerably sim-pler than the case D = 10, because the Pfaffiancan be reduced to a determinant for D = 4, 6 [1].In D = 10, this possibility does not exist gener-ically (for an exception for SU(3) cf. [1]). Wehave now developed new methods to computePfaffians which should allow computations forD = 10 in the near future. It is possible if tediousto work out the predictions of the BRST defor-mation technique for indD=100 (G) cf [13], whichagain differ from the conjectures of [12]. It wouldbe interesting to check the results of [13] by ourMonte Carlo methods.A further strength of the Monte Carlo ap-proach is to allow the calculation of quantitiesother than just the integral Z. We briefly reviewas a second illustration of the here advocatedapproach the study of the correlation functions< TrXkµ} >, where Xµ is an arbitrary single ma-trix. This correlation function allows to infer theeigenvalue distribution of the matrices. Indeed,denoting the normalized eigenvalue density of in-dividual matrices by ρ(λ), one has< TrXkµ} >=∫ ∞−∞dλ ρ(λ) λk. (4.2)Here the calculation was immediately feasible alsofor D = 10 case, since we only needed to test forabsolute convergence, i.e. it suffices to considera simplified measure obtained from the absolutevalue of the original measure:µ′({XAµ}) = TrXkµ∣∣∣F({Xkν })∣∣∣. (4.3)This is algorithmically far more efficient, becausenow the problem may again be reduced to thecomputation of the square of a Pfaffian, which isreadily available, avoiding the calculation of thePfaffian itself.The final result for the asymptotic eigenvaluedensities as λ → ∞ supersymmetric systems inD = 4, 6, 10, for the supersymmetric system withgauge groups SU(N), isρSUSYD(λ) ∼λ−3 D = 4λ−7 D = 6λ−15 D = 10. (4.4)These laws were obtained by applying the aboveMarkov chain random walk tool to the measureeq.(4.3), i.e. we established divergence of eq.(4.2)iff k ≥ 2, 6, 14 (respectively for D = 4, 6, 10),leading to eq.(4.4). Note that these power lawsare independent of N . They demonstrate thatthe present matrix models are very different fromthe classic Wigner-type models. It would be in-teresting to obtain the generalization of eq.(4.4)to other gauge groups.AcknowledgmentsThis work was supported in part by the EU underContract FMRX-CT96-0012.5Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias StaudacherReferences[1] W. Krauth, H. Nicolai and M. Staudacher,Monte Carlo Approach to M-Theory, Phys. Lett.B431 (1998) 31, hep-th/9803117.[2] P. Yi,Witten Index and Threshold Bound Statesof D-Branes, Nucl. Phys. B505 (1997) 307,hep-th/9704098; S. Sethi and M. Stern, D-Brane Bound State Redux, Commun. Math.Phys. 194 (1998) 675, hep-th/9705046.[3] A.V. Smilga, Witten Index Calculation in Su-persymmetric Gauge Theory, Yad. Fiz. 42(1985) 728, Nucl. Phys. B266 (1986) 45;A.V. Smilga, Calculation of the Witten Indexin Extended Supersymmetric Yang-Mills The-ory, (in Russian) Yad. Fiz. 43 (1986) 215.[4] G. Moore, N. Nekrasov and S. Shatashvili, D-particle bound states and generalized instan-tons, Commun. Math. Phys. 209 (2000) 77,hep-th/9803265.[5] M. B. 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