Statistical Physics Approach to M-Theory Integrals

arXiv:cond-mat/0010127v1 [cond-mat.stat-mech] 9 Oct 2000 Abstract: Staudacher Matthias and Krauth Werner ttsia hsc praht -hoyIntegrals M-theory to Approach Physics Statistical h generators the ehnclguetere,adapa nmulti- large of in calculations quantum appear instanton of and supersymmetric theories, part gauge of bulk mechanical index the yield Witten impli- integrals the important the have as They cations, dimensions. zero to D-dimen- theory Yang-Mills of continuum Euclidean reduction sional dimensional a from stem scrutiny recently. close under come also have groups ceptional nitgaso non-linearly clas- focusses D the now of integrals of interest on difference models, the Wigner-type At sic models. gauge-invariant of matrix class new exis- important the an revealed of tence has theory field in work Recent Introduction 1. Keywords: re man analytical In conflicting between integrals. here. controversies resolve supersymmetric to and possible bosonic eigenvalu multiple asymptotic of to applications evaluations mat near-s methods, of of Markov-chain class computability by new Carlo tion ubiquitous Monte an versus integrals, convergence Yang-Mills Absolute of study the in ful SU ino ie i ler Lie( algebra Lie given a of tion X with µ ∗ E-mail: Golm D-14476 Mühlenberg 1 Am für Gra Max-Planck-Institut Albert-Einstein-Institut, E-mail: 05 Cedex Paris F-75231 Lhomond S rue Normale 24, Ecole Statistique, Physique de CNRS-Laboratoire µ , ( akpeetdb .Krauth W. by presented Talk hs riay utpeReanintegrals Riemann multiple ordinary, These N A ,btteotooa,smlci n ex- and symplectic orthogonal, the but ), 1 = PROCEEDINGS 1 = D . . . , matthias@aei-potsdam.mpg.de [email protected] , . . . , eepantecnet fcmuainlsaitclpyiswhich physics statistical computational of concepts the explain We ot al ehd,Mter,Mti oes agMlsTheory. Yang-Mills Models, Matrix M-theory, Methods, Carlo Monte T The . A dim( ftefnaetlrepresenta- fundamental the of G X .Tegroup The ). µ r osrce from constructed are N G coupled ): uyYang-Mills susy X µ G = matrices a be may X µ A T A , vitationsphysik rsmn-audfrincmatrices fermionic Grassmann-valued integrat- the after out obtained ing is which integral, bosonic effective the supersym- only for theories) Yang-Mills (even metric down write we definitions), mech- Eguchi-Kawai anism. the via theories informa- gauge on non-perturbative tion further contain whether elucidated they be to remains model) It IKKT M-theory. (the and theory proposed string of in formulations appear they Furthermore, theories. rmtergru xc ouinfor solution ranging exact integrals, rigorous these the on from bear to brought been nti equation, this In h don ersnaino the of representation adjoint the matrix tain o h aeo rvt c.[]frcomplete for [1] (cf. brevity of sake the For uigtels e er,itneeothas effort intense years, few last the During upérieure Z oprubtv unu ffcs2000 Effects Quantum Nonperturbative D,G N e 4 g 1 M 2 N nua nerl,snuaiydetec- singularity integrals, ingular ut sn h ehd presented methods the using sults = r[ Tr =nme fsupersymmetries) of number (= hc a ecntutdfrom constructed be can which , itiuin n onumerical to and distributions e Z X P µ i oes sustetdare: treated Issues models. rix ( A,µ Y ,X ae led,i a been has it already, cases y { X ν ][ dX √ } X stePaa facer- a of Pfaffian the is ) 2 µ µ π ,X A aepoe eyhelp- very proven have × ν ] ( P X X µ µ SU A .

∗ ) 2 [2],[3] (2) . (1.1) 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 computation 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. gets stuck. If so, it has become attracted by a Besides its “consulting” role within the Monte point x0 with Carlo framework (as explained in the caption of figures 2 and 3), we have used it extensively to es- x0+ǫ dxµ′(x)= (2.5) tablish the existence conditions for bosonic and ∞ Zx0 susy Yang-Mills integrals, which have not been obtained analytically beyond the 1-loop level. We i. e. a non-integrable singularity. In figures 2 and have also adopted the method to obtain impor- 3 , we show x for Markov chains with stationary t tant information on the asymptotic behavior of distributions µ′(x) and µ′′(x), respectively. Fig- integrals. ure 3, in particular, implies that the variance of the integral eq.(2.1) is infinite so that the result We conclude the discussion of our toy prob- of fig. 1 cannot be trusted, while figure 2 assures lem by showing how, after all, the integral eq.(2.1) us that the integral exists. can be computed by Monte Carlo methods. Con-

3 Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacher sider first generalization of the approach eq.(2.6) was found to be wanting: the mismatch between and µ measure operator F could only be smoothed with a very large number 1 x−α2 dx 1 x−α1 xα1−α2 dx 0 0 of steps (α , α ) in eq.(2.6). Q(α2, α1)= 1 = 1 − 1 2 −α1 z }| { αz1 }| { R0 x dx R 0 x dx A much better approach has come from the measure observation that can be compactified onto the R R (2.6) F surface of a hypersphere, because both the action According to the discussion in section| {z 3,}Q(α , α ) 2 1 and the Pfaffian are homogeneous functions of can be computed from random numbers distributed A as µ(x)= x−α1 as long as the radius R = Xµ , which can therefore be integrated out.q IntroducingP polar coordinates ∞ d−1 2α2 α1 < 1. (2.7) R, Ω and noting ˜ = dRR (Ω, R), we − F 0 F arrive at the ultimate formulation of the integral R α1 α2 Q(α2, α1) error (in %) N = dΩ ˜(Ω). (3.2) 0.0 0.15 1.179 0.2 % ZD,G F 0.15 0.55 1.881 1 % Z This means that the integrand ˜ is compared to 0.55 0.75 1.799 1 % F 0.75 0.85 1.656 0.7 % the constant function on the surface of a hypersphere in dimension d = dim(G) . 0.85 0.9 1.508 0.6 % N The integral eq.(3.2) can still not be eval- Table 1: 1 2 Values of α , α , for which the quantity uated directly, so that the strategy of eq.(2.6) Q(α2, α1) is computed according to eq.(2.6). The has to be used. Here, we simply compute a few integral eq.(2.1) is well approximated by Q. α ratios of the integrals dΩ ˜(Ω) for differ- F Q ent values 0 <α< 1. In this case, of course, It is easy to see that all of the pairs (α1, α2) R h i in table 1 satisfy the bound of eq.(2.7), and the pairs (α2, α1) are tested by the qualitative Monte Monte Carlo data for Q(α2, α1) can thus be trusted, Carlo algorithm, as analytical convergence con- just as the final result ditions in the spirit of eq.(2.7) are lacking. Af- ter having expended an extraordinary amount of 1 dx x−0.9 = Q =9.98 0.16. (2.8) rigor on these very difficult integrals, we never- 0 ± theless obtain well-controlled predictions, to be Z Y surveyed below. 3. “Measurement” = “Comparison”

After these preliminary steps, we finally confront 4. Synopsis the Monte Carlo measurement of the Yang-Mills integrals. In this context, we recall from our ba- The methods presented in the previous sections sic physics training the heading of this section. were used to compute a number of results which Translated to the context of a Monte Carlo calcu- are fully discussed in [1], [6], [7], [8]. For comple- lation, the measure/compare equivalence means mentary Monte Carlo studies, using somewhat that the integral eq.(1.1) has always to be written different techniques, see [9], [11]. To give an in- as dication of the scope and the quality of the data, we present here our recent calculations for gauge dXA ( XA ) N = µ µ( XA ) F { µ } . groups other than SU(N), as well as an intrigu- ZD,G √ { µ } µ( XA ) A,µ 2π ( µ ) ing qualitative result concerning the asymptotics Z Y { } (3.1) of the eigenvalue distributions. In the in eq.(3.1), we compare to the The first example concerns the evaluation of {} F measure, which we are free to choose (but which the integrals for the gauge groups SO(N),Sp(2N) we have to be able to integrate analytically). As and G2. These calculations can be connected to mentioned before, the Gaussians of eq.(1.2) are other theoretical work essentially by dividing Z too different from to work. A straightforward by the volume G of the group G. In this way, F F

4 Nonperturbative Quantum Effects 2000 Werner Krauth and Matthias Staudacher we arrive at a numerical value for the bulk contri- be interesting to check the results of [13] by our bution to the quantum-mechanical Witten index, Monte Carlo methods. which is given by A further strength of the Monte Carlo approach is to allow the calculation of quantities D 1 N ind0 (G)= D,G. (4.1) other than just the integral . We briefly review G Z Z F as a second illustration of the here advocated In table 3 we list our Monte Carlo results approach the study of the correlation functions k < Tr X >, where Xµ is an arbitrary single ma- for this bulk index, obtained by the methods ex- µ } plained above, for groups up to rank three. We trix. This correlation function allows to infer the furthermore compare these data to analytic pre- eigenvalue distribution of the matrices. Indeed, dictions from the generalization of the deforma- denoting the normalized eigenvalue density of in- tion method of Moore et al. [4] to these groups dividual matrices by ρ(λ), one has ∞ [8]. Note the excellent precision (2% statistical k k < Tr Xµ >= dλ ρ(λ) λ . (4.2) error) for groups up to SO(7), where the inte- } −∞ Z gral eq.(1.1) lies in 84 dimensions. Intriguingly, Here the calculation was immediately feasible also both our numerical and analytical results are at for D = 10 case, since we only needed to test for variance with a previous conjecture [12] for non- absolute convergence, i.e. it suffices to consider unitary groups. In the special case of SO(7), a simplified measure obtained from the absolute e.g., ref. [12] obtains the fraction 15/128 which value of the original measure: is incompatible with our data. µ′( XA ) = Tr Xk ( Xk ) . (4.3) µ µ F { ν } Group Monte Carlo Exact This is algorithmically far more efficient, because D=4 G ind0 (G) now the problem may again be reduced to the SO(3) 0.2503 0.0006 1/4 computation of the square of a Pfaffian, which is ± SO(4) 0.0627 0.0013 1/16 readily available, avoiding the calculation of the ± SO(5) 0.1406 0.001 9/64 Pfaffian itself. ± SO(6) 0.0620 0.001 1/16 The final result for the asymptotic eigenvalue ± SO(7) 0.0966 0.0017 25/256 densities as λ supersymmetric systems in ± → ∞ Sp(2) 0.2500 0.0002 1/4 D =4, 6, 10, for the supersymmetric system with ± Sp(4) 0.139 0.0015 9/64 gauge groups SU(N), is ± Sp(6) 0.0973 0.003 51/512 −3 ± λ D =4 SUSY −7 G2 0.173 0.003 151/864 ρD (λ) . (4.4) ± λ D =6 ∼  −15 Table 2: Monte Carlo results versus proposed  λ D = 10 (BRST deformation method) exact values for the These laws were obtained by applying the above D = 4 bulk index. Markov chain random walk tool to the measure eq.(4.3), i.e. we established divergence of eq.(4.2) Let us mention that at present the calcula- iff k 2, 6, 14 (respectively for D = 4, 6, 10), ≥ tions for D = 4 and D = 6 are considerably sim- leading to eq.(4.4). Note that these power laws pler than the case D = 10, because the Pfaffian are independent of N. They demonstrate that can be reduced to a determinant for D =4, 6 [1]. the present matrix models are very different from In D = 10, this possibility does not exist gener- the classic Wigner-type models. It would be in- ically (for an exception for SU(3) cf. [1]). We teresting to obtain the generalization of eq.(4.4) have now developed new methods to compute to other gauge groups. Pfaffians which should allow computations for D = 10 in the near future. It is possible if tedious Acknowledgments to work out the predictions of the BRST defor- D=10 mation technique for ind0 (G) cf [13], which This work was supported in part by the EU under again differ from the conjectures of [12]. It would Contract FMRX-CT96-0012.

5 Nonperturbative Quantum Eﬀects 2000 Werner Krauth and Matthias Staudacher

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