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: matthias@aei-potsdam.mpg.de

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.TheXµare 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- G X XAT tion of a given Lie algebra Lie( ): µ = µ A, sonic integral, which is obtained after integrat- with A =1,...,dim(G). The group G may be ing out the N (= number of supersymmetries) SU(N), but the orthogonal, symplectic and ex- Grassmann-valued fermionic matrices ceptional groups have also come under close scrutiny Z Y A dXµ recently. ZN = √ × D,G π These ordinary, multiple Riemann integrals A,µ 2 1 stem from a dimensional reduction of D-dimen- Tr [Xµ ,Xν ][Xµ,Xν ] A e 4g2 (P X ). (1.1) sional Euclidean continuum Yang-Mills theory to µ zero dimensions. They have important impli- In this equation, P({X}) is the Pfaffian of a cer- cations, as the integrals yield the bulk part of tain matrix M, which can be constructed from the Witten index of supersymmetric quantum the adjoint representation of the Xµ. mechanical gauge theories, and appear in multi- During the last few years, intense effort has N instanton calculations of large 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 Z grals, both in the supersymmetric and the purely 1 I(α)= dxµ(x)x−α (2.1) bosonic case (where the Pfaffian in eq.(1.1) is 0 simply omitted). We have also obtained very pre- with a constant weight function µ(x) = 1, which cise values (statistical estimates) of Z for several 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 Z A 2 A Y dX XA F X N µ − µ ( µ ) 2 Z √ e 2σ2   , 0 5000 10000 D,G = P 2 π XA MC-time µ,A 2 − µ e 2σ2 (1.2) Figure 1: Partial sums St (cf. eq.(2.2)) for α =0.9 A where F( Xµ ) is the integrand in the second vs Monte Carlo time t. The numerical estimate for 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 x ≤ x ≤ this distribution. The straightforward approach uniformly distributed points t with 0 t 1 is thwarted by the fact that the integrals eq.(1.2) and compute - and, equivalently, eq.(1.1) - only barely con- Xt −α verge, if at all. I ∼ St =1/t xi , (2.2) The reason for this bad convergence lies in i=1 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)givesrisetoa Monte Carlo calculation for α =0.9isshown 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 Z. There is presently no mathemat- I(α =0.9) = 6.13 0.46, without emitting any 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 O = x−α, 0.3 the variance is given by Position 0.2 Z Z 0.1 Var dxµ x O x 2 − dxµ x O x 2. = ( ) ( ) [ ( ) ( )] 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 (2.3) MC-time

The error inp the Monte Carlo evaluation eq.(2.2) Figure 2: Position xt vs Monte Carlo time t for the behaves like Var/t, and, for α ≥ 0.5, is infinite. Markov chain with stationary distribution µ0(x)= −0.9 This situation is virtually impossible to diagnose x . The time evolutionR of xt does not get stuck, 1 dxx−0.9 from within the simulation itself. because the integral 0 exists. We have developed a highly efficient tool to 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 µ0(x)=µ(x)x−α and µ00(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 tributed between + and −, and to go from xt 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 MC-time to xt+1 according to the following probability ta- 00 −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 ,µ0 x /µ x . t+1 =  min(1 ( t+1) ( t)) xt else 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 xt, neither used nor useful to compute the integral 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 Z figures 2 and 3), we have used it extensively to es- x0+ dxµ0(x)=∞ (2.5) tablish the existence conditions for bosonic and x0 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,weshowxt for Markov chains with stationary tant information on the asymptotic behavior of distributions µ0(x)andµ00(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 operator to be wanting: the mismatch between F and µ measurez }| { z }| { R1 R 1 −α2 −α1 α1−α2 could only be smoothed with a very large number 0 x dx 0 x x dx Q α ,α R R of steps (α1,α2) in eq.(2.6). ( 2 1)= 1 = 1 −α x−α1 dx x 1 dx 0 0 | {z } A much better approach has come from the measure observation that F can be compactified onto the (2.6) surface of a hypersphere, because both the action According to the discussion in section 3, Q(α2,α1) and the Pfaffian are homogeneous functions of can be computed from random numbers distributed qP R XA as µ(x)=x−α1 as long as the radius = µ , which can therefore be integrated out. IntroducingR polar coordinates α − α < . ˜ ∞ d−1 2 2 1 1 (2.7) R, Ω and noting F = 0 dRR F(Ω,R), we arrive at the ultimate formulation of the integral Z α1 α2 Q(α2,α1) error (in %) ZN = dΩF˜(Ω). (3.2) 0.0 0.15 1.179 0.2 % D,G 0.15 0.55 1.881 1% 0.55 0.75 1.799 1% This means that the integrand F˜ is compared to 0.75 0.85 1.656 0.7 % the constant function on the surface of a hyper- 0.85 0.9 1.508 0.6 % sphere in dimension d = dim(G)N . The integral eq.(3.2) can still not be eval- α ,α Table 1: Values of 1 2, 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 Q has to be used. Here, we simply compute a few integral eq.(2.1) is well approximated by Q. R h iα ratios of the integrals dΩ F˜(Ω) for differ- <α< It is easy to see that all of the pairs (α1,α2) ent values 0 1. In this case, of course, 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 Z 1 Y 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 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, Z ( ) we present here our recent calculations for gauge Y dXA F {XA} N µ A ( µ ) SU N Z = √ µ({X }) . groups other than ( ), as well as an intrigu- D,G π µ µ {XA} A,µ 2 ( µ ) ing qualitative result concerning the asymptotics (3.1) of the eigenvalue distributions. In the {}in eq.(3.1), we compare F to the The first example concerns the evaluation of 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 F to work. A straightforward by the volume FG of the group G. In this way,

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 1 D G ZN . other than just the integral Z. We briefly review ind0 ( )=F D,G (4.1) G as a second illustration of the here advocated In table 3 we list our Monte Carlo results approach the study of the correlation functions < X k } > X for this bulk index, obtained by the methods ex- Tr µ ,where µ is an arbitrary single ma- 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- ρ λ tion method of Moore et al. [4] to these groups dividual matrices by ( ), one has Z ∞ [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- −∞ 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. 0 A k k µ ( Xµ )=TrXµ F({Xν}) . (4.3) Group Monte Carlo Exact This is algorithmically far more efficient, because G D=4 G ind0 ( ) 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  λ−3 D =4 SUSY G2 0.173 0.003 151/864 ρ λ ∼ λ−7 D . D ( )  =6 (4.4) λ−15 D Table 2: Monte Carlo results versus proposed =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), D D tions for =4and = 6 are considerably sim- leading to eq.(4.4). Note that these power laws D pler than the case = 10, because the Pfaffian are independent of N. They demonstrate that D , can be reduced to a determinant for =4 6[1]. the present matrix models are very different from D In = 10, this possibility does not exist gener- the classic Wigner-type models. It would be in- SU ically (for an exception for (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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