Quantum billiards in multidimensional models with branes

Gravitational D-dimensional model with l scalar fields and several forms is considered. When cosmological type diagonal metric is chosen, an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed the conformally covariant Wheeler-DeWitt (WDW) equation for the model is studied. Under certain restrictions asymptotic solutions to WDW equation are found in the limit of the formation of the billiard walls which reduce the problem to the so-called quantum billiard on the (D+ l -2)-dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with 9-dimensional quantum billiard for D = 11 model with 330 four-forms which mimic space-like M2- and M5-branes of D=11 supergravity. The second one deals with the 9-dimensional quantum billiard for D =10 gravitational model with one scalar field, 210 four-forms and 120 three-forms which mimic space-like D2-, D4-, FS1- and NS5-branes in D = 10 II A supergravity. It is shown that in both examples wave functions vanish in the limit of the formation of the billiard walls (i.e. we get a quantum resolution of the singularity for 11D model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behaviour when all 120 electric branes are present.


Introduction
This paper deals with the quantum billiard approach for D -dimensional cosmological-type models defined on the manifold (u − , u + ) × R D−1 , where D ≥ 4 .
The billiard approach in classical gravity originally appeared in the dissertation of Chitré [1] for the explanation the BLK-oscillations [2] in the Bianchi-IX model [3,4]. In this approach a simple triangle billiard in the Lobachevsky space H 2 was used.
In [5] the billiard approach for D = 4 was extended to the quantum case. Namely, the solutions to the Wheeler-DeWitt (WDW) equation [6] were reduced to the problem of finding the spectrum of the Laplace-Beltrami operator on a Chitré's triangle billiard. Such approach was also used in [7] in the context of studying the large scale inhomogeneities of the metric in the vicinity of the singularity.
A straightforward generalization of the Chitré's billiard to the multidimensional case was performed in [8,9,10] where multidimensional cosmological model with multicomponent "perfect" fluid and n Einstein factor spaces was studied. In [10] the search of oscillating behaviour near the singularity was reduced to the problem of proving the finiteness of the billiard volume. This problem was reformulated in terms of the problem of the illumination of the sphere S n−2 by point-like sources. In [10] the inequalities on the Kasner parameters were found and the "quantum billiard" approach was considered; see also [11,12]. The classical billiard approach for multidimensional models with fields of forms and scalar fields was suggested in [13], where the inequalities for the Kasner parameters were also written. For certain examples these inequalities have played a key role in the proof of the never-ending oscillating behaviour near the singularity which takes place in effective gravitational models with forms and scalar fields induced by superstrings [14,15,16]. It was shown in [17] that in these models the parts of billiards are related to Weyl chambers of certain hyperbolic Kac-Moody (KM) Lie algebras [18,19,20,21]. This fact simplifies the proof of the finiteness of the billiard volume. Using this approach the well-known result from [22] on the critical dimension of pure gravity was explained using hyperbolic algebras in [23]. For reviews on the billiard approach see [16,24].
In recent publications [25,26] the quantum billiard approach for the multidimensional gravitational model with several forms was considered. The main motivation for this approach is coming from the quantum gravity paradigm; see [27,28] and references therein. It should be noted that the asymptotic solutions to the WDW equation presented in these papers are equivalent to the solutions obtained earlier in [10]. The wave function ( Ψ KKN ) from [25,26] corresponds to the harmonic time gauge, while the wave function ( Ψ IM ) from [10] is related to the "tortoise" time gauge. (These functions are connected by a certain conformal transformation Ψ KKN = ΩΨ IM .) In [10,25,26] a "semiquantum" approach was used: the gravity (of a toy model) was quantized but the matter sources (e.g. fluids, forms) were considered at the classical level. 3 Such a semi-quantum form of the WDW equation for the model with fields of forms and a scalar field was suggested earlier in [33].
In our previous publication [34] we have used another form of the WDW equation with enlarged minisuperspace which includes the form potentials [35]. We have suggested another version of the quantum billiard approach by deducing the asymptotic solutions to WDW equation for the model with fields of forms when a non-composite electric brane ansatz has been adopted.
In [34] we have considered an example of a 9 -dimensional quantum billiard for D = 11 model with 120 four-forms which mimic space-like M2 -brane solutions ( SM2 -branes) in D = 11 supergravity. 4 It was shown in [34] that the wave function vanishes as y 0 → −∞ (i.e. at the singularity), where y 0 is the "tortoise" time-like coordinate in the minisuperspace.
In this paper we substantially generalize the approach of [34] to the case when scalar (dilatonic) fields and dilatonic couplings are added into consideration. Here the composite electromagnetic ansatz for branes is considered instead of non-composite electric one from [34]. We present new examples of quantum billiards with electric and magnetic S -branes in D = 11 and D = 10 models, which are non-composite analogues of truncated bosonic sectors of D = 11 and D = 10 supergravitational models. In both examples of billiards magnetic branes do not participate in the formation of the billiard walls since magnetic walls are hidden by electric ones. The adding of magnetic branes does not change the classical asymptotic oscillating behaviour of scale factors and scalar field (for D = 10 ). In the quantum case adding of magnetic branes changes the asymptotic behaviour of the wave function, but nevertheless, as in [34], the wave function vanishes as y 0 → −∞ . For D = 11 this means a quantum resolution of the singularity for the model with electric and magnetic branes which mimic (space-like) SM2 -and SM5 -branes in 11D supergravity.

The setup
Here we study the multidimensional gravitational model governed by the action , θ a = 0 , and we have where ∆ is some finite set of (colour) indices and S YGH is the standard (York-Gibbons-Hawking) boundary term [40,41]. In the models with one time and the usual fields of forms all θ a > 0 when the signature of the metric is (−1, +1, . . . , +1) . For such a choice of signature θ b < 0 corresponds to a "phantom" form field F b .

Ansatz for composite brane configurations
We consider the manifold with the metric Here one may replace R n in (2.2) by R k × (S 1 ) n−k , 0 ≤ k ≤ n , without any change of all the relations as presented below.
Although in what follows all examples deal with cosmological ( S -brane) solutions with w = −1 and ε(i) = +1 for all i , we reserve here general notations for signs just keeping in mind possible future applications to static configurations with w = 1 , ε(1) = −1 and ε(k) = +1 for k > 1 (e.g. fluxbranes, wormholes etc.) and solutions with several time-like directions.
Here and below for I ∈ Ω a,e and J ∈ Ω a,m , a ∈ ∆ , i.e. in electric and magnetic case, respectively.

Sigma-model action
Here we present two restrictions on the sets of branes which guarantee the diagonal form of the energy-momentum tensor and the existence of the sigmamodel representation (without additional constraints) [42] (see also [43]). The first restriction deals with any pair of two (different) branes both electric ( ee -pair) or magnetic ( mm -pair) with coinciding color index: for any I, J ∈ Ω a,v , a ∈ ∆ , v = e, m (here d(I) = d(J) ).
The second restriction deals with any pair of two branes with the same color index, which include one electric and one magnetic brane ( em -pair): (2.14) where I ∈ Ω a,e , J ∈ Ω a,m , a ∈ ∆ . These restrictions are satisfied identically in the non-composite case, when there are no two branes corresponding to the same form F a for any a ∈ ∆ .
It follows from [42] that the equations of motion for the model (2.1) and the Bianchi identities, dF s = 0 , s ∈ S m , for fields from (2.3), (2.7)-(2.10), when restrictions (R1) and (R2) are imposed, are equivalent to the equations of motion for the σ -model governed by the action whereẋ ≡ dx/du , (σ A ) = (φ i , ϕ α ) , µ = 0 , the index set S is defined in (2.11) and is a truncated target space metric with and co-vectors Here χ e = +1 and χ m = −1 ; is the indicator of i belonging to I : δ iI = 1 for i ∈ I and δ iI = 0 otherwise; and In the electric case (F (a,m,I) = 0) when any factor space with the coordinate x i is compactified to a circle of length L i , the action (2.15) coincides with the action In what follows we will use the scalar products of is the matrix inverse to the matrix (2.16). Here The scalar products (2.21) read [42] ( The action (2.15) may also be written in the form where X = (XÂ) = (φ i , ϕ α , Φ s ) ∈ R N , N = n + l + m , m = |S| is the number of branes and minisupermetric G = GÂB(X)dXÂ ⊗ dXB on the minisuperspace M = R N is defined as follows: The minisuperspace metric (2.26) may also be written in the form This vector is time-like and all (U s , U Λ ) < 0 , since [42] ( (2.30)

Quantum billiard approach
In this section we develop a quantum analogue of the billiard approach which deals with asymptotical solutions to Wheeler-DeWitt (WDW) equation.

Restrictions.
First we outline restrictions on parameters which will be used in derivation of the "quantum billiard" for all s . These restrictions are necessary conditions for the formation of infinite "wall" potential in certain limit (see below). The first restriction reads (see (2.24)) The second restriction means that the matrix (h αβ ) is positive definite, i.e. the so-called phantom scalar fields are not considered.

Wheeler-DeWitt equation
Now we fix the temporal gauge as follows: where f : M → R is a smooth function. Then we obtain the Lagrange system with the Lagrangian and the energy constraint The set of Lagrange equations with the constraint (3.7) is equivalent to the set of Hamiltonian equations for the Hamiltonian where PÂ = µe 2f GÂB(X)ẊB are momenta (for fixed gauge) and (GÂB) = (GÂB) −1 .
Here we use the prescriptions of covariant and conformally covariant quantization of the hamiltonian constraint H f = 0 which was suggested initially by Misner [44] and considered afterwards in [45,46,47,48] and some other papers.
We obtain the Wheeler-DeWitt (WDW) equation 5 is the wave function corresponding to the f -gauge (3.5) and satisfying the relation 6 In (3.10) we denote by ∆[G f ] and R[G f ] the Laplace-Beltrami operator and the scalar curvature corresponding to the metric respectively. The choice of minisuperspace covariant form for the Hamiltonian operator H f (3.10) with arbitrary real number a is one of the solutions to the operator ordering problem in multidimensional quantum cosmology [49,50,51,52]. The Lapalace-Beltrami form of WDW equation was considered previously in [6,53,54,55,56]. Similar prescription appears in quantization of a point-like particle moving in a curved background, for a review see [57,58].
It was shown in [47,48] by rigorous constraint quantization of parametrized relativistic gauge systems in curved spacetimes that the privileged choice for a in cosmological case is given by (3.11). For this value of a and N > 1 there is one-to-one correspondence between solutions to WDW equations for any two choices of temporal gauges given by (3.5) with smooth functions f 1 and f 2 instead of f , respectively. This fact follows from (3.12) and the following relation: (3.14) We note that the coefficients a N and b N are the well-known ones in the conformally covariant theory of a scalar field [59].
Now we put f = f (σ) . Then we get Here we deal with a special class of asymptotical solutions to WDWequation. Due to restrictions (3.2) and (3.3) the (minisuperspace) metricŝ G , G have a pseudo-Euclidean signatures (−, +, ..., +) . We put Here we use a diagonalization of σ -variables We restrict the WDW equation to the lower light cone V − = {z = (z 0 , z)|z 0 < 0, η ab z a z b < 0} and introduce the Misner-Chitré-like coordinates where y 0 < 0 and y 2 < 1 . In these variables we have f = y 0 .
Using the relation f ,A =Ḡ AB σ B , following from (3.19), we obtain These relations just follow from the relation For the wave function we suggest the following ansatz: where the prefactor e C(σ) is chosen for the sake of cancellation the terms linear in derivatives Ψ * ,A arising in calculation of ∆[G f ]Ψ f . This takes place for With this choice of the prefactor we obtain Using relations (3.22) we get The calculation of the scalar curvature R e 2f G gives us the following formula: Collecting relations (3.28), (3.29) and (3.30) we obtain the following identity: Here we denote In what follows we call A as A -number.
Now we put where the parameters Q s = 0 correspond to the charge densities of branes and e iQ s Φ s = exp(i s∈S Q s Φ s ) . Using (3.31) we get Here and in what follows U s (σ) = U s A σ A . It was shown in [13] that as y 0 = f → −∞ . Here V ∞ is the potential of infinite walls which are produced by branes where we denote θ ∞ (x) = +∞ , for x ≥ 0 and θ ∞ (x) = 0 for x < 0 . The vectors v s , s ∈ S , which belong to where the N 0 -dimensional vectors u s = (u s0 , u s ) = (u sa ) are obtained from U s -vectors using the diagonalization matrix (S A a ) from (3.19) for all s . In what follows we use a diagonalization (3.19) from [13] obeying for all s ∈ S . The inverse matrix (S a A ) = (S A a ) −1 defines the map which is inverse to (3.19) z see (2.30). This choice of U with k = 1 was done in [13]. Remark 1. Conditions (3.42) (or (3.47)) may be relaxed. In this case we obtain a more general definition of the billiard walls (e.g. for u s0 < 0 and u s0 = 0 ) described in [24].
Thus, we are led to the asymptotical relation for the function Ψ 0,L (y 0 , y) we obtain the following asymptotical relation (for y 0 → −∞ ) We assume that the minus Laplace-Beltrami operator (−∆ L ) with the zero boundary conditions has a spectrum obeying the following inequality: The examples of billiards obeying this restriction were considered in [25,26] (see also the next section).
Here we restrict ourselves to the case of negative A -number Solving equation (3.51) we get for A < 0 the following set of basis solutions: where B iω (z) = I iω (z), K iω (z) are the modified Bessel functions and By using the asymptotical relations for z → +∞ , we find for y 0 → −∞ . Here C ± are non-zero constants, "plus" corresponds to B = I and "minus" -to B = K . Now we evaluate the prefactor e C(σ) in (3.34), where Here we denote In what follows we use the vector U = (U A ) as a time-like vector in the relation for z 0 in (3.45). Thus, we need to impose the restriction (3.46) ( (U, U ) < 0 ) .
Using (3.20), (3.45) and f = y 0 we obtain as y 0 → −∞ for any fixed y ∈ B and C ± = 0 . Here we denote where "plus" corresponds to the solution with B = I and "minus" -to B = K . Relation (3.33) may be rewritten in the following form: where we have used the identity following from the definition of U in (3.60). It should be noted that restrictions (U, U ) < 0 and (U s , U s ) > 0 , s ∈ S , imply A < 0 . Now we study the asymptotical behaviour of the wave function (3.34) Remark 2. It should be noted that solution (3.55) is similar to those which were found in quantum cosmological models with Λ -term, perfect fluid etc., see [60,61] and references therein. For these solutions we have v = e qz 0 instead of e −y 0 , where v is the volume [60] or the "quasi-volume" [61] scale factor. Our restriction A < 0 corresponds to the restriction Λ < 0 for the solutions from [60].

Examples
Here we illustrate our approach by two examples of quantum billiards in dimensions D = 11 and D = 10 . In what follows we use the notation Ω(n, k) for the set of all subsets of {1, . . . , n} , which contain k elements . Any element of Ω(n, k) has the form I = {i 1 , . . . , i k } , 1 ≤ i 1 < . . . < i k ≤ n , The number of elements in Ω(n, k) is C k n = n! k!(n−k)! . In this section we deal with (n + 1) -dimensional cosmological metric of Bianchi-I type, where u ∈ (u − , u + ) .

9 -dimensional billiard in D = 11 model
Let us consider an 11 -dimensional gravitational model with several 4-forms, which produce non-composite analogues of SM -brane solutions in D = 11 supergravity [62]. The action reads as follows: where first we put L = L e , where Here F I 4,e is "electric" 4-form with the index I ∈ Ω(10, 3) . The number of such forms is C 3 10 = 120 .
The action (4.2) with L from (4.3) describes non-composite analogues of SM2 -brane solutions which are given by the metric (4.1) with n = 10 and Consider the non-trivial case when all charge densities of branes Q s , s ∈ S e , are non-zero. In the classical case we get a 9 -dimensional billiard B ∈ H 9 with 120 "electric" walls [34]. This classical billiard coincides with the 9d billiard from [17,16]. B has a finite volume. It is a union of several identical "small" billiards which have finite volumes as they correspond to the Weyl chamber of the hyperbolic Kac-Moody algebra E 10 [17].
Remark 3. In [34] we have used 120 form fields and non-composite ansatz for branes to avoid the appearance of the set of 45 constraints which arise for composite solutions with diagonal metric [42,63]. These constraints are coming from the relations T ij = 0 , 1 ≤ i < j ≤ 10 , where T ij are spatial components of the stress-energy tensor. We note that in [16,17] this problem was circumvented by considering non-diagonal metrics from the very beginning.
Then we get (see (2.23)) in agreement with our restriction (3.46). Since N = 130 ( m = 120 ) and (U s , U s ) = 2 we obtain from (3.65) the following value for the A -number [34]: . The minus Laplace-Beltrami operator (−∆ L ) on B with the zero boundary conditions imposed has a spectrum obeying restriction (3.53) with N 0 = 10 [26].
We get from the previous analysis the asymptotical vanishing of the wave function Ψ f → 0 as y 0 → −∞ . Now we consider the electromagnetic case, which mimics solutions with SM2 -and SM5 -branes.
We put in (4.2) L = L e + L m , where Here F J 4,m is a "magnetic" 4-form with the index J ∈ Ω(10, 6) . The number of such forms is C 6 10 = 210 . We extend the cosmological electric ansatz by adding the following relations: J ∈ Ω(10, 6) , a = (4, m, J) , where F (a,m,J) are defined in (2.9). For charge densities we put Q s = 0 , s ∈ S .
In the electromagnetic case we get the same 9 -dimensional billiard B ∈ H 9 as in the electric case, since magnetic walls are hidden by electric ones. This could be readily verified using the billiard chamber belonging to the lower light cone and the fact that any magnetic U -vector is the sum of two electric ones. The Lobachevsky space H 9 may be identified with the hypersurface y 0 = 0 in the lower light cone. Then the billiard B may be obtained just by the projection of W onto H 9 : (y 0 , y) → y . Adding into our consideration a magnetic brane with U m = U 1e +U 2e , where U 1e and U 2e correspond to electric branes, gives a new inequality in the definition of W : U m (σ) < 0 , which is satisfied identically due to relations U 1e (σ) < 0 and U 2e (σ) < 0 from the definition of W in the electric case. Thus, the addition of any magnetic SM5 -brane does not change the electric billiard chamber nor the electric billiard.
We obtain The analysis carried out in the previous section implies the asymptotical vanishing of the wave function Ψ f → 0 as y 0 → −∞ .
We put Q s = 0 , s ∈ S e . In the classical case we get the same 9 -dimensional billiard B ∈ H 9 with 120 "electric" walls as in the SM2 -brane case [15,16].
Let us us calculate (U, U ) , where U = U e = s∈S e U s . We get The first term in the sum C 2 8 = 28 is the number of sets I 1 ∈ Ω(9, 3) which contain i and the second term C 1 8 = 8 is the number of sets I 2 ∈ Ω(9, 2) which contain i ( i = 1, . . . , 9 ). Thus, U i = 36 for all i . For the ϕcomponent we get (see (2.18)) (4.17) Then we get the same value as in the SM2 -case. Since N = 130 ( m = 120 ) and (U s , U s ) = 2 , s ∈ S e , we obtain from (3.65) the same value for the A -number as in the SM2 -case A = A e (D2, F S1) = − 1340 43 (4.19) and the asymptotical vanishing of the wave function Ψ f → 0 as y 0 → −∞ . Now we consider the electromagnetic case, which mimics solutions with SD2 -, SF S1 -, SD4 -and SN S5 -branes in D = 10 IIA supergravity.
The number of "magnetic" 4-forms is C 5 9 = 126 , while the number of "magnetic" 3-forms is C 6 9 = 84 . We extend the cosmological electric ansatz by adding the following relations: J 1 ∈ Ω(9, 5) , a 1 = (4, m, J 1 ) and J 2 ∈ Ω(9, 6) , a 2 = (3, m, J 2 ) , where F (a,m,J) are defined in (2.9). For the charge densities we put Q s = 0 , s ∈ S . In the electromagnetic case the 9 -dimensional billiard is the same as in the pure electric case, i.e. B = B e ∈ H 9 , since magnetic walls are hidden by electric ones. (This may be readily proved along a similar line to that was followed for M -branes in the previous subsection.) The calculation of (U, U ) in the electromagnetic case U = U em = s∈S U s gives Here C 4 8 = 70 is the number of sets I 1 ∈ Ω(9, 5) which contain i and C 5 8 = 56 is the number of sets I 2 ∈ Ω(9, 6) which contain i ( i = 1, . . . , 9 ). Thus, U i = 162 for all i . For the ϕ -component we get (see (2.18)) (4.23) We obtain Thus, we are led to the same values of the scalar product (U, U ) and the A -number as for the model which mimics SM2 -and SM5 -branes. For the wave function we obtain the asymptotic vanishing Ψ f → 0 as y 0 → −∞ . 7 Remark 5. The coincidence of the A -numbers is not surprising since there is a one-to-one correspondence between the sets of space-like branes: (SM2, SM5) and (SD2, SF S1, SD4, SN S5) , which preserves the scalar products (U s , U s ′ ) . The number N is the same in both cases.

Conclusions
Here we have continued our approach from [34] by considering the quantum billiard for the cosmological-type model with n 1-dimensional factor spaces in the theory with several forms and l scalar fields. After adopting the electromagnetic composite brane ansatz with certain restrictions on brane intersections and parameters of the model we have deduced the Wheeler-DeWitt (WDW) equation for the model, written in the conformally covariant form. It should be noted that in our previous paper [34] we were dealing with a gravitational model which contains fields of forms without scalar fields. In [34] only electric non-composite configurations of branes were considered. Thus, the generalization of the model from [34] is rather evident. By imposing certain restrictions on the parameters of the model we have obtained the asymptotic solutions to the WDW equation, which are of the quantum billiard form since they are governed by the spectrum of the Lapalace-Beltrami operator on the billiard with the zero boundary condition imposed. The billiard is a part of the (N 0 − 1) -dimensional Lobachevsky space H N 0 −1 , where N 0 = n + l .
In both cases we have shown the asymptotic vanishing of the basis wave functions Ψ f → 0 , as y 0 → −∞ , for any choice of the Bessel function B = K, I . For D = 11 model this result may be interpreted as a quantum resolution of the singularity. It should be noted that in the approach of [25,26] asymptotic (basis) solutions to WDW equation in the harmonic gauge are vanishing as ρ = e −y 0 → +∞ .
In the examples presented above the magnetic walls change the asymptotical behaviour of the wave function Ψ f . Thus, hidden magnetic walls which do not contribute to the asymptotical behaviour of the classical solutions for y 0 → −∞ should be taken into account in the quantum case. This is the first lesson from this paper. The second one is related to the use of the conformally covariant version of the WDW equation. Here we were able to develop the quantum billiard approach for the model with branes only for a special conformal choice of the parameter a = (N − 2)/(8(N − 1)) in the WDW equation, where N = n + l + m and m is the number of branes. The study of the asymptotical behaviour of the wave function (as y 0 → −∞ ) for the non-conformal choice of the parameter a = (N − 2)/(8(N − 1)) should be a subject of a separate publication.
It should be noted that in the two examples presented here we have considered non-composite branes while initially we had formulated the quantum billiard approach for the composite branes with rather severe restrictions on brane intersections. Unfortunately, these restrictions exclude the possibility of efficiently applying the formalism to cosmological models with diagonal metrics in 11D and 10D IIA supergravities (the relaxing of these restrictions will lead to quadratic constraints on the brane charge densities Q s [63]). In the classical case this obstacle was avoided in [16] by considering the ADM type approach for non-diagonal cosmological metrics and using the Iwasawa decomposition. In this case the Chern-Simons terms were irrelevant for the classical formation of the billiard walls [16]. But in the quantum case the consideration of the Chern-Simons contributions needs a separate investigation. This (and some other topics) may be a subject of future publications.