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Non-Collapsing Membrane Instantons in Higher Dimensions

We introduce a particular embedding of seven dimensional self-duality membrane equations in C^3\times R which breaks G_2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C^3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a finite one and then they bounce to infinity in finite time. Their motion is periodic. These generalized complex Nahm equations, in the axially symmetric case, lead to extensions of the continuous Toda equation to complex space.


Introduction
Some years ago, we introduced the notion of the self-duality for supermembrane in 4 + 1-dimensions and in the light-cone gauge. The corresponding self-duality (s-d) equations proved to be an integrable system with an infinite number of conservation laws and particular solutions were found [1,2] which were collapsing configurations of membrane instantons to point-like or string-like objects. Similar covariant self-duality equations have been introduced before for 2 + 1-dimensions [3] and later generalized to 6 + 1 dimensions in [4].
These objects, represent world-volume instantons of the supermembrane. In the light cone gauge, the world-volume time and the target time are identical, so these configurations are space-time membrane instantons and they provide quantum mechanical tunnelling through the membrane self-interaction potential moving with velocities bigger than light. Thus, they can travel infinite distances in finite time.
Their equations of motion, which are Nahm's type equations for the area-preserving diffeomorphism group of the membrane, lead for axially symmetric configurations to continuous Toda equations relating thus the membrane instantons with the self-dual Einstein metrics with isometries [5,6].
The basic ingredients for the study of covariant membrane instantons in higher than (4 + 1)-dimensions, were contained in the pioneering paper of ref [4], however this work was until recently overlooked. The authors in [7] introduced higher dimensional self-duality equations for the light-cone membranes as well as in the case of the quantized Poisson (i.e. Moyal) bracket. The detailed properties of the octonionic lightcone membrane instantons where studied in [8,9], where the invariance of the seven dimensional equations under the exceptional group G 2 was exploited. The invariance under this group has as a consequence one remaining supersymmetry consistent with the membrane background. In three dimensions there are eight remaining supersymmetries [10].
In a parallel development, octonionic self-duality for seven and eight dimensional gravity was proposed [11,12] and explicit seven and eight gravitational instantons which generalize four-dimensional ones (satisfying first order equations) were found [12,13]. These higher dimensional gravitational instantons where among the first few explicitly known self-dual metrics with exceptional holonomies G 2 and Spin (7) which were also lifted in 10 and 11-dimensional supergravity. Recently, exceptional holonomy higher dimensional instantons were studied for their rôle in string and M-theory and an important activity around this subject has been created [14].
In this letter, we introduce the complexified self-duality equations of the membrane in seven dimensions and represent them as generalized Nahm equations. We show that the extra dimensions manifest themselves in the solutions through the appearance of an non-zero conserved charge which prevents the collapse of the membrane. We integrate completely the three-dimensional complex Nahm's equations for S 2 and T 2 topologies, assuming factorization of time. We find periodic non-collapsing instantons. Starting from infinite size they contract, with increasing angular velocity, to a minimum size and then they bounce back to infinity in finite time.
2. The Self-Duality Membrane equations in seven Dimensions.
Choosing fixed values for the 8th and 9th membrane coordinates, the seven-dimensional self-duality equations [4,7,8] where Ψ ijk is the completely antisymmetric tensor that defines the multiplications of octonions [15]. The Gauss law results automatically by making use of the Ψ ijk cyclic symmetry The Euclidean equations of motion are obtained as follows where use has been made of the identity and of the cyclic property of the symbol φ ijlm [15].
At this point we would like to make a general remark on the nature of the motion described by (1). These In what follows, we proceed to the complexification of the self-duality equations. We embed the seven dimensional space R 7 into C 3 × R in a very specific way which depends on the particular definition of the octonionic structure constants used in ref [8] which assume the following multiplication table [15] Thus, if we define then the self-duality equations become where I, J, K take the values 1, 2, 3, whereas D t is the 'covariant' derivative These strikingly simple equations of self-duality, break the G 2 invariance down to SU (3). The SU (3) invariance comes from the unique cross product existing in C 3 which is a remnant of the octonionic cross product in seven dimensions. One consequence is that, the three dimensional world-volume manifolds described by (7) are SU (3) special lagrangian sub-manifolds of C 3 [16].
In the next section, we will consider the factorization of time and the restriction to three complex dimensions of the above first order equations. Before that, we would like to observe that it is possible to generalize the connection of the three dimensional self-duality equations with the continuous Toda equations [1,6,5]. This is possible if we consider axially symmetric solutions of the above system. Indeed, the axially symmetric Ansatz, where, R, z complex functions, impliesȧ 0 = 0 for all times and thus a 0 can be fixed to zero by an area preserving transformation. For the equations (7) we obtaiṅ and the index σ 2 refers to the derivative with respect to σ 2 . This system of equations has as integrability condition the following non-linear equation which extends the continuous Toda equation to three complex dimensions.
obtain the formΨ In the real three dimensional case [1], we have R * = R and z = z * , while the continuous Toda equation for Ψ = Ψ * reads [1,5,6], In the next section, among other things we find the complete solution of the above (10) system or of the generalized continuous Toda equation (11) restricting the functions R, z so that R(σ 2 , t) = sin σ 2 ζ(t),

Membrane instantons in three complex dimensions
As we show below, it is possible to extend the known three dimensional instanton solutions into six dimensions where, apart from the radial expansion of the instanton, we observe rotational motion in all of the three planes (X 1 , X 4 ), (X 2 , X 5 ), (X 3 , X 6 ) of the six-dimensional space (which we choose to call them I,II,III complex planes).
We assume factorization of time which will lead to a coherent motion of all the membrane points. These solutions are analogous (but for Euclidean time) to the real time solutions of second order equations of motion for toroidal and spherical membranes recently studied in [17] where f i are three complex functions on the surface. First we observe that the Poisson bracket i are functions of the same combination σ 1 , σ 2 . From the equation for a 0 we findȧ 0 = 0 and therefore by an appropriate area preserving transformation we may fix a 0 to be zero. So we are left with the three complex Nahm's equations for z i . We shall examine in detail two topologies, -spherical (S 2 ) and toroidal (T 2 ).
Up to now only three-dimensional solutions of the self duality equations are known [1]. In order to factorize the time dependence we choose for the case of S 2 the three functions f i to be The three functions for the algebra SU (2) under Poisson bracket satisfy For the three complex functions of time, we find the complex Euler equations 1 In the case of T 2 we choose the following three functions Now we observe that the factorization of time is implemented for any three n i 's such that In this case, we obtain for the corresponding ζ i (t) where n = n 11 n 22 − n 12 n 21 ∈ Z.
In both cases (S 2 and T 2 ) the equations for the time evolution are essentially the same. The T 2 case is obtained from the equations of S 2 if we make the replacement t → nt for n integer. Therefore we only need to investigate the equation (17) which in component form is writteṅ There is an obvious symmetry of the above system where q k , k = 1, 2, 3 are real and q 1 + q 2 + q 2 = 0. This invariance leads to the conservation of the three On the other hand, the equations of motion (22) imply that all three charges Q i are equal to There are two additional constants of motion in analogy with the Euler equations for the rigid body, where c ij are constants. In polar coordinates we obtain Theen, equations (22) reduce toṙ where φ = φ 1 + φ 2 + φ 3 . For simplicity we define s 1 = r 2 1 . We further combine (31) with (28) to obtain the following differential equationṡ After substitutions, the differential equation obtains a unique form in the right-hand side for all s i which where s 1 = s s 2 = s − a, s 3 = s − b, where a = c 12 , b = c 12 + c 23 .
If we define a new function of time U(t) = s(t) − a+b 3 , the differential equation becomeṡ which is recognized as the standard form of the Weierstrass equation with solution the (doubly periodic) Weierstrass function U(t) = P(t; g 2 , g 3 ), with Before we proceed to the analysis of the solution, we would like to point out that there is an isomorphism between the membrane and the matrix model solutions with factorization of time for the spherical and toroidal topologies. This implies that the above solutions have isomorphic matrix model instanton solutions similar to those examined recently in ref [18].

Analysis of the non-collapsing instanton solutions
The equation of motion (33) for the membrane radii (r 2 1 ≡ s, r 2 2 = s − a and r 2 3 = s − b), is analogous to the motion of a particle in a potential V . This is indeed the motion of any point of the Euclidean There is a critical value Q 2 = Q 2 c for which the V (s, Q) has a double root which is the s max . Q c is determined as follows In this case we calculate the maximum root to be The physical region is this case is beyond s c 1 .
When Q 2 > Q 2 c there are two complex conjugate roots (the maximum of V is below the real axis) the physical region is s > s 1 where s 1 is the real root. The three cases described above are presented in figure (1). In what follows, we proceed in the detailed description of the dynamics of the membrane instanton in the three cases discussed above (Q = 0, Q = 0, Q = Q c ).
• When Q = 0 it is possible to redefine the time-independent phases (29) to zero values and the selfduality equation reduces to the ones of the three dimensional case [1]. Because of the conservation laws we only need the equation for s = r 2 1 which readṡ We distinguish the following cases [1]: Figure 1: The 'Potential' for three characteristic Q-values: In all cases, the allowed region is beyond the highest root to the infinity.
• a = b = 0. The solution is a spherical membrane with the radius varied with time as There is a critical value t c = t 0 − 1/r 0 where r → ∞, whilst for t → ∞ the radius shrinks to zero.
• a = 0, b = 0. The equation becomesṡ = −2s √ s − b and the solution obtained is r0 the membrane has an infinite radius. On the contrary, when the configuration collapses to a string.
• a > b > 0. This is the most general four-dimensional case. The s-d equation readṡ In order to write it in a more familiar form, we make the transformations x = √ a r , k 2 = b a < 1, with r > a and therefore x < 1 √ a . Then, separating variables we have The right-hand side is the elliptic integral [20] F (φ, k), and the radius r is given Here, we assumed as initial condition t = 0 and r 0 = r(t = 0) = ∞. The positivity of r restricts the t-range in a half period of the elliptic sn. The real period is the complete elliptic integral, i.e., when the upper limit of (44) is equal to unity, sin φ = 1, With the above initial conditions, at t = 0 the volume of the ellipsoidal membrane is infinite, whereas at time t = T 4 reaches a minimum value with r min = √ a sn √ a T 4 and the membrane collapses to an elliptic disc.
It is worth mentioning that in three dimensions, assuming simple factorization of time, we do not find noncollapsing membranes. On the other hand we find all possible collapsed configurations for the membrane, that is, points, strings and discs.
• Now we consider the case Q = 0. This case differentiates from the Q = 0 case because it exists only in dimensions higher than three ( see eq.29).
As we shall see, a remarkable fact is that the dynamics of the membrane in higher dimensions is encoded in the higher dimensional angular momentum Q which, from the point of view of three dimensions, it behaves like a charge.
In the case of spherical topology, there are three different geometries, spherical, ellipsoidal with axial symmetry, and anisotropic ellipsoidal ones. These three cases correspond to the degeneracy of the roots of the polynomial in the right-hand side of equation (33).
If the degeneracy g is g = 3, we have the spherically symmetric membrane which from any initial condition it approaches the radius equal to the largest real root of the equation (33) in finite time and it goes back to infinity.
If g = 2, we have the axially symmetric ellipsoid which from an initial configuration it decreases its volume until a limiting one which is determined also by the largest real root. The same also happens to the anisotropic ellipsoidal membrane. The general solution in terms of elliptic Jacobi functions or Weierstrass function of the equation (34), can be found in a similar way with the case Q 2 = 0 in eq (41) where x = s − e 3 and e ij = e i − e j . If the topology is toroidal, the radii r 1,2,3 of the torus T 3 inside which the T 2 toroidal membrane is embedded, at any moment of time they are equal to √ s, √ s − b, √ s − a respectively. We note that in this case there are no non-degenerate solutions below four dimensions.
In the following, we discuss the Q = 0 spherical case (a = b = 0). Integrating the equation we find the solution in terms of the incomplete beta function [20] t = 1 6Q 1/3 Beta( Q 2 r 6 ; We assume here the following initial conditions: At t = 0 the spherical membrane has infinite volume and in finite time T = 1 6Q 1/3 Beta( 1 6 , 1 2 ) contracts at the minimum permitted radius r 0 = Q 1/3 and goes back to infinity. From the angular velocity equation (29), at t = 0 or infinite radius, the angular velocity is zero and contracting it develops at the limiting time T angular velocity ω T =φ = Q 1/3 . The ellipsoidal cases follow similar pattern and we parametrize the solution in terms of the Weierstrass function [20]: where g 2,3 are functions of a, b given by (35,36) above.
Due to the non-zero angular momentum the brane obtains a minimum size given by the radii squared, s 1 , s 1 − a, s 1 − b where as discussed in the beginning of this section, s 1 stands for the largest root of V (s, Q).
In the solution (34) we assume initial conditions s(0) = ∞ and s 1 is given by s 1 = P( T 2 ; g 2 , g 3 ) + a+b 3 , where T is the real period of the Weierstrass function. In the special case of Q = Q c , we have a simple algebraic solution (similar to the Q = 0 case), and s 1 = s c 1 with s c 1 given by (40).

Conclusions
Breaking the G 2 invariance of the octonionic self-duality equations for the membrane in seven dimensions down to SU (3), we found explicit solutions of non-collapsing rotating membrane instantons which they have periodic motion starting at some initial moment from infinite size, shrinking down to a finite one in a half period and then bouncing back to infinity. The rôle of these instantons for the quantum mechanical vacuum of the membrane depends on the period which is the inverse temperature in membrane plasma of finite temperature. In the case of infinite period (zero temperature) the membrane instantons collapse to point-, string-and disc-like objects which represent the vacua of the quantum mechanical membrane.
Since up to now it is not known how to quantize the supermembrane, we hope that the information we provided in this work is a step towards this direction.