Spinning membranes

We present new solutions of the classical equations of motion of bosonic (matrix-)membranes. Those relating to surfaces in spheres provide spinning membrane solutions in AdSp × Sq , as well as in flat space–time. Nontrivial reductions the BMN matrix model equations are also given.  2004 Published by Elsevier B.V.


Introduction
Starting from the premise that 'membranes are to M-theory what strings are to string theory' the search for classical solutions of membrane dynamics needs almost no justification. Given the additional fact that promising approaches to M-theory are within the context of matrix mechanics, solutions to its equations of motion are equally relevant. The observation that a discretized formulation of membrane dynamics is matrix mechanics [1] links the two.
In the context of string theory, the study of classical solutions was recently revived in [2] (see [3] for a review of further interesting subsequent developments). Relating time-dependent classical solutions of the string sigmamodel in an AdS 5 × S 5 target space-time to the dual conformal field theory, extends the testable features of the duality between string theory and N = 4 SYM, i.e., of the AdS/CFT correspondence.
A likely extension of these ideas to M-theory is to consider their motion on maximally supersymmetric backgrounds which, aside from eleven-dimensional Minkowski space, are AdS 7 × S 4 and AdS 4 × S 7 . The former is the near-horizon limit of a stack of N coincident M5 branes with 1 2 R AdS = R S = l p (πN) 1/3 and the latter is the near-horizon limit of a stack of N M2 branes with 2R AdS = R S = l P (32π 2 N) 1/6 . The dualities between classical supergravity on these background and the conformal field theories on the world-volume of the branes which create them has been studied. In particular for the AdS 7 × S 4 case, if the duality holds, nontrivial information about the (0, 2) conformal field theory of N interacting tensor multiplets in six dimensions has been obtained, e.g., its conformal anomaly has been computed [4,5]. Direct verifications have, however, so far been impossible, mainly due to the lack of knowledge of the interacting (0, 2) theory.
One of the open problems in string theory is its quantization in nontrivial backgrounds, such as AdS 5 × S 5 . An exception is the gravitational plane wave background which is obtained as the Penrose limit of the AdS 5 × S 5 vacuum of type IIB string theory. In this background light-cone quantization leads to a free theory on the worldsheet whose spectrum is easily computed [6]. This opens the way to the duality between string theory and another sector of large-N SYM, which is characterized by large R-charge (∼ √ N ) and conformal weight (∼ √ N ). The extensive activity to which this has led was initiated in [7].
The difficulties related to quantization are much more severe in M-theory where quantization on any background is still elusive. The semiclassical analysis, which in the case of string theory provides valuable nontrivial information about the dual conformal field theory, can, however, be extended to M-theory. While the equations of motion of strings on AdS 5 × S 5 reduce, for special symmetric configurations, to classical integrable systems [8,9], this is not as simple for membranes. Also, the integrable spin-chains which appear in the discussion of the dual gauge theory [10,11], have so far no known analogue in the (0, 2) tensor theory. However, the matrix model of the discrete light cone description of M-theory on plane waves obtained as Penrose limits of AdS 4 × S 7 and AdS 7 × S 4 is known [7] and has been studied (see, e.g., [12]).
In this Letter we present new solutions to bosonic matrix model equations (in Minkowski space, and of the BMN matrix model), as well as make a first step towards the semi-classical analysis of M-theory in AdS p × S q backgrounds, where we will find that the equations of motion, upon imposing a suitable ansatz, may be reduced to the equations describing minimal embeddings of 2-surfaces into higher spheres (as well as generalizations thereof).

The bosonic matrix model equations
The time evolution of spatially constant SU(N) gauge fields in R 1,d as well as of regularized membranes in R 1,d+1 [1] is governed by equations of motion [X i , X j ], X j involving d Hermitean traceless N × N time-dependent matrices, with the constraint ('Gauss law', respectively, reflecting a residual diffeomorphism invariance in a light cone orthonormal gauge description of relativistic membranes) As shown in [13], solutions of these equations may be found by making the ansatz Inserting the ansatz (3) into (1) yields, under the assumption that ϕ and x are related throughφx 2 = L (= const), Before we turn to the construction of solutions of the matrix equations, let us note that given any solution of (7) there are always trivial ways to solve the contraint (8). Given a solution M of (7) one can define M := ( M , 0) (by adding d zeroes) and choose A such that A M = ( 0, − M ). In this way each term in the sum (8) will be identically zero. Clearly, M is a solution of (7) with d = 2d. Another way to satisfy (8) is by letting M = ( M , M ). Below we will find solutions which do not rely on this "doubling mechanism".

Solutions of the matrix equation for d = 8
A very simple way to solve (7) is in terms of the Hermitian generators T a of any semi-simple Lie algebra If we choose the basis such that the Cartan-Killing metric is κ ab = c 2 δ ab , M = (T a ) solves (8) with λ = c 2 .
If we require d 9 and even, the only physically interesting case, apart from SU(2) (the 'fuzzy sphere') is SU(3) with d = 8. However, since the discussion can be easily generalized to any SU(N) with N odd, we will give the solution of (8) for the general case.
To solve (8) with A as given in (4), we choose a particular basis for the su(N) Lie algrebra. A standard basis is It is not difficult to verify that (7) and (8) and satisfies M 2 = N 2 −1 N . This being a consequence of the algebra, not its particular representation means, that higher-dimensional representations of SU(N) yield higher-dimensional solutions of (7) and (8). In particluar we obtain a solution for d = 8 for any representation of SU (3).
Another way to present solutions related to SU(N), which has the advantage of allowing to pass to a continuum limit, is as follows. For arbitrary odd N > 1, define N 2 independent N × N matrices providing a basis of the Lie algebra gl(N, C), with [14] (14) (for the moment, we will put M(N) = 1, as only when N → ∞, M(N) N → Λ ∈ R, this "degree of freedom" is relevant). Using (14), it is easy to see that (15) Let now N = 3 and The components of M form a basis of hermitian 3 × 3 matrices, and thus of the Lie algebra su (3). It is straightforward to relate this basis to the basis (10) but perhaps one should note that (10) is not invariant under general linear transformations. It is also easy to check that (16) satisfies (7) (4), (8) is also satisfied. We therefore obtain a solution of (1), satisfying the constraint (2) for N = 3 and d = 8, by letting with x(t) and ϕ(t) satisfying (6).
The above construction can be generalized to yield other solutions with d = 8. It is straightforward to verify that is a solution of (7) and (8) if m 2 = n 2 with N arbitrary. The reason is that, by using (15) the "discrete Laplace operator" when acting on any of the components of M, in each case yields the same scalar factor ("eigenvalue") In the general case (18) is a solution for fixed N = m 2 + n 2 , which we assume to be odd. Higher-dimensonal representations can be obtained if we expand the eight N × N matrices in terms of a basis of gl(N, C)  (22). We want to stress that these generalizations of (16) are not higher-dimensional representations of SU (3); the set of matrices M does not form a closed commutator algebra.

The continuum limit of matrix solutions as minimal surfaces in S 7
As mentioned in [13], (7) (with M 2 = ) is a discrete version of the equations for a minimal surface in a (higherdimensional) sphere. In [15], such surfaces in S 3 were proven to exist for arbitrary genus.
Equations for a minimal surface m(ϕ 1 , ϕ 2 ) in a sphere can be obtained by varying the integral √ g − µ m 2 − 1 dϕ 1 dϕ 2 with g = det(g rs ) and g rs = ∂ r m · ∂ s m. One obtains the equations where ∆ is the Laplace-Beltrami operator on scalar functions gg rs ∂ s .
While γ , in this form, becomes irrelevant (insofar each of the 4 arguments in x + := x [0] + , as well as those in − can have an arbitrary phase-constant), not only their sum, (28), but (due to the mutual orthogonality of gives a minimal torus in S 7 .
To see this, one could recall (41), which shows that (48), rewritten as is identical to the standard 'minimal surface' equation This, incidentally, justifies calling (20) 'discrete Laplace operator'. Eq. (50) is the Euler-Lagrange equation which one obtains if one varies w.r.t. the embedding coordinates r a (ϕ 1 , ϕ 2 ) and the local Lagrange multiplier µ(ϕ) (which guarantees r 2 = 1). Another way to show the equivalence of (49) (hence (48)) to (50) is as follows: the results of Ref. [18] allow one to choose the coordinates ϕ s in the diffeomorphism invariant equation (50) such that g/(ω 2 0 − ω 2 ) is equal to any given density with the same 'volume' (i.e., integral over d 2 ϕ). Choosing it to be ρ shows that solutions of (50) give solutions of (49). To show the converse, one notes that (49) automatically implies that g ρ 2 = ω 2 0 − ω 2 (multiply (49) by r, and use r 2 = 1 three times: once on the r.h.s., once for r · ∂ u r = 0 and, finally, to write r · ∂ s ∂ u r as −g su ).
One of the reasons for making the ansatz (3) was to find solutions that do not collapse to zero. In (54) we have mass-terms and hence, we are not forced to only consider "rotating" solutions, as we did for (1).
Apart from the trivial static (known) solutions, (L = 0, z = 0; x = 0, m or 2m), and genuinely timedependent solutions of (63), there are several "intermediate" solutions, for which z is constant, but nonzero (making ϕ(t) linear in t): 2 for which x = 0, z = ±z 0 , as well as those corresponding to the roots of the quintic equation obtained via z 2 = 3mx − x 2 − 2m 2 . Replacing M a by M a , respectively, M a , in the second part of (62) leads to yet other solutions. One can consider both the m → 0 (m → ∞) limit of these solutions as well as their N → ∞ continuum limit.
It is easy to show that all four reductions lead to systems of ordinary differential equations which are in a canonical way Hamiltonian, e.g., for (65) w.r.t.
Even though exact solutions of these systems of equations are as yet unknown and probably may not exist in terms of known functions, they can be easily solved numerically.

Note added
After this paper was submitted, we became aware of Refs. [19,20] where simple solutions to the membrane equations on AdS 7 × S 4 were found.