On the Octonionic Self Duality equations of 3-brane Instantons
Emmanuel Floratos George K. Leontaris

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        		CERN-TH-2017-045
        
        
        
        
        On the Octonionic Self Duality equations  of 3-brane  Instantons 
        
        				Emmanuel Floratos ^a,b,c[E-mail: ] and 
        			George K. Leontaris^c,d[E-mail: ]
        
        
        ^a
        					 Institute of Nuclear Physics, NRCS Demokritos,
        							Athens, Greece
        ^b
        			 Department of Physics, University of Athens,
        					Athens, Greece
        ^c
        			Theory Department, CERN,
        			CH-1211, Geneva 23, Switzerland 
        ^d
        					Physics Department, Theory Division, Ioannina University,
        					GR-45110 Ioannina, Greece 
         We study the octonionic selfduality equations  for p=3-branes in the light cone gauge and we construct explicitly, instanton solutions for spherical and toroidal topologies in various flat spacetime dimensions (D=5+1,7+1,8+1,9+1), extending previous results for p=2 membranes. Assuming factorization of time  we reduce the self-duality equations to  integrable systems and we determine explicitly periodic, in Euclidean time, solutions  in terms of the elliptic functions.  These solutions describe 4d associative and non-associative calibrations in D=7,8 dimensions.  It turns out that for spherical topology the calibration is non compact while for the toroidal topology  is compact. We discuss possible applications of our results to the problem of  3-brane  topology change and its implications for a non-perturbative definition of the  3-brane interactions.
        	
        
        § INTRODUCTION
        
        
          The tremendous progress of  understanding of perturbative superstring theory as well as its duality symmetries in various spacetime backgrounds <cit.> has led to a well substantiated proposal of M theory, the unifying theory of all superstring
          theories <cit.>-<cit.>.
          The new  objects contained in M-theory which are solitonic gravitational back-reactions of various D-branes, are the M2 and M5-branes.
          These objects were expected naturally from the eleven dimensional (11d) supergravity and, in this framework, they are
           considered as fundamental as the strings are for the various 10d supergravities.  The basic obstacle in understanding these objects as fundamentals, lies in  the absence of a coupling constant and the consequent problem of the definition of their self-interactions.
        
         An interesting proposal to define the self-interactions of the branes is to use their Euclidean instantons to interpolate between vacua (asymptotic states) with different number of branes <cit.>. The simplest ones could be instantons interpolating between states of one and two branes. The study of the simplest such instantons is already a difficult problem but one hopes that finding explicit solutions and trying to understand their moduli space would be and interesting approach. The classification of all such instantons and the geometry of their moduli space is probably beyond the present day capabilities.
        
         Almost three decades ago the problem of determining the corresponding self-duality equations for membranes has been solved in the case of
         2+1 dimensional embedding spacetime in ref <cit.> and in the case of 4+1 in ref <cit.>. In the latter, using results from the study of Nahm equations <cit.> for Yang-Mills (YM) monopoles it was shown that the self-duality equations form an integrable model and its Lax  pair and conserved quantities were determined. An important work by Ward <cit.> clarified further the situation reducing the self-duality  equations to Laplace equation in 3d flat space, for the time function of the surface. This function is a level set function -as it is called  in Morse theory- and the problem of determining topology changing membrane instantons is reduced into the search for non-trivial (multi-valued) time functions.
        
        
        Recent progress in this direction has been made  in the works <cit.>. The classification of  self-duality (SD) equations  in various dimensions  has shown that the possible SD equations are determined by the existence of cross products
        of vectors which in turn is related to the four division algebras <cit.>. Apart from the trivial cases of  branes of dimension p embedded in p+1 space dimensions, the interesting cases concern the p=2 brane (membrane) and the p=3 brane.
        
         In references <cit.> and <cit.> the study of membrane instanton equations in higher
         dimensions has provided some insight in the  difficulty of the problem and certain explicit solutions have been obtained
         (see also references <cit.>-<cit.>).
         In the present work we extend the study to the case of 3-branes in d=8 spatial dimensions using the octonionic cross product for
         three 8-dimensional vectors.  We obtain a convenient form of a four complex set for the SD equations which enable us to reduce the
         SD equations in six dimensions.
        
         The SD equations imply the Euclidean second order equations for the 3-branes and satisfy automatically the Gauss Law of volume preserving diffeomorphisms symmetry of the theory. In the case of external fluxes in the theory such as the one coming from pp-wave  supersymmetric gravitational backgrounds, it is possible to redefine the time and to reduce the  SD equations to the case of flat background, and find explicit solutions for the 3-sphere and the 3-torus. In the case of the sphere the instantons are periodic in Euclidean time going from a finite radius to infinity and back in finite time, while in the case of  the three torus the periodic solution interpolates between finite radii.
        
          The layout of this paper is as follows. In section 2 we review the Hamiltonian formalism and the Equations of Motion (EoM) for 3-branes
          in the light cone gauge and in flat spacetime, as well as,  the corresponding Gauss Law. We discuss the symmetry in this gauge which is
          the volume preserving diffeomorphisms of the 3-brane and which gives the possibility to describe the 3-brane as an incompressible fluid.
          In sections 3 and 4, we derive the first order self-duality equations in eight dimensions and its various lower dimensional reductions and we present them in a very useful and suggestive complex form in four-dimensional complex space. We show that these equation imply  the second order equations and the Gauss Law in Euclidean signature spacetime. In section 5,   assuming factorization of time, we solve analytically the 3-brane self-duality equations for the case of spherical (S^3) and toroidal (T^3) topology of the brane.
          Finally, in section 6 we present our conclusions and discuss the application of our methods to the issue of topology change of three branes, a problem which is relevant to cosmological brane models.
        
        
        
        
        
        
        
        
        § S^3 AND T^3 BRANES IN THE LIGHT-CONE GAUGE
        
        
        In this section we briefly present the Hamiltonian system in terms of the Nambu 3-brackets. For the
        S^3 brane in 9+1 Minkowski spacetime the Hamiltonian in the light-cone gauge is
         H= T_3/2∫ dΩ_3 (Ẋ_i^2+1/3!{X_i, X_j, X_k}^2)
        
        where the indices run the 1,2,… ,8 transverse dimensions.
        The corresponding EoM are
        Ẍ_i = 1/2{{X_i, X_j, X_k}, X_j, X_k} ,
        
        and the Gauss Law takes the form
        
            {Ẋ_i, X_i}_ξ_α,ξ_β=0, α, β=θ_1,2,3, αβ
        
        The volume element is
        
        dΩ_3 =sinθ_2 sin^2θ_3 dθ_1dθ_2dθ_3 ,
        
        and the Nabu 3-bracket for S^3 is defined as follows
        {X_i, X_j, X_k} = 1/sinθ_2sin^2θ_3 ϵ^αβγ∂_αX_i
        ∂_βX_j
        ∂_γX_k ·
        For S^3 there are four functions, namely the polar coordinates of the unit four-vectors,
        
         e_1=cosθ_1sinθ_2sinθ_3 , 
         e_2=sinθ_1sinθ_2sinθ_3 , 
         e_3=cosθ_2sinθ_3 , 
         e_4=cosθ_3 ·
        They satisfy the SO(4) relations
        {e_a,e_b,e_c} = ϵ^abcde_d, 
        where the indices take the values 1,2,3,4.
        These equations are instrumental for the factorization of the time from the internal coordinates of
        the 3-brane.
        
        
        It is possible to write down explicitly the infinite dimensional algebra of the volume preserving diffeomorphisms S^3,
        {Y_α, Y_β, Y_γ}   =   f^δ_αβγY_δ ,
        
        using as a basis the hyper-spherical harmonics
        
        Y_α(Ω)   =   Y_nlm(θ_1,θ_2, θ_3),  α =(nlm),  m=-l,…, l,  l=0,1,2,…, n-1 .
        
        The structure constants f^δ_αβγ can be expressed in terms of the 6-j  symbols of SU(2)
        since SO(4) ≃ SU(2)× SU(2).
        
        
        
        When the brane has the toroidal topology T^3, the global symmetries are U(1)× U(1)× U(1) and three translations
        along the cycles of the torus. The basis of functions on T^3 is taken to be
        
        e_n⃗=e^in⃗·ξ⃗, ξ⃗∈ [0, 2π]^3,  n⃗ =(n_1,n_2,n_3)∈ℤ ·
        This basis   defines an infinite dimensional volume preserving diffeomorphism group of the torus
        through the Nabu-Lie 3-algebra
        { e_n⃗_1, e_n⃗_2, e_n⃗_3}= -i  det (n⃗_1,n⃗_2,n⃗_3 ) 
        e_n⃗_1+n⃗_2+n⃗_3 ·
        It is important to notice that ∀ A∈ SL_3(ℤ) the above algebra remains invariant under the
        transformations n⃗→ An⃗. Thus,
        for this 3-dimensional brane there is an important discrete symmetry, namely the 3-dimensional modular group SL_3(ℤ)
        which could have implications in the quantum mechanical spectrum of this object.
        The EoM for the 3-torus are formally the same as in S^3 but it is much more convenient to complexify the
        eight transverse coordinates to four complex ones as we shall see later in the next section.
        
        
        
        
        
        
        §  SELF-DUALITY AND OCTONIONS 
        
        
        
        
        It is known that  p-fold, n-dimensional vector cross products are defined only in the following cases
         any  p=1,2,3,…       n=p+1
        
        p=2      n=7
        
        p=3      n=7
        
        p=3      n=8 .
        In the above cases, for any set of p vectors, v∈ℝ^n, their p-fold cross product, which we denote by
        Π_p(v_1,v_2,… , v_p),  is a linear map which satisfies the following properties
        Π_p(v_1,v_2,… , v_p)· v_k    =   0,  k=1,2,3,… , p ,
        
        while the norm of the cross product satisfies the important property,
        |Π_p(v_1,v_2,… , v_p)|^2   =    det(v_i· v_j) ·
        In the present work we will focus on the last case p=3, n=8 of (<ref>) since the other cases can be obtained
        by appropriate reductions to lower dimensions. Now, we proceed to derive the light-cone self-duality equations
        for 3-branes living in eight transverse flat dimensions. We note in passing that it is possible after toroidal compactification
        to obtain interesting equations for charged 3-branes in lower dimensions.
        
        It is easy to observe that the potential energy term of  the Hamiltonian (<ref>)  can be rewritten in terms
        of the determinant of the induced metric
        
            det(∂_αX_i∂_βX_i) =1/3!∑_i,j,k=1^8 {X_i,X_j,X_k}^2 .
        
        The cross product of three 8-dimensional vectors of case (<ref>) is explicitly given as <cit.>Π_i(v_1,v_2, v_3)  := ϕ_ijklv_1^jv_2^kv_3^l,   i,j,k,l=1,2,⋯ 8 ·
        where the definition of ϕ_ijkl and conventions used in the paper are given in the appendix.
        Setting v_α^i=∂_αX_i where α=1,2,3 and i=1,2,…, 8, we find that
        the potential energy of the 3-brane is the norm squared of the above defined cross product of its
        tangent vectors (see eq (<ref>) ).
         Another way to see that is to use directly the identity (<ref>) given in the appendix  for the tensor ϕ_ijkl.
        It is obvious now that the Hamiltonian (<ref>) of the 3-brane, can be written as
         H= T_3/2∫ dΩ_3 (Ẋ_i^2+1/3!{X_i, X_j, X_k}^2)
                ≡T_3/2∫ dΩ_3 (Ẋ_k-i Π_k)(Ẋ_k+i Π_k) ,
        
        where i,j,k=1,2,… , 8 and
        
            Π_k ≡Π_k(∂_1X,∂_2 X, ∂_3X) .
        
        For vacuum configurations, H=0, and in Euclidean time we find the self-duality equations
        Ẋ_i =±1/3!ϕ_ijkl{X_j, X_k, X_l} ·
        
        § THE  3-BRANE INSTANTONS IN VARIOUS DIMENSIONS
        
        
        
        
        
        There is a natural  generalization of the self-duality equations in 8 dimensions in terms of the symbol
        ϕ_ijkl,
        Ẋ_i = 1/3!ϕ_ijkl{X_j, X_k, X_l} . 
        where now the indices i,j,k,l run for 1 to 8. For convenience, we introduce the shorthand notation
        
        X_ijk≡{X_i, X_j, X_k}
        where the indices run from 1 to 8. We know however that the same equation exists in seven dimensions that is,
        i,j,k,l=1,2,… 7, because of the existence of the cross product p=3, n=7 given  in section 3.
        Then, employing the definition of ϕ_ijkl given in the appendix, we find that
        eq (<ref>) implies  the following eight  non-linear (first order) differential equations
        Ẋ_1   =    X_823+X_865+X_847+X_735+X_762+X_524+X_346
        Ẋ_2   =    X_183+X_684+X_857+X_743+X_167+X_635+X_154
        Ẋ_3   =    X_812+X_485+X_867+X_472+X_751+X_265+X_416
        Ẋ_4   =    X_862+X_835+X_871+X_567+X_732+X_512+X_136
        Ẋ_5   =    X_816+X_845+X_872+X_647+X_371+X_623+X_146
        Ẋ_6   =    X_851+X_824+X_873+X_457+X_172+X_325+X_314
        Ẋ_7   =    X_814+X_836+X_825+X_546+X_531+X_432+X_612
        Ẋ_8   =    X_321+X_615+X_642+X_534+X_174+X_376+X_275 ·
        We notice that the self-duality equations of 3-branes in seven dimensions mentioned before, are simply
        obtained from the above system, by eliminating the last equation and the terms containing the index 8
        on the right-hand side of the remaining seven equations.
        
        Next, we proceed to the  complexification  of the general system  (<ref>) by choosing specific
        pairing of the 8 real coordinates, as follows
        
         Z_1= X_1+i X_4,   Z_2= X_2+i X_5,   Z_3= X_3+i X_6,   Z_4= X_8+i X_7 ·
        After some elaborate manipulation of the original system of  equations,
        we arrive at the following simple  form
        Ż_1   =   -1/2{Z_1,Z_k,Z̅_k}+ {Z̅_2,Z̅_3,Z̅_4}
        Ż_2   =   -1/2{Z_2,Z_k,Z̅_k}+ {Z̅_3,Z̅_4,Z̅_1}
        Ż_3   =   -1/2{Z_3,Z_k,Z̅_k}+ {Z̅_4,Z̅_1,Z̅_2}
        Ż_4   =   -1/2{Z_4,Z_k,Z̅_k}+ {Z̅_1,Z̅_2,Z̅_3}
        We observe that the  equations (<ref>-<ref>) have an SU(4) symmetry acting on the complex vector (Z_1,Z_2,Z_3,Z_4).
        It is possible from the four complex equations to obtain various interesting reductions to lower dimensions.
        For example, if we demand that Z_k, k=1,2,3,4 are real (i.e., X_4=X_5=X_6=X_7=0), or if Z_k are
        all pure imaginary, that X_1=X_2=X_3=X_8=0, then we obtain the self-duality equation of three branes
        in four real dimensions
        Ẏ_i=1/3!ϵ_ijkl{Y_j, Y_k, Y_l } ,
        
        where Y_k represent  the non-zero coordinates of Z_k  for each of the above  two cases.
        
        A more detailed study  of   toroidal compactifications and  double dimension reduction
         to lower dimensional branes, as well as their relation to extended continuous
         Toda systems, will be discussed in a forthcoming work <cit.>.
        
        
        
        
        
        
        § EXPLICIT SOLUTIONS FOR 3-BRANE INSTANTONS IN EIGHT  DIMENSIONS
        
        
        We now study the solutions of the four complex non-linear differential equations (<ref>-<ref>).
        This system can be considerably simplified if we seek solutions with time factorization
        
         Z_a= ζ_a(t) f_a(σ_1,σ_2,σ_3) .
        In the subsequent analysis, we work out the cases of spherical and toroidal topology.
        
        
        
        
         §.§ Spherical Topology
        
        For  spherical topology of the closed 3-brane, S^3, we  choose the functions f_a
        to be f(a)=e_a, a=1,2,3,4, where e_a are the polar coordinates of unit 4-vectors (<ref>)
        in R^4.
        
        In this factorization, the first term of the right-hand side of the complex equations (<ref>-<ref>)
        is proportional to {e_i,e_k,e_k}
        which is identically zero.  The simplified equations now read
        ζ̇_1   =   ζ̅_2ζ̅_3ζ̅_4
        ζ̇_2   =   ζ̅_3ζ̅_4ζ̅_1
        ζ̇_3   =   ζ̅_4ζ̅_1ζ̅_2
        ζ̇_4   =   ζ̅_1ζ̅_2ζ̅_3 ·
        We notice that the equations are invariant under  a global U(1) symmetry ζ_k→ e^iq_kζ_k
        with  ∑_k=1^4 q_k=0.
        
        In the following, we  consider the conjugate equations and form the four combinations
        
            Q_k= i/2(ζ̅_k ζ̇_k -ζ̇̅̇_k ζ_k) = i/2(
            ζ̅_1ζ̅_2ζ̅_3ζ̅_4-ζ_1ζ_2ζ_3ζ_4) ,
        
        for k=1,2,3,4. Assuming  polar form
        ζ_k = r_k(t) e^iϕ_k(t) ,
          we find
        
        r^2_kϕ̇_k   =   Q ,
        
        where Q is a conserved quantity
        
        Q=-r_1r_2r_3r_4sinϕ,  ϕ=∑_i=1^4ϕ_i ·
        Furthermore, subtraction for two different values of k gives three new constants of motion
        
        |ζ_k|^2-|ζ_l|^2 =c_kl ,
        
        which are the analogue of Euler's equations for the rigid body.  Substituting (<ref>)  in the sums,
        ζ̅_k ζ̇_k +ζ̇̅̇_k ζ_k  we observe
        dr_k^2/dt=2 r_1r_2r_3r_4 cosϕ ·
        For r_1^2=s, using  the conservation laws we write this equation  as follows
        ṡ=√(s(s-a)(s-b)(s-c)-Q^2) ·
        
        
        
        This equation can be solved using standard elliptic functions. Depending on the initial conditions,
        we divide the solutions into two classes: those with Q=0 where there is no rotation, and those with Q 0
         where we have rotating instantons. In figure <ref> we plot the `effective potential'
        (i.e. the square of the right-hand side of the differential equation) for Q=0 and Q 0.
        In the first case (left plot) we show two admissible cases. The upper curve stands for four distinct real roots (radii),
        and the second curve corresponds to a double root, i.e., when two radii are equal. Curves below this one
        do not cross the real axis and correspond to complex radii which is not admissible.
        In the second plot, we fix the value of Q=Q_0 and draw three characteristic cases.
        
        First, we consider the case Q=0 and the possibility of static solutions, ṡ=0. If one of the radii
        is zero (s=a, s=b, s=c or s=0), then we obtain static solutions  living in the other six dimensions.
        In general,  we obtain a solution in terms of the elliptic integral of the first kind F(ϕ|m).
        In particular, the positivity of all roots requires s≥ max(a,b,c).
        Assuming a>b>c>0 while taking  s=a at the initial time  t=t_0,
        integration of (<ref>) gives
        
        t-t_0 = 2/√(c (a-b))( i K(b (a-c)/c(a-b))
        - F(sin ^-1(√((a-b) (s-c)/(b-c)
           (a-s)))|a (c-b)/c (a-b) )) ·
        In order to find s(t)  we  need the inverse of  the elliptic integral of the first kind
          F(ϕ|k)=∫_0^sinϕds/√((1-ks^2)(1-s^2))  which is given by
            the Jacobi Amplitude  t= F(ϕ|k) →ϕ= F^-1(t, k) = am(t,k).
             Furthermore,  in order to absorb unimportant factors, we redefine time
            t =2/√(c (a-b)) t'  and make the definitions
                 
                     k   =   a (c-b)/c(a-b)
        σ   =   √((a-b) (c-s)/(c-b) (a-s))=sin(  am(t,k)) ,
                
            (also written in the literature as sn(t,k))  and finally solve (<ref>) for s(t).
         In figure <ref> (left side) we plot time as a function of s=r^2 for a=1 and three sets of
         b,c values. In all cases, the radius goes to infinity at finite time.
        
         For the sake  of simplicity, we exemplify the above for the case of  a double root  b=c<a, i.e,
         when two of the radii are equal.
         Then
           ∫_a^sdu/√(u (u-a)) (u-b)= 2 /√(b (a-b))( π/2- tan^-1(√(s (a-b)/b (s-a))) ) ·
          If we define x=b/a <1, a new time a t/π→ t, and
          
            q(x,t)= x tan ^2(1/2π(1-t √((1-x) x))) ,
        
        we obtain
          ŝ(t) ≡s(t)/a=q(x,t)/x-1+q(x,t) ·
          This takes its  minimum value for ŝ(0)=1 and grows rapidly to infinity in finite time,
          at the zeros of the denominator. A few representative cases are plotted in the right side
          of figure <ref>.
          
          
          
          
        The case Q 0 can be separated into two classes:  first, pure uniform  rotational motion with
         time independent radii (Euler tops), and second, when we have both rotation and bounce  motion.
         In the second class we consider three different cases according to the initial conditions
         for the radii and the value of Q. Without loss of generality we can order the initial values of the
         radii in degreasing magnitude, that is r_1>r_2>r_3>r_4, which implies  a>b>c>0.
         The solution can be written in terms of the elliptic integral of the first kind, once we use the
         four roots e_1, e_2, e_3, e_4 which are  algebraic functions of a,b,c and Q. From the geometry of the
         potential plot, we see that when Q=0, e_1=0, e_2=c, e_3=b, e_4=a. When Q 0 but smaller than
         a critical value, Q_c, the root e_1 is negative, e_2 and e_4 increase while e_3 decreases.
        When Q=Q_c, two roots become equal,  e_2=e_3, and the potential has a local maximum V(Q_c)=0.
        
        
        
        
        
        
         §.§ Toroidal Topology
        
        For the case of toroidal topology, we define
        
        Z_a = ζ_a(t) e ^n⃗_a·ξ⃗ ·
        The four  vectors n⃗_a∈ℤ^3 are not completely arbitrary.
        When we
        substitute (<ref>) into the system of the four complex equations (<ref>-<ref>) we find that
        the above Ansatz provides a solution only if
        n⃗_1+ n⃗_2+ n⃗_3+ n⃗_4 = 0⃗ ·
        Then we get
        ζ̇_1   =   +i n
        ζ̅_2ζ̅_3ζ̅_4
        ζ̇_2   =   -i  n
        ζ̅_3ζ̅_4ζ̅_1
        ζ̇_3   =   +i ζ̅_4ζ̅_1ζ̅_2
        ζ̇_4   =   -i  ζ̅_1ζ̅_2ζ̅_3 ,
        where n stands for the determinant
        
        n= det (n⃗_j, n⃗_k, n⃗_l) .
        
        From these and their complex conjugates we readily get,
        d/dt |ζ_1|^2= d/dt |ζ_3|^2   =   +i n (ζ̅_1ζ̅_2ζ̅_3 ζ̅_4-ζ_1ζ_2 ζ_3 ζ_4)
        
        d/dt |ζ_2|^2= d/dt |ζ_4|^2   =   -i n (ζ̅_1ζ̅_2ζ̅_3 ζ̅_4-ζ_1ζ_2 ζ_3 ζ_4)
        ·
        Subtraction and addition of the appropriate equations gives
        
         |ζ_1|^2-|ζ_3|^2   =    c_13
        
          |ζ_2|^2-|ζ_4|^2   =    c_24
        
          |ζ_1|^2+|ζ_2|^2   =    c_12
        
           |ζ_3|^2+|ζ_4|^2   =    c_24 ·
        We define the currents
        
        Q_a=-1/2( ζ̇_aζ̅_a-ζ̇̅̇_a ζ_a) ,
        
        and find that
        
         Q_1=Q_3 =Q≡n/2(J̅+J),   Q_2=Q_4=-Q=-n/2(J̅+J)
        where  J= ζ_1ζ_2 ζ_3 ζ_4 and J̅ its complex conjugate.
        We observe now that J̇̅̇ = -J̇,  which implies
        
            d Q/dt=n/2d /dt(J̅+J)=0 ,
        
        so that the charge Q is conserved.  Writing ζ_k = r_k e^iϕ_k,
        we find that Q= r_1r_2r_3r_4cosϕ. Furthermore, using the redefinitions Q'=Q/n and t'= 2 n t, 
        while taking r_1^2=s, we obtain the differential equation
        ds/dt   =   √(s(a-s)(s-b) (c-s)-Q^2) ,
        
        with a=c_12, b=c_13, c=c_13+c_34.  This is  exactly the same equation as in the case of spherical topology, but
        notice that r_k  are bounded form equations (<ref>-<ref>).
        
        
        We note that with the same method it is also possible to solve the four-dimensional 3-brane self-duality equations presented before,
        where we shall end up with the real system in the place of equations (<ref>-<ref>).
        In this case, the system of equations is the elegant integral system of refs <cit.>.
        
        
          The solution for the case of toroidal topology is expressed for both cases Q=0 and Q 0 by the same
        elliptic integral as in the spherical topology, but from the conservation laws we find the interesting result
        that the motion is oscillatory and bounded and so, the 4-dimensional manifold is compact. This is
         contrary to the case of spherical topology where we have always   non-compact world-volumes.
        Since this Euclidean world volumes are calibrations of the (8+1)-dimensional embedding spacetime,
        our solution is among the first examples of compact non-associative (octonionic) calibrations.
        
        Indeed, ordering the four radii in decreasing order r_k+1<r_k,  inspection of the equations (<ref>-<ref>)
        shows that
        the constants a,b,c,d must satisfy b<a<c with a,c positive while the variable s is bounded
        between a and b,  where now b<s<a<c.
        In figure <ref>, the potential is plotted for three characteristic values of Q.
        The time evolution of the four radii, is also shown in figure <ref>. As already emphasized,
        all four radii are bounded and interpolate between a minimum and a maximum value.
        
        
        
        
        
        
        
        
        § CONCLUSIONS AND OPEN QUESTIONS
        
        
        The present work provides a method for solutions of the self-duality equations for 3-branes in higher dimensions.
        The factorization of time exploits the finite sub-algebras of the volume preserving diffeomorphisms  and reduces the SD equations
         to well known integrable systems with explicit solutions in terms of the standard elliptic functions. The new result is that
         the 3-sphere instanton interpolates between flat space-time at infinity and a finite radii 3-sphere. This is similar to
          the Einstein-Rosen wormhole solutions  of General Relativity. The minimization of the Euclidean  4-volume in 8-dimensions,
           which is the origin of the SD equations, classifies their solutions as calibrations of the background geometry.
           For the sphere this calibration is non-compact while for the 3-torus the calibration is compact <cit.>.
           It is possible to solve with the same method SD equations in maximally supersymmetric geometry backgrounds such
           as pp-waves with fluxes in 8+1 spacetime dimensions  by redefining the time variable <cit.>,
           a trick which has also a physical implication in the interpretation of the  time as the group renormalization
           scale and the SD equations as flow equations between various geometries [We thank Costas Bachas for this observation.].
             It would be interesting to apply similar ideas with those of ref <cit.> to transform the nonlinear
             three-brane SD equations into a linear Laplace equation problem for 8+1 space time dimensions to
         study possible topology changes <cit.>. Another interesting
          direction is to abandon the factorization of time and try to solve the three-brane SD equations in 8+1 dimensions requiring
          axial symmetry and proceed in analogy with the study of real time BPS states of 3-branes.
        
        
        
         Acknowledgments. 
        The authors   would like to thank CERN theory division for their
        kind hospitality and the stimulating atmosphere during which the main part
        of this work was realized. EGF would like to thank also the theory
        department of ENS in Paris for their kind hospitality and stimulating
        atmosphere during which the last parts of this paper were finished.
        
        
        
        
        
        
        
        
        § THE OCTONIONIC STRUCTURE CONSTANTS
        
        The structure constants of the octonionic multiplications  Ψ_ijk and its dual ϕ_ijkl,
        which measures the non-associativity of octonions, are given by (for a textbook, see for example <cit.>)
        Ψ_ijk   =   {[ 1 2 3 4 5 6 7; 2 4 6 3 7 5 1; 3 6 7 5 2 1 4 ].
        ϕ_ijkl   =   {[ 4 3 5 6 1 7 2; 5 7 2 1 3 4 6; 6 5 1 7 4 2 3; 7 1 4 2 6 3 5;   ].
        
        Below, we provide the simplest identities between these tensors used in our computations
        and by summing more pairs of indices we may obtain additional ones.
        
        
        When two indices are summed, a useful  multiplication rule connecting the two symbols is
        Ψ_ijkΨ_lmk=δ_ilδ_jm-δ_imδ_jl+ϕ_ijlm
        The corresponding identity for the ϕ_ijkl symbols is written as follows <cit.>ϕ_abclϕ_ijkl   =   (δ_aiδ_bj-δ_ajδ_bi)δ_ck
           +   (δ_biδ_cj-δ_bjδ_ci)δ_ak
           +   (δ_ciδ_aj-δ_cjδ_ai)δ_bk
           +   ϕ_abij δ_ck+ ϕ_bcij δ_ak+ ϕ_caij δ_bk
           +   ϕ_abjk δ_ci+ ϕ_bcjkδ_ai+ ϕ_cajk δ_bi
           +   ϕ_abki δ_cj+ ϕ_bcki δ_aj+ ϕ_caki δ_bj ·99
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