Supermembranes are hypothesized objects that live in the 11-dimensional theory called M-Theory and should also exist in eleven-dimensional supergravity. Supermembranes are a generalisation of superstrings to another dimension. Supermembranes are 2-dimensional surfaces. For example, they can be spherical or shaped like a torus. As in superstring theory the vibrations of the supermembranes correspond to different particles. Supermembranes also exhibit a symmetry called supersymmetry without which the vibrations would only correspond to bosons and not fermions.
Energy The energy of a classical supermembrane is given by its surface area. One consequence of this is that there is no difference between one or two membranes since two membranes can be connected by a long 1 dimensional string of zero area. Hence, the idea of 'membrane-number' has no meaning. A second consequence is that unlike strings a supermembrane's vibrations can represent several particles at once. In technical terms this means it is already 'second-quantized'. All the particles in the Universe can be thought to arise as vibrations of a single membrane.
Spectrum When going from the classical theory to the quantum theory of supermembranes it is found that they can only exist in 11 dimensions, just as superstrings can only exist in 10 dimensions. When examining the energy spectrum (the allowed frequencies that a string can vibrate in) it was found that they can only be in discrete values corresponding to the masses of different particles. It has been shown:
The energy spectrum for the classical bosonic membrane is continuous. The energy spectrum for the quantum bosonic membrane is discrete. The energy spectrum for the quantum supermembrane is continuous. At first the discovery that the spectrum was continuous was thought to mean the theory didn't make sense. But it was realised that it meant that supermembranes actually correspond to multiple particles. (The continuous degrees of freedom corresponding to the coordinates/momenta of the additional particles).
Action The action for a classical membrane is simply the surface area of the world sheet. The quantum version is harder to write down, is non-linear and very difficult to solve. Unlike the superstring action which is quadratic, the supermembrane action is quartic which makes it exponentially harder. Adding to this the fact that a membrane can represent many particles at once not much progress has been made on supermembranes.
Low energy sector It has been proven that the low energy vibrations of the supermembrane correspond to the particles in 11 dimensional supergravity.
Topology A supermembrane can have multiple thing tubes or strings coming out of it with little or no extra energy cost since strings, for example, have no area. This means that all orientable topologies of membranes are physically the same. Also, joined and disjointed supermembranes are physically the same. Thus the topology of a supermembrane has no physical meaning.
Mathematics The infinite supermembrane can be described in terms of an infinite number of patches. The coordinates of (each patch of) a supermembrane at any casual slice of time are 11 dimensional and depend on two continuous parameters ( σ , θ ) {\displaystyle (\sigma ,\theta )} and a third integer parameter (k) denoting the patch number:
X μ k ( σ , θ ) = x μ k + O ( σ , θ ) {\displaystyle X_{\mu }^{k}(\sigma ,\theta )=x_{\mu }^{k}+O(\sigma ,\theta )}
Therefore, the super membrane can describe an infinite number of particles if we associate somehow the coordinate of each particle with some topological property of the patches - perhaps holes in the membrane or closed loops.
Supermembrane Field Theory Since supermembranes correspond to multiple particles the field theory of membranes correspond to a Fock space. Informally, let a(x) denote the continuous degrees of freedom in the energy spectrum:
Φ [ X ] = Φ [ x , a , . . ] = ϕ ( x ) + ∫ a ( y ) ϕ ( x , y ) d y + ∫ a ( y ) a ( z ) ϕ ( x , y , z ) d y d z + . . . {\displaystyle \Phi [X]=\Phi [x,\mathbf {a} ,..]=\phi (x)+\int \mathbf {a} (y)\phi (x,y)dy+\int \mathbf {a} (y)\mathbf {a} (z)\phi (x,y,z)dydz+...}
The action can be written as
S = ∫ Φ [ X ] Q Φ [ X ] D [ X ] {\displaystyle S=\int \Phi [X]\mathbf {Q} \Phi [X]D[X]}
where Q is the kinetic operator. No interaction terms are needed since there is no concept of membrane number. Everything is the same membrane. The action is not quite the same type as the one for superstrings or particles since it involves terms with multiple particles. The terms relating to single fields must recover the classical field equations of Dirac, Maxwell and Einstein. The propagator to get from a state with membrane X to one at another conformal slice with membrane Y is:
G ( X , Y ) = Q − 1 ( X , Y ) {\displaystyle G(X,Y)=\mathbf {Q^{-1}} (X,Y)}
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