Are D-branes or M-branes described by wave functions? I know that point particles and strings are, but what about branes?
 A: $D$-branes can be thought of as similar to a Fermi surface. The Fermi surface for electrons is defined by the occupation number 
$$
\langle n_i\rangle~=~\frac{1}{e^{-(E_i-\mu)\beta}~+~1}
$$
for $\mu$ a chemical potential that is the maximum energy a particle can have that defined the Fermi surface. As the temperature $t~\rightarrow~0$ or $\beta~\rightarrow~\infty$ the energy of an electron defines the surface as $\langle n_i\rangle~=~1$ for $E_i~\le~\mu$ and zero for $E_i~>~0$. The chemical potential defines the Fermi energy surface and for elementary case in momentum space occurs on a sphere in momentum space of radius $k_f~=~\frac{\sqrt{2mE_f}}{\hbar}$. For a D-brane we have instead of electrons we have strings or $D0$-branes. In addition these branes occur in the bulk space of $10$ or $11$ dimensions.
This the averaged value of an operator on the vacuum state, which in general is a condensate. For a Yang-Mills gauge field the field tensor $F_{\mu\nu}$ may define $\langle F_{\mu\nu}F^{\mu\nu}\rangle~\ge~0$ on the vacuum or as the temperature in a Euclideanized sense $E/kT~=~E\tau/\hbar$ so $T~=~(\hbar/k)\tau^{-1}$ and $\tau~=~it$ the Euclidean time. This is a parameter for a type of phase change, which is similar to the idea of the occurrence of electrons on the Fermi surface at the transition temperature of superconductivity. 
This condensate may be a type of symmetry breaking, just as superconductivity breaks $U(1)$ symmetry, but in general it is simply a reduction on the gauge group at low energy to some space or surface. The occurrence of the condensate occurs on the quotient of the gauge group $G$ with a subalgebran $K$ sush that quotient are transformations of a projective coordinates of goldstone(like) particles. This may define a type of space or surface, such as the case this is a Hermitean symmetric space $SO(n)/SO(n-2)\times SO(2)$ or $SO(n)/SO(n-1)~=~S^{n-2}$. 
We generally think of this occurring for a large number of particles. For a $D$ brane we may think of there being a large number of $D0$-branes, which are point-like particles $S$-dual to string endpoint (Chan-Paton factors) or what we call particles. In a large $N$ limit, say a condensate of goldeston(like) particles or pions, similar to a Higgs condensate, this is then really a classical-like object. 
In the early days it was thought that membranes would be quantized objects for waves on a distributed region. These attempts did not work well. Witten in his 1995 M-theory showed that membranes as large classical-like objects could serve as ways to transform different string types between each other. They are often called $D$-branes where the endpoints of an open string satisfy Dirichlet boundary conditions. One can also work with $N$-branes that have Neumann boundary conditions at string endpoints. The $M$-theory is then a low energy effective theory of sorts, where low means relative to Planck energy or the Hagedorn temperature of strings. That there is this classical-like structure is a clue that in some ways this is not the final theory. There are likely wave functions or functionals underlying $D$-branes, but currently they are not known. There is some as yet not well understood physics underlying the structure of these mysterious $D$-branes.
A: This definition is clear enough:

In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For example, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. The word "brane" comes from the word "membrane" which refers to a two-dimensional brane

The underlying framework of nature  is quantum mechanics. Quantum mechanics is a probabilistic theory, the probability resulting from the complex conjugate square of a wavefunction, which is built up for the problem at hand ( scattering or decay) with various mathematical tools. So of course any extension  of the elementatry particle problem has to contain the basic postulates of quantum mechanics, to start with, in addition of having to embed all the paraphernalia of the standard model that encapsulates decades of data in particle physics.
