The D-brane in n dimensions fixes the endpoints of strings in Dirichlet boundary conditions. If we let σ be the parameterization of an open string, the Dirichlet boundar condition is for the string X(σ = 0) = X(σ = 2π) = 0. Here the string length is parameterized for σ \in [0, 2π]. This turns out to be dual to the Neumann boundary condition, where the momentum is held fixed. The Neumann boundary conditions are ∂X(0)/∂σ = ∂X(2π)/∂σ = 0. The brane may be wrapped around a compactified space, say for a D-8 brane 4 of these dimensions might be wrapped around a Calabi-Yau (CY) manifold. A string on the brane will then respect this and effectively the string is also wrapped around the CY. If we let the string have length X then n/r = dy/dτ
Suppose a closed string (loop) is then on the CY manifold, but there is no wrapping so the string may dynamically evolve much as a particle does. The CY manifold has periodic structure (similar to a cylinder or torus) and has zero net Ricci curvature. The reason for R_{ij} = 0 is somewhat technical, but strings on a surface with Ricci curvature expand themselves in a divergent manner. The string then orbits a region of the CY, say along the x_i coordinate direction with radius r with a momentum p = nħ/r. I will from now on make ħ = 1. If we let the string have length X then n/r = dX/dτ, and the momentum is then p = dX/dτ. Again we have the string length parameterized for σ \in [0, 2π]. This can be compared to a string that is wrapped around that coordinate direction of the CY manifold. This is like a rubber band slipped on a tube of radius r, and if I double the band winding the tension increases as does the string length 2πσ and so the length along the string is given by X = rσ. Hence the force (tension) increases as does the length; the energy spectrum also increases. The string has a tension T, which is a force which is the gradient of a potential with respect to the string parameter. The energy and force of the string obeys a Hooke's law and the force is E = Tr. The winding number is then a quantization of the energy spectrum of the string so E ~ wr = Tr. The winding number is then w = (1/r)dX/dσ. The one thing which is clear is that if I exchange R = 1/r the spectrum of the string with n quantum modes orbiting the string and the string with winding number w are simply exchanged. This is T-duality and it tells us
dX/dσ = dX/dτ --- under r --- > 1/r interchange
The string orbiting around the radius of compactification is equivalent to the string wound around that direction.
This then connects to open strings on the brane wrapped on a CY space. The closed string wrapped on a CY manifold may split into an open string with endpoints attached to the brane. The above condition for closed strings is for open strings a statement of duality between strings with Dirichlet boundary conditions and those with Neumann boundary conditions. This T-duality means that a charge associated with a string with Dirichlet BC is dual to the charge with Neumann BC. This is a way of getting a duality between say the electric charge and its magnetic monopole dual.
Let the coordinates y_i correspond to those on the wrapped brane. The motion of the string X with respect to these coordinates is governed by dX/dτ which is expanded as
dX/dτ = sum_i (∂X/∂y_i)(∂y_i/∂τ).
We may write this as
dX/dτ = ∇_yX•v,
where v, for v_i = ∂y_i/∂τ, is a tangent vector on the brane. We physically have the interchange r --- > 1/r for the T-duality. Appealing to the notion of reciprocal space in solid state physics and other areas, this is then expanded according to the momentum representation on the brane
dX/dσ = ∇_pX•u
where u, for u_i = ∂p_i/∂σ, is a tangent vector on the momentum representation of the brane. The D-brane is a large object with many modes, indicated below, so this representation involves for now the phase space representation (y_i, p_i) for i = 1, 2, ..., dim. Now appeal to quantum mechanics where we know that p_{op} = -i∇_y and the position operator x_{op) = -i∇_p, and the motion of the string is quantized.
A D-brane is built up from strings, and we may then consider the dynamics of the brane. Consider the unitary operators U and V corresponding to an N-dimensional Hilbert space so
UV = e^{2πi/N}VU,
with U^N = 1, V^N = 1. If we set U = e^{ip} and V = e^{ix} then
e^{ip + ix} = e^{ip)e^{ix}e^{-[p, x]/2}
and so
e^{ip + ix} e^{-[p, x]/2} = e^{2πi/N}e^{ip + ix} e^{-[x, p]/2}
or
e^{ip + ix} = e^{2πi/N}e^{ip + ix} e^{-[x, p]/2}
and so [x, p] = 2πi/N. For the x and p in a one dimensional subspace of the Hilbert space N = 1 and we clearly have quantum mechanics.
We now consider something additional. The brane or CY manifold is Ricci flat, but may have additional curvature for dim > 3. We then multiply
e^{i∇_i}e^{i∇_j} = e^{∇_i +∇_j + R_{ijkl}y^iy^k}.
The Riemann curvature is the Weyl curvature for Ricci flatness, R_{ijkl} = C_{ijkl}. In addition the curvature here is in O(ħ^2) and is then a quantized effect. We then make a similar argument again for the U and V operator with a commutation given by UV = e^{2πi/N}VU and find the Weyl curvature is then in units of ħ.
The braney dynamics in string theory then necessitates noncommutative coordinates. Here it was worked with momentum, where we have [p_i, p_j] =/= 0. This can just as easily be worked in position coordinates as well.
Cheers LC