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Finally, the equations for A1 and y i give, respectively, ∂2 B R R2 + Π2 B 2 + R2 ∂2 2y ′i = 0, R2 R4 − Π2 B 2 (R2 + Π2 )(R2 + B 2 ) = 0. 47) The Hamiltonian density is given by H = EΠ − L = 1 R (R2 + Π2 )(B 2 + R2 ) . 50) which can be saturated only if R2 = y ′i yi′ = ±ΠB ⇔ E2 = 1 . 47). Note that the Poynting vector generated by the electromagnetic field is always tangent to the curve C and its modulus is precisely |ΠB|. We can then use exactly the same arguments as in [47]. 51) tells us that, once we set E 2 = 1, and regardless of the value of B(σ 2 ), the Poynting vector is automatically adjusted to provide the required centripetal force that compensates both the tension and the gravitational effect due to the background curvature at every point of C.

74) again. 97). At this point we have to choose U, K and A1 so that they describe a D2-brane with worldvolume R1,1 × C, with C an arbitrary curve in M8 . 97) with such a source term in the right hand sides. If this has to correspond to the picture of D0/F1 bound states expanded into a D2 by rotation, the boundary conditions of the Laplace equations must be such that the solution carries the right conserved charges. In the appropriate units, q0 = ∂M8 ∗˜8 dC1 , qs = ∂M8 ∗˜8 dB2 , ∂M A1 −→8 Lij y j dy i .

G. to motivate the AdS/CFT correspondence. Many other purposes, however, require the consideration of more complicated configurations in less trivial backgrounds. The gauge/string correspondence and the appearance of NC gauge theories are examples of this. One can think of different ways of complicating the picture, like 1. considering non-flat D-branes, 2. putting them in non-flat backgrounds, 3. intersecting D-branes of (possibly) different dimensions. All three issues have been intensively studied in the literature and they have led to many interesting insights in different areas of physics.