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More than just strings

    To understand the presence of objects in string theory that are not strings, but higher dimensional objects, or even points, it helps to know the formulation of Maxwell's equations in the language of differential forms, because this is what tells us that the sources of charge in the Maxwell equations are zero-dimensional objects. Gauge field strengths that are p+2-forms turn out to have sources that are p-dimensional objects. We call these p-branes.
    In the regular maxwell equations in d=4 spacetime dimension, the electric and magentic fields are packed together into the field strength F, which satisfies the equation F=dA, d is the exterior derivative, and A is the vector potential, a one-form. The two-form *F is the dual of F relative to the spacetime volume four-form v. (The subscripts on F, etc, below are just to indicate the degree of the differential form.)

Maxwell equations, d=4

    The charge sources enter through the equation d*F=*J, where *J is the three-form dual to the current four-vector J=(r,j). In the rest frame of the charge density r, J=(r,0), so *J is r times the the volume element for three-dimensional space. In a three-dimensional space, a surface that can be localized in three dimensions (has codimension three) must be a zero-dimensional surface, also known as a point.
    This is the math that tells us that the Maxwell equations couple electrically to sources that are points, or zero-branes, as zero-dimensional objects are now called in string theory. (For magnetic couplings, the roles of F and *F are interchanged, but that won't be covered here.) This same math works for two-forms in any spacetime dimension, so we know that Maxwell's equations couple to point charges in any spacetime dimension.
    Superstring theories contain electromagnetism, but they also contain field strengths that are three-forms, four-forms and on up. These field strengths obey equations just like the Maxwell equations, and their sources can be analyzed in the same manner as above.
     Suppose we start in d spacetime dimensions with a vector potential A that is a p+1-form. Then F is a p+2-form, v is a d-form (because it's the volume element of d-dimensional spacetime), *F is a (d-p-2)-form, and d*F is a (d-p-1)-form. (Once again, the subscripts are just to indicate the degree of the differential form.)

Maxwell equation, d,p version

    The equations of motion tell us that the source term *J is also a (d-p-1)-form. In the rest frame of an isolated source, *J is proportional to a volume element of a (d-1-p)-dimensional subspace of (d-1)-dimensional space. The codimension of the source is therefore (d-p-1), and since space has dimension d-1, the charges that serve as sources must be objects with p dimensions, known as p-branes. So a (p+2)-form field strength couples to sources that are p-branes. This little fact has turned out to be extremely important in string theory.
    Superstring theories are theories with gravity, so these p-dimensional localizations of charge must lead to spacetime curvature. A p-brane spacetime whose metric solves the equations of motion for a (p+2)-form field strength in d spacetime dimensions can be described using p space coordinates {yi} along the p-brane and (d-1-p) space coordinates {xa} orthogonal to the p-brane.

p-brane metric

The isometries of this spacetime consist of translations (shifting the coordinate by a constant) and Lorentz transformations in the (p+1)-dimensional worldvolume, plus spatial rotations in the (d-1-p)-dimensional space orthogonal to the p-brane.
    There's a problem with adding gravity, however. Most p-brane spacetimes turn out to be unstable. Supersymmetry stabilizes p-branes, but only for the certain values of p and d. Two of the most important p-branes in string theory are the two-brane in d=11 and the five-brane in d=10.
    Since we're talking about a spacetime metric, we're obviously in the low energy limit of string theory. But p-branes can be protected from quantum corrections by supersymmetry, if they satisfy an equality between mass and charge known as the BPS condition. These branes are then known as BPS branes.

Next: From p-branes to D-branes>>

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