Understanding M-Theory?

Question

Could someone explain in detail how p-branes interact with each other?

Answer 1

p-brane is a spatially extended, mathematical concept that appears in string theory and its relatives (M-theory and brane cosmology). The variable p refers to the spatial dimension of the brane. That is, a 0-brane is a zero-dimensional pointlike particle, a 1-brane is a string, a 2-brane is a "membrane", etc. Every p-brane sweeps out a (p+1)-dimensional world volume as it propagates through spacetime.

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.)

F = d A , dF=0

_
w = dt^dx^dy^dz

d*F=4*22/7*J

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.)

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.

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.

Originally string theory was a theory of 1-branes called strings. By the mid-1990s it became apparent that the theory could be extended to also include higher dimensional objects. Typically these objects are non-perturbative features of the theory (meaning they do not appear in perturbation theory) which is partially why early string theorists were unaware of them.

Besides the fundamental string (or F-string) of string theory and its magnetic dual, the NS5-brane, the most important type of branes that appear are the D-branes. Different types of D-branes appear in different theories. Even Dn-branes for n appear in type IIA string theory where as the odd Dp-branes appear in type IIB string theory.

With the development of M-theory, an extra dimension appeared and the fundamental string of string theory became a 2-dimensional membrane called an M2-brane or supermembrane). Its magnetical dual is an M5-brane. The various branes of string theory are thought to be related to these higher dimensional M5-branes wrapped on various cycles.

Answer 2

i think it's on the Quantum level this stuff. You will probably need a mathematical description of how they interact.

but you should phone this guy:

http://www.damtp.cam.ac.uk/user/ngt1000/