Let’s do a small post on D-Branes and their charges. Before starting, we will review some familiar concepts so that it is easy to understand what follows. A very basic understanding of electromagnetism and Quantum Field Theory (QFT) is assumed.
In electrodynamics (using Newtonian space and time) the magnetic field is represented by a vector potential A. In order to describe the electric field, we need another scalar quantity called the potential.
Using special relativity, the scalar and the vector potential can be combined to define a vector potential but this potential is a four-vector (i.e. it has an index that has four possible values corresponding to one time and three space directions). This potential which has one index is called a 1-form (or one-form).
Now we can have a generalized version of the vector potential which we call p-forms i.e. they are mathematical objects with p indices (a technical but unimportant point here is that these objects should be antisymmetric in all the indices).
Point particles have no extent in space and thus, have zero space dimensions. They are sources of a 1-form field (i.e. electromagnetic field). Similarly, objects with p-space dimensions are sources of a (p+1)-form field. For example, a string has a single space dimension and thus, it should be the source of a 2-form field.
Let’s talk about Type II A and Type II B string theory now. Both of these theories have a 2-form field called the Kalb-Ramond field or simply, the B field. The fundamental strings are the sources of this B-field. However, both of these theories contain additional p-form fields called the R-R fields.
Type II A has a 1-form, 3-form, 5-form, 7-form, and 9-form field. Therefore in this theory, we must have objects with 0,2,4,6, and 8 dimensions which are sources of these fields. These objects are the D-p branes. A D-p brane has p space dimensions. So, type II theory must have D0, D2, D4, D6, and D8 branes.
Similarly, type II B has a 0-form, 2-form, 4-form, 6-form, 8-form, and 10-form field. Therefore in this theory, we must have objects with -1,1,3,5,7 and 9 dimensions which are sources of these fields. These sources are D(-1), D1, D3, D5, D7 and D9 branes. D(-1) might look nonsense but it is a sensible object called the D instanton (with no extent in time and space).
The branes discussed above are the stable D-p branes in the type II theories. They are stable because they carry R-R charges and they must be conserved (just like an electrically charged particle can’t just decay without leaving anything because that will violate electrical charge conservation).
There are some branes in string theory and M theory that I didn’t talk about. These include NS5 brane (which has magnetic charge under the Kalb-Ramond field), and M2 and M5 branes (which are found in M theory).
Hassaan Saleem 