By Blackburn Ch. M.

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**Example text**

We just switched left– and right–moving fermions. 83). The expansion can be characterized by brµ d˜nν with ´r · 12 µ n ¾ . ¯ Neveu–Schwarz Neveu–Schwarz boundary conditions (NS NS). 83), but of course the left– and right–moving Fourier modes are now independent. Both expansions are thus characterized by half–integer modes labelled by brµ b˜ νs with ´r · 1 1 2 µ ´n · 2 µ ¾ . We are now ready to quantize the superstring. Besides the bosonic creation and annihilation operators we now have fermionic creation and annihilation operators which can be in different sectors determined by the fermionic boundary conditions.

The same holds for the rank 5 central charge Y with one timelike index. The corresponding rank 6 central charge Y˜ will have only spacelike indices and gives rise to a BPS state with 6 extended spatial directions. The 6–brane can again be constructed as a soliton solution in the corresponding supergravity theory. The 9–brane however, usually referred to as the M9–brane, is special. Although progress has been made in identifying and constructing this state in the supergravity model [20], its properties are still not completely understood.

An important generalization involves adding U ´N µ Chan–Paton factors to the open string theory. Along the compact direction we could introduce vacuum expectation values for the gauge fields (usually called Wilson lines) that will break the gauge group to U ´1µN . We will just mention the result that the T–dual theory will involve N Dp–branes positioned in the D p 1 compact direction(s). g. U ´1µ ¢ U ´1µ U ´1µ ¢ SU ´2µ. We finally would like to point out that we can define D–branes in a decompactified spacetime by taking the limit R 0 which takes R˜ ∞, formally decompactifying the T–dual target spacetime.