Superstring theory & D-brane
Supersymmetry
Generally, "supersymmetry(SUSY)" is a ideal symmetry relating bosons with fermions. The presence of supersymmetry implies that the existence of "superpartner", which is the supersymmetry particles corresponding to normal particles. The value of supersymmetry generator N is determined by "spinor" as a figure below:
Verbally, combining supersymmetry with gravity named "supergravity". So "superstring theory" is a theory which made up of string theory and supersymmetry.
Up to now, we haven't discovered any obvious evidences about supersymmetry. However, it's undeniable that this is a subtle symmetry which can lead to the pretty mathematical results actually.
Up to now, we haven't discovered any obvious evidences about supersymmetry. However, it's undeniable that this is a subtle symmetry which can lead to the pretty mathematical results actually.
GSO projection
To introduce fermions to bosonic string theory, physicists developed "GSO projection", which also eliminated tachyon and preserved spacetime supersymmetry.
Beforehand, there are two formalism as following :
- RNS (Ramond-Neveu-Schwartz) formalism: Introducing fermions through the means of supersymmetry into string theory, so that there's worldsheet supersymmetry exists in the theory.
- GS (Green-Schwartz) formalism: GS formalism is similar to RNS formalism but it has spacetime supersymmetry. Exactly, it's equivalent to RNS formalism which has been GSO projection.
In conclusion, through the GSO projection, spacetime has supersymmetry and also project out tachyon. To preserve consistency, superstring theory requires spacetime to has 10 dimensions.
Beforehand, there are two formalism as following :
- RNS (Ramond-Neveu-Schwartz) formalism: Introducing fermions through the means of supersymmetry into string theory, so that there's worldsheet supersymmetry exists in the theory.
- GS (Green-Schwartz) formalism: GS formalism is similar to RNS formalism but it has spacetime supersymmetry. Exactly, it's equivalent to RNS formalism which has been GSO projection.
In conclusion, through the GSO projection, spacetime has supersymmetry and also project out tachyon. To preserve consistency, superstring theory requires spacetime to has 10 dimensions.
Superstring theory
On the whole, there are five types of superstring theory, namely, type I, type IIA, type IIB, HO, HE.
It's worth noting that Michael Green and John Schwartz cancelled "anomaly" of type I string theory by introducing gauge group SO(32). On the other hand, there are gauge anomalies and gravitational anomalies arise from the Feynman diagram in string theory. However, for the special choice of gauge group SO(32) as well as E8 × E8 the anomalies will be cancelled by a tree diagram. These processes called "Green-Schwartz (GS) mechanism", which is a representative discovery during the first superstring revolutionary.
It's worth noting that Michael Green and John Schwartz cancelled "anomaly" of type I string theory by introducing gauge group SO(32). On the other hand, there are gauge anomalies and gravitational anomalies arise from the Feynman diagram in string theory. However, for the special choice of gauge group SO(32) as well as E8 × E8 the anomalies will be cancelled by a tree diagram. These processes called "Green-Schwartz (GS) mechanism", which is a representative discovery during the first superstring revolutionary.
Fig 2.1 John Schwartz (left) and Michael Green (right) are the founder of superstring theory.
For 10-dimensional superstring theory, it has 32-component Majorana spinor, which can be decomposed into a pair of 16-component Majorana-Weyl spinor. There are copious ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities as below:
- Type IIA invariant:
- Type IIB invariant:
- Heterotic invariant:
Besides, type IIA theory is non-chiral, whereas type IIB theory is chiral, namely a parity violating theory.
Five types of superstring theories and bosonic string theory have plenty of different features:
D-brane
During superstring theory development, Joseph Polchinski and some string theorists proposed an extended object, which open string can end with Dirichlet boundary conditions, named "D-brane".
As we discussed in the former section, there are two main types of boundary conditions: Neumann boundary condition corresponding to free endpoint moving in spacetime at the speed of light, Dirichlet boundary condition let open string attached to an object, namely D-brane.
As we discussed in the former section, there are two main types of boundary conditions: Neumann boundary condition corresponding to free endpoint moving in spacetime at the speed of light, Dirichlet boundary condition let open string attached to an object, namely D-brane.
Fig 2.2 The illustration of D-brane
There are numerous stable D-brane existed with p-dimensions, which notes "Dp-brane". For instance, there is a D25-brane in bosonic string theory. Generally, there are different dimensional D-brane presents in different types of superstring theory. The stable D-branes have even dimensions in IIA string theory, and have odd dimensions in IIB string theory. The different types of stable Dp-brane in superstring theories as the following:
- Type I: p=1,5,9
- Type IIA: p=0,2,4,6,8
- Type IIB: p= -1,1,3,5,7,9
- HO: none
- HE: none
In type IIB string theory, one D-brane has -1 dimension called "instanton", which is a classical solution to the equations of motion in classical field theory.
Branes are the solitonic solutions exists in string theory. Superstring theories, heterotic string theories , and D-brane are the important development during the first superstring revolutionary. Nevertheless, there are more advanced theories beyond 10-dimensional superstring theory.