M-theory & Dualities
Duality
"Duality" of different theories means they seem like different but are equivalent actually. There are numerous dualities existed in string theory. For instance:
- T-duality: Compatifying one theory on a circle with radial R, equivalent to the other theory compatified on a circle with 1/R.
- S-duality(Strong-weak duality): When one theory's coupling constant increases to strong coupled limit, it equivalent to the other theory which coupling constant decline to weak coupled limit.
- U-duality: Roughly, one D-dimensional theory equivalent to the other theory with different dimensions.
There are different dualities existed in superstring theory, and connecting between with different theories.
IIA<---T-dual--->IIB ; HO<---T-dual--->HE
IIB<---S-dual--->IIB ; I<---S-dual--->HE
IIA<---U-dual--->M ; HE<---U-dual--->M
As above, for instance, type IIA theory compactified on a circle with R is equivalent to type IIB theory compactified on a circle with 1/R, and vice versa.
On the other hand, it's noticeable that type IIB theory has "self-duality", namely, D-string in the theory under strong coupled limit is equivalent to F-string(fundamental string) in the same theory under weak coupled limit, and vice versa.
Furthermore, when we rise the coupling constant of 10-dimensional type IIA or HE theory, they will equivalent to 11-dimensional M-theory, namely, a new dimension opened. As a result, U-duality is a duality combined dimension as well as coupling constant.
- T-duality: Compatifying one theory on a circle with radial R, equivalent to the other theory compatified on a circle with 1/R.
- S-duality(Strong-weak duality): When one theory's coupling constant increases to strong coupled limit, it equivalent to the other theory which coupling constant decline to weak coupled limit.
- U-duality: Roughly, one D-dimensional theory equivalent to the other theory with different dimensions.
There are different dualities existed in superstring theory, and connecting between with different theories.
IIA<---T-dual--->IIB ; HO<---T-dual--->HE
IIB<---S-dual--->IIB ; I<---S-dual--->HE
IIA<---U-dual--->M ; HE<---U-dual--->M
As above, for instance, type IIA theory compactified on a circle with R is equivalent to type IIB theory compactified on a circle with 1/R, and vice versa.
On the other hand, it's noticeable that type IIB theory has "self-duality", namely, D-string in the theory under strong coupled limit is equivalent to F-string(fundamental string) in the same theory under weak coupled limit, and vice versa.
Furthermore, when we rise the coupling constant of 10-dimensional type IIA or HE theory, they will equivalent to 11-dimensional M-theory, namely, a new dimension opened. As a result, U-duality is a duality combined dimension as well as coupling constant.
Fig3.1 Dualities between five superstring theories as well as M-theory.
M-theory & Supergravity
It's interesting that when type IIA/ HE superstring theory under strong coupling limit, a new dimension will opened, and the it will become 11-dimensional "M-theory".
M-theory's low energy description is "11-D supergravity". The theory was purposed by Edward Witten in 1995, who is one important figure in the field of string/M-theory. In fact, there are numerous things about mathematical/theoretical physics were dedicated to him.
M-theory's low energy description is "11-D supergravity". The theory was purposed by Edward Witten in 1995, who is one important figure in the field of string/M-theory. In fact, there are numerous things about mathematical/theoretical physics were dedicated to him.
Fig 3.2 Edward Witten, who putted forward M-theory in 1995
By the way, a letter "M" has plenty of definitions, for example, "membrane", "mystery", "mother", "matrix", "magic", and so on. The most important value of M-theory is that it "unified" five different types of superstring theory in 11 dimension.
As a consequence, the discovery of M-theory is regarded as the commencement of the second superstring revolution.
As a consequence, the discovery of M-theory is regarded as the commencement of the second superstring revolution.
Fig3.3 Five different 10-D superstring theories are the different parts of 11-D M-theory actually, and each of theories are connected by dualities.
M-brane
As the discussion in previous section, there are also stable branes existed in M-theory.
- M2-brane: carries electric charge
- M5-brane: carries magnetic charge, analogous to NS5-brane in string theory.
The dual description of M5-brane under electric-magnetic duality is M2-brane. Besides, M5-more is more mysterious and recondite than M2-brane.
- M2-brane: carries electric charge
- M5-brane: carries magnetic charge, analogous to NS5-brane in string theory.
The dual description of M5-brane under electric-magnetic duality is M2-brane. Besides, M5-more is more mysterious and recondite than M2-brane.