Whatever it is otherwise, string theory turns out to be an organizational principle that subsumes a wealth of effective quantum field theories together with hints for their UV-completion. As such, string theory reveals a multitude of equivalences between superficially very different-looking (classes of) quantum field theories, or between various limits (notably strong/weak coupling limits) of various different-looking quantum field theories. Large classes of these equivalences go by the name of “dualities”:
Then there are the “string-string dualities”, notably the
Then there are the
There is also
the relation between 11-dimensional supergravity and the various string theories, see
under construction
reduction from 11d | electric σ-model | weak/strong coupling duality | magnetic σ-model |
---|---|---|---|
M2-brane in 11d sugra EFT | $\leftarrow$electric-magnetic duality$\rightarrow$ | M5-brane in 11d sugra EFT | |
HW reduction | |||
$\downarrow$ on orientifold K3$\times S^1//\mathbb{Z}_2$ | $\downarrow$ on orientifold K3$\times S^1//\mathbb{Z}_2$ | ||
F1-brane in heterotic supergravity | $\leftarrow$S-duality$\rightarrow$ | black string in heterotic sugra | |
HW reduction | |||
$\downarrow$ on orientifold T4$\times S^1//\mathbb{Z}_2$ | $\downarrow$ on orientifold T4$\times S^1//\mathbb{Z}_2$ | ||
F1-brane in heterotic supergravity | $\leftarrow$S-duality$\rightarrow$ | black string in type IIA sugra | |
KK reduction | |||
$\downarrow$ on K3$\times S^1$ | $\downarrow$ on K3$\times S^1$ | ||
F1-brane in IIA sugra | $\leftarrow$S-duality$\rightarrow$ | black string in heterotic sugra | |
KK reduction | |||
$\downarrow$ on $T^4\times S^1$ | $\downarrow$ on $T^4 \times S^1$ | ||
F1-brane in IIA sugra | $\leftarrow$S-duality$\rightarrow$ | black string in type IIA sugra | |
F-reduction | $\updownarrow$ T-duality on $S^1$ | ||
F1-brane in IIB sugra | $\leftarrow$S-duality$\rightarrow$ | D1-brane in 10d IIB sugra | |
U-duality | $\updownarrow$ T-duality on $T^2$ | ||
D3-brane in IIB sugra | $\leftarrow$S-duality$\rightarrow$ | D3-brane in IIB sugra |
gauge theory induced via AdS-CFT correspondence
M-theory perspective via AdS7-CFT6 | F-theory perspective |
---|---|
11d supergravity/M-theory | |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$ | compactificationon elliptic fibration followed by T-duality |
7-dimensional supergravity | |
$\;\;\;\;\downarrow$ topological sector | |
7-dimensional Chern-Simons theory | |
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality | |
6d (2,0)-superconformal QFT on the M5-brane with conformal invariance | M5-brane worldvolume theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | double dimensional reduction on M-theory/F-theory elliptic fibration |
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence | D3-brane worldvolume theory with type IIB S-duality |
$\;\;\;\;\; \downarrow$ topological twist | |
topologically twisted N=2 D=4 super Yang-Mills theory | |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface | |
A-model on $Bun_G$, Donaldson theory |
$\,$
gauge theory induced via AdS5-CFT4 |
---|
type II string theory |
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$ |
$\;\;\;\; \downarrow$ topological sector |
5-dimensional Chern-Simons theory |
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality |
N=4 D=4 super Yang-Mills theory |
$\;\;\;\;\; \downarrow$ topological twist |
topologically twisted N=4 D=4 super Yang-Mills theory |
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface |
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence |
The original article suggesting the relation of 11d sugra to string theory is
An original article collecting all the weak/strong electric/magnetic dualities is
Mike Duff, chapter 6 of The World in Eleven Dimensions: Supgergravity, Supermembranes and M-theory, IoP 1999
Joseph Polchinski, Dualities (arXiv:1412.5704)
A detailed discussion of dualities induced on K3-compactifications is in
Also