nLab duality in string theory

Idea

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”:

Examples

duality in string theory

general mechanisms

• T-duality (topological T-duality, non-abelian T-duality)

  mirror symmetry

• S-duality

  electric-magnetic duality, Montonen-Olive duality, geometric Langlands duality

• U-duality

  exceptional generalized geometry

string-fivebrane duality

• dual heterotic string theory

string-string dualities

• HET - I

• HET - II

• HET - M

M-theory

• M - IIA.

• M- IIB

F-theory

• M - F

• F - HET

string-QFT duality

• AdS-CFT duality

  (an open/closed string duality)

  • AdS2/CFT1 duality

  • AdS3/CFT2 duality

    • p-adic AdS/CFT duality

  • AdS-QCD correspondence

• Chern-Simons theory as topological string theory

QFT-QFT duality:

• effective QFT incarnations of open/closed string duality,

  relating (super-)gravity to (super-)Yang-Mills theory:

  • KLT relations

  • BCJ relations

  • classical double copy

• Kramers-Wannier duality

• Seiberg duality (swapping NS5-branes)

• AGT correspondence (wrapped M5-branes)

• 3d-3d correspondence (wrapped M5-branes)

Overview

under construction

S-duality in string theory

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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

Related concepts

• duality in physics

• Gopakumar-Vafa duality

• mysterious duality

• magic pyramid

References

Original articles

The original article suggesting the relation of 11d sugra to string theory is

• Edward Witten, String Theory Dynamics In Various Dimensions, Nucl.Phys.B443:85-126,1995 (arXiv:hep-th/9503124)

An original article collecting all the weak/strong electric/magnetic dualities is

• Mike Duff, M-theory (the theory formerly known as string), Int.J.Mod.Phys. A11 (1996) 5623-5642 (arXiv:hep-th/9608117)

Surveys and lecture notes

• John Schwarz, Lectures on Superstring and M Theory Dualities, Nucl. Phys. Proc. Suppl. 55B:1-32, 1997 (arXiv:hep-th/9607201)

• Michael Dine, String Theory Dualities [arXiv:hep-th/9609051]

• Luis Alvarez-Gaumé, Frederic Zamora, Duality in Quantum Field Theory (and String Theory), AIP Conference Proceedings 423, 46 (1998) (arXiv:hep-th/9709180)

• Stefan Forste, Jan Louis, Duality in String Theory, Nucl. Phys. Proc. Suppl. 61A:3-22, 1998 (arXiv:hep-th/9612192)

• Ashoke Sen, Duality symmetries in string theory, Current Science Vol. 77, No. 12, 25 December 1999 (jstor:24104472)

• Mike Duff, chapter 6 of The World in Eleven Dimensions: Supgergravity, Supermembranes and M-theory, IoP 1999

• Joseph Polchinski, Dualities (arXiv:1412.5704)

More on dualities induced on K3-compactifications:

• Paul Aspinwall, K3-surfaces and string duality (arXiv:hep-th/9611137)

More in view of the landscape of string theory vacua:

• T. Daniel Brennan, Federico Carta, Cumrun Vafa, The String Landscape, the Swampland, and the Missing Corner (arXiv:1711.00864)

Also

• Cumrun Vafa, around 3:30, 12:00 of On Mathematical Aspects of String Theory (video)

Discussion amplifying the role of category theory, and higher geometry is in

• Eric Sharpe, Categorical Equivalence and the Renormalization Group, Proceedings of LMS/EPSRC Symposium Higher Structures in M-Theory, Fortschritte der Physik 2019 (arXiv:1903.02880)