nLab string theory results applied elsewhere

Idea

Beyond the speculative hypothesized role of string theory as a physical theory of fundamental strings that constitute the observed fundamental particles in the standard model of particle physics, the theory has shed light on many aspects of quantum field theory as such, both on the conceptual structure of QFT as well as on concrete theories and their concrete properties. This entry lists such instances of string theory results having led to insights in non-stringy physics and in particular into experimentally confirmed physics, such as QCD in the standard model of particle physics.

Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.

Jacques Hadamard (source)

The construction of efficient computational methods for field theory scattering amplitudes has benefited substantially from string theory input.

Roiban & Volovich 2004

Examples

The two basic theories that underlie observed fundamental physics – and which string theory unifies at least qualitatively and in perturbation theory – are Yang-Mills theory and Einstein gravity/general relativity.

1. Applications to Yang-Mills theory

2. Applications to Gravity

Many of the insights are based on the gauge/gravity duality in string theory:

1. General relation between Yang-Mills theory and gravity

Worldline formalism

The worldline formalism for expressing QFT scattering amplitudes in an effective gauge invariant way (different from but equivalent to the Feynman rules) was originally found by taking the point-particle limit of the expressions for string scattering amplitudes. See at worldline formalism for more.

Example:

The first calculation along these lines was actually done earlier in (Metsaev-Tseytlin 88), where the 1-loop beta function for pure Yang-Mills theory was obtained as the point-particle limit of the partition function of a bosonic open string in a Yang-Mills background field. This provided a theoretical explanation for the observation, made earlier in (Nepomechie 83) that when computed via dimensional regularization then this beta function coefficient of Yang-Mills theory vanishes in spacetime dimension 26. This of course is the critical dimension of the bosonic string.

• R.I. Nepomechie, Remarks on quantized Yang-Mills theory in 26 dimensions, Physics Letters B Volume 128, Issues 3–4, 25 August 1983, Pages 177-178 Phys. Lett. B128 (1983) 177 (doi:10.1016/0370-2693(83)90385-4)

• Ruslan Metsaev, Arkady Tseytlin, On loop corrections to string theory effective actions, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-9)

General relation between Yang-Mills theory and gravity

AdS/CFT correspondenceopen/closed string duality

• Spenta R. Wadia, Gauge/Gravity Duality and Some Applications (arXiv:1009.0212)

talks at

• Zvi Bern, Thomas Gehrmann, Frank Petriello, Anastasia Volovich, The Harmony of Scattering Amplitudes KITP Program (2011) (web)

Applications to Yang-Mills theory

Worldline formalism

The worldline formalism for expressing QFT scattering amplitudes in an effective gauge invariant way (different from but equivalent to the Feynman rules) was originally found by taking the point-particle limit of the expressions for string scattering amplitudes. See at worldline formalism for more.

Example:

The first calculuation along these lines was actually done earlier in (Metsaev-Tseytlin 88), where the 1-loop beta function for pure Yang-Mills theory from the partition function of a bosonic open string in a Yang-Mills background field. This provided a theoretical explanation for the observation, made earlier in (Nepomechie 83) that when computed in via dimensional regularization then this beta function coefficient of Yang-Mills theory vanishes in spacetime dimension 26. This of course is the critical dimension of the bosonic string.

• R.I. Nepomechie, Phys. Lett. B128 (1983) 177

• {MetsaevTseytlin88} Ruslan Metsaev, Arkady Tseytlin, On loop corrections to string theory effective actions, Nuclear Physics B Volume 298, Issue 1, 29 February 1988, Pages 109-132 (doi:10.1016/0550-3213(88)90306-990306-9))

QFT Duality and specifically Montonen-Olive electric/magnetic duality

By embedding quantum field theories in string theory (typically as the worldvolume theories of various branes) the various dualities of string theory will relate different QFTs in ways that are typically far from obvious from just looking at these QFTs themselves.

The investigation specifically of N=1 D=4 super Yang-Mills theory and N=2 D=4 super Yang-Mills theory in this fashion has come to be known as geometric engineering of quantum field theory.

Montonen-Olive duality of (super) Yang-Mills theory derives from conformal invariance of the 6d (2,0)-supersymmetric QFT (see there) compactified on a torus.

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

See also

• Matthew Strassler, Millenial Messages for QCD from the Superworld and from the String (arXiv:hep-th/0309140)

Application to QCD and experimental particle physics

The realization of Yang-Mills theory that describes quarks and their interaction by the strong nuclear force carried by gluons is quantum chromodynamics (QCD).

1. QCD Scattering amplitudes

2. Quark-gluon plasma

Scattering amplitudes

The string scattering amplitudes exhibit certain relations due to the extended nature of the string which are retained in the point particle limit and hence explain and serve to discover subtle relations in QFT scattering amplitudes.

• twistor string theory explains some (super) Yang-Mills theory scattering amplitudes

• Precision scattering amplitudes in QCD use twistor string theory on-shell recursion methods,

This also goes by the term “on-shell methods”. See also at amplituhedron.

Reviews include

• Matthew Strassler, From string theory to the large hadron collider (blog post)

• Lance Dixon, Calculating Amplitudes, December 2013 (web)

• Rutger Boels, On-shell recursion for string theory amplitudes on the disk and the sphere (pdf)

Original articles include

• Zvi Bern, Lance Dixon, David Kosower, On-Shell Methods in Perturbative QCD (arXiv:0704.2798)

• Joseph Polchinski, Matthew Strassler, Hard Scattering and Gauge/String Duality, Phys.Rev.Lett.88:031601,2002, (arXiv:hep-th/0109174)

See also

• Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi, Elliptic polylogarithms and Feynman parameter integrals (arXiv:1902.09971)

  reviewed in

  Lorenzo Tancredi, Feynman integrals and higher genus surfaces, talk at Amplitudes 2019 (pdf)

See also below Application to gravity – Scattering amplitudes.

Quark-gluon plasma

Properties of quark-gluon plasma from AdS/CFT-dual type II string theory

• Pavel Kovtun, Quark-Gluon Plasma and String Theory, RHIC news (2009) (blog entry)

• Makoto Natsuume, String theory and quark-gluon plasma (arXiv:hep-ph/0701201)

• Steven Gubser, Using string theory to study the quark-gluon plasma: progress and perils (arXiv:0907.4808)

To the confinement problem

Discussion of confinement in the context of the AdS-CFT correspondence is in

• David Berman, Maulik K. Parikh, Confinement and the AdS/CFT Correspondence, Phys.Lett. B483 (2000) 271-276 (arXiv:hep-th/0002031)

• Henrique Boschi Filho, AdS/QCD and confinement, Seminar at the Workshop on Strongly Coupled QCD: The confinement problem, November 2011 (pdf)

To super Yang-Mills theory

Seiberg duality in super Yang-Mills theory is conceptually explained by type II string theory on certain D-brane configurations (…)

Applications to gravity

Scattering amplitudes

• open/closed string duality in string scattering amplitudes allows to compute gravity scattering amplitudes in terms of Yang-Mills theory scattering amplitudes: the KLT relations

• Zvi Bern, Perturbative Quantum Gravity and its Relation to Gauge Theory, Living Rev Relativ. 2002; 5(1): 5. (arXiv:gr-qc/0206071, doi:10.12942/lrr-2002-5)

• more on this string-organizatioon of graviton scattering amplitudes is in

  David C. Dunbar, Paul S. Norridge, Calculation of Graviton Scattering Amplitudes using String-Based Methods, Nucl.Phys. B433 (1995) 181-208 (arXiv:hep-th/9408014)

• KLT relations used for instance to demonstrate:

  • multi-loop finiteness of N=8 d=4 supergravity

Gravitational wave signatures

Application of the KLT relation/double copy-technique to computation of gravitational wave-signature of relativistic binary mergers:

• Zvi Bern, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng, Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett. 122, 201603 (2019) (arXiv:1901.04424)

Black hole entropy

Semi-classical QFT computations suggest that there should be entropy associated with black holes, the Bekenstein-Hawking entropy, without however providing microscopic degrees of freedom of which this would be an entropy in the ordinary sense.

Since the quantum dynamics of general black holes is outside the reach of perturbative methods in string theory, certain supersymmetric black hole solutions in supergravity have properties independent of the coupling and are known to be the strong-coupling limit of what at weak coupling is a certain configuration of branes in flat space. Therefore the ordinary entropy of these brane configurations should match the Bekenstein-Hawking entropy of the corresponding black holes, and this has been confirmed to good precision.

While this argument does not give direct information about the origin of the BH-entropy of physically observed black holes, it does show conceptually, in the general context of black holes in theories of gravity, BH-entropy can be accounted for by microscopic degrees of freedom in a theory of quantum gravity.

Reviews include

• Ofer Aharony, Steven Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz, Chapter 5 of: Large $N$ Field Theories, String Theory and Gravity, Phys. Rept. 323:183-386, 2000 (arXiv:hep-th/9905111)

Application to pure mathematics

string theory “knows more mathematics than we do” and its study has led to the development of a large quantity of new mathematics: mirror symmetry (enumerative formulas, homological mirror symmetry, Bridgeland stability), new invariants (Gromov-Witten, Donaldson-Thomas, Gopakumar-Vafa, etc.), topological field theory and topological quantum gravity, etc. (Douglas 19, slide 30)

Reviews include

• Gregory Moore, The Impact of D-Branes on Mathematics (2014), Physical Mathematics and the Future (2014)

• Mina Aganagic, String Theory and Math: Why This Marriage May Last, (arXiv:1508.06642)

• Ibrahima Bah, Daniel Freed, Gregory Moore, Nikita Nekrasov, Shlomo S. Razamat, Sakura Schafer-Nameki, A Panorama Of Physical Mathematics c. 2022 (arXiv:2211.04467)

For a review of prospects of computerized support for string theory, see

• Michael Douglas, Physics and Formal Methods, (slides)

See also Fields medal (and other) work related to string theory

• twisted K-theory

• monstrous moonshine

• The string orientation of tmf was directly motivated from the index of the heterotic string (“Witten genus”).

• The relation between Bruhat-Tits buildings and zeros of the Riemann zeta function, as described in Huang-Stoica-Yau.

• mirror symmetry took complex geometers by complete surprise, but is easily seen from the 2d (2,0)-sigma model with target space the respective CY manifolds.

• Morse theory of loop spaces via loop space supersymmetric quantum mechanics describing superstring propagation.

(…)