In string theory, a heterotic string is a closed string (or loop) which is a hybrid ('heterotic') of a superstring and a bosonic string. There are two kinds of heterotic string, the heterotic SO(32) and the heterotic E8 × E8, abbreviated to HO and HE. Heterotic string theory was first developed in 1985 by David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm[1] (the so-called "Princeton String Quartet"[2]), in one of the key papers that fueled the first superstring revolution.

Overview

In string theory, the left-moving and the right-moving excitations are completely decoupled,[3] and it is possible to construct a string theory whose left-moving (counter-clockwise) excitations are treated as a bosonic string propagating in D = 26 dimensions, while the right-moving (clockwise) excitations are treated as a superstring in D = 10 dimensions.

The mismatched 16 dimensions must be compactified on an even, self-dual lattice (a discrete subgroup of a linear space). There are two possible even self-dual lattices in 16 dimensions, and it leads to two types of the heterotic string. They differ by the gauge group in 10 dimensions. One gauge group is SO(32) (the HO string) while the other is E8 × E8 (the HE string).[4]

These two gauge groups also turned out to be the only two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in 10 dimensions. (Although not realized for quite some time, U(1)496 and E8 × U(1)248 are anomalous.[5])

Every heterotic string must be a closed string, not an open string; it is not possible to define any boundary conditions that would relate the left-moving and the right-moving excitations because they have a different character.

String duality

String duality is a class of symmetries in physics that link different string theories. In the 1990s, it was realized that the strong coupling limit of the HO theory is type I string theory — a theory that also contains open strings; this relation is called S-duality. The HO and HE theories are also related by T-duality.

Because the various superstring theories were shown to be related by dualities, it was proposed that each type of string was a different limit of a single underlying theory called M-theory.

References

Gross, David J.; Harvey, Jeffrey A.; Martinec, Emil; Rohm, Ryan (1985-02-11). "Heterotic String". Physical Review Letters. American Physical Society (APS). 54 (6): 502–505. doi:10.1103/physrevlett.54.502. ISSN 0031-9007.

Dennis Overbye (2004-12-07). "String theory, at 20, explains it all (or not)". The New York Times. Retrieved 2020-03-15.

Becker, Katrin; Becker, M.; Schwarz, J. H. (2007). String theory and M-theory : a modern introduction. Cambridge New York: Cambridge University Press. p. 253. ISBN 978-0-521-86069-7. OCLC 607562796.

Joseph Polchinski (1998). String Theory: Volume 2, p. 45.

Adams, Allan; Taylor, Washington; DeWolfe, Oliver (2010-08-10). "String Universality in Ten Dimensions". Physical Review Letters. American Physical Society (APS). 105 (7): 071601. arXiv:1006.1352. doi:10.1103/physrevlett.105.071601. ISSN 0031-9007.

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String theory

Background

Strings History of string theory

First superstring revolution Second superstring revolution String theory landscape

Calabi-Yau-alternate

Theory

Nambu–Goto action Polyakov action Bosonic string theory Superstring theory

Type I string Type II string

Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism

String duality

T-duality S-duality U-duality Montonen–Olive duality

Particles and fields

Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon

Branes

D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition

Conformal field theory

Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model

Gauge theory

Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G2, F4, E6, E7, E8) ADE classification Dirac string p-form electrodynamics

Geometry

Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold

Calabi–Yau manifold Hyperkähler manifold

K3 surface G2 manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten domain wall K-theory (physics) Twisted K-theory

Supergravity Superspace Lie superalgebra Lie supergroup

Holography

Holographic principle AdS/CFT correspondence

M-theory

Matrix theory Introduction to M-theory

String theorists

Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

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