The full non-abelian action was first derived in 1983 by George Chapline and Nicholas Manton.
[1] Classically the theory can admit any gauge group, but a consistent quantum theory resulting in anomaly cancellation only exists if the gauge group is either
Supergravity was much studied during the 1980s as a candidate theory of nature.
As part of this it was important to understand the various supergravities that can exist in different dimensions, with the possible supergravities being classified in 1978 by Werner Nahm.
[2] Type I supergravity was first written down in 1983, with Eric Bergshoeff, Mees de Roo, Bernard de Wit, and Peter van Nieuwenhuizen describing the abelian theory,[3] and then George Chapline and Nicholas Manton extending this to the full non-abelian theory.
[1] An important development was made by Michael Green and John Schwarz in 1984 when they showed that only a handful of these theories are anomaly free,[4] with additional work showing that only
result in a consistent quantum theory.
[5] The first case was known at the time to correspond to the low-energy limit of type I superstrings.
Heterotic string theories were discovered the next year,[6] with these having a low-energy limit described by type I supergravity with both gauge groups.
[nb 1] Its field content consists of the
The right-hand side must have the same chirality as the supercharges and must also be symmetric under an exchange of the spinor indices.
The second term is the only central charge that is admissible under these constraints up to Poincare duality.
[10]: 37–48 [nb 2] The central charge corresponds to a 5-brane solution in the supergravity which is dual to the fundamental string in heterotic string theory.
[11] The action for type I supergravity in the Einstein frame is given up to four-fermion terms by[12]: 325 [nb 3] Here
The spacetime index gamma-matrices are position-dependent fields
These transformation rules are useful for constructing the Killing spinor equations and finding supersymmetric ground states.
At a classical level the supergravity has an arbitrary gauge group, however not all gauge groups are consistent at the quantum level.
[13]: 98–101 The Green–Schwartz anomaly cancellation mechanism is used to show when the gauge, mixed, and gravitational anomalies vanish in hexagonal diagrams.
[4] In particular, the only anomaly free type I supergravity theories are ones with gauge groups of
It was later found that the latter two with abelian factors are inconsistent theories of quantum gravity.
[14] The remaining two theories both have ultraviolet completions to string theory, where the corresponding string theories can also be shown to be anomaly free at the string level.
heterotic string theory reduce to type I supergravity with an
heterotic string theory reduces to type I supergravity with an
[13]: 92–93 There are additional corrections that the supergravity receives in string theory, notably the Chern–Simons term becomes a linear combination of the Yang–Mills Chern–Simons three-form found at tree-level and a Lorentz Chern–Simons three-form
To maintain supersymmetry of the action when this term is included, additional higher-derivative corrections must be added to the action up to second order in
In type I string theory, the gauge coupling constant is related to the ten-dimensional Yang–Mills coupling constant by
, while the coupling constant is related to the string length
Instead, one has to transform the action into the various string frames through a Weyl transformation and dilaton redefinition[13]: 93 S-duality between type I string theory and
heterotic string theory can be seen at the level of the action since the respective string frame actions are equivalent with the correct field redefinitions.
[16] Similarly, Hořava–Witten theory, which describes the duality between