In supersymmetry, type I supergravity is the theory of supergravity in ten dimensions with a single supercharge. It consists of a single supergravity multiplet and a single Yang–Mills multiplet. The full non-abelian action was first derived in 1983 by George Chapline and Nicholas Manton. 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 SO ( 32 ) {\displaystyle {\text{SO}}(32)} or E 8 × E 8 {\displaystyle E_{8}\times E_{8}} . Both these supergravities are realised as the low-energy limits of string theories, in particular of type I string theory and of the two heterotic string theories.
History 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. 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, and then George Chapline and Nicholas Manton extending this to the full non-abelian theory. 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, with additional work showing that only SO ( 32 ) {\displaystyle {\text{SO}}(32)} and E 8 × E 8 {\displaystyle E_{8}\times E_{8}} result in a consistent quantum theory. 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, with these having a low-energy limit described by type I supergravity with both gauge groups.
Theory Type I supergravity is the ten-dimensional supergravity with a single Majorana–Weyl spinor supercharge. Its field content consists of the N = 1 {\displaystyle {\mathcal {N}}=1} supergravity supermultiplet ( g μ ν , ψ μ , B , λ , ϕ ) {\displaystyle (g_{\mu \nu },\psi _{\mu },B,\lambda ,\phi )} , together with the N = 1 {\displaystyle {\mathcal {N}}=1} Yang–Mills supermultiplet ( A μ a , χ a ) {\displaystyle (A_{\mu }^{a},\chi ^{a})} with some associated gauge group. Here g μ ν {\displaystyle g_{\mu \nu }} is the metric, B {\displaystyle B} is the two-form Kalb–Ramond field, ϕ {\displaystyle \phi } is the dilaton, and A μ a {\displaystyle A_{\mu }^{a}} is a Yang–Mills gauge field. Meanwhile, ψ μ {\displaystyle \psi _{\mu }} is the gravitino, λ {\displaystyle \lambda } is a dilatino, and χ a {\displaystyle \chi ^{a}} a gaugino, with all these being Majorana–Weyl spinors. The gravitino and gaugino have the same chirality, while the dilatino has the opposite chirality.
Algebra The superalgebra for type I supersymmetry is given by
{ Q α , Q β } = ( P γ μ C ) α β P μ + ( P γ μ ν ρ σ δ C ) α β Z μ ν ρ σ δ . {\displaystyle \{Q_{\alpha },Q_{\beta }\}=(P\gamma ^{\mu }C)_{\alpha \beta }P_{\mu }+(P\gamma ^{\mu \nu \rho \sigma \delta }C)_{\alpha \beta }Z_{\mu \nu \rho \sigma \delta }.}
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