In supersymmetry, type IIB supergravity is the unique supergravity in ten dimensions with two supercharges of the same chirality. It was first constructed in 1983 by John Schwarz and independently by Paul Howe and Peter West at the level of its equations of motion. While it does not admit a fully covariant action due to the presence of a self-dual field, it can be described by an action if the self-duality condition is imposed by hand on the resulting equations of motion. The other types of supergravity in ten dimensions are type IIA supergravity, which has two supercharges of opposing chirality, and type I supergravity, which has a single supercharge. The theory plays an important role in modern physics since it is the low-energy limit of type IIB string theory.
History After supergravity was discovered in 1976, there was a concentrated effort to construct the various possible supergravities that were classified in 1978 by Werner Nahm. He showed that there exist three types of supergravity in ten dimensions, later named type I, type IIA and type IIB. While both type I and type IIA can be realised at the level of the action, type IIB does not admit a covariant action. Instead it was first fully described through its equations of motion, derived in 1983 by John Schwartz, and independently by Paul Howe and Peter West. In 1995 it was realised that one can effectively describe the theory using a pseudo-action where the self-duality condition is imposed as an additional constraint on the equations of motion. The main application of the theory is as the low-energy limit of type IIB strings, and so it plays an important role in string theory, type IIB moduli stabilisation, and the AdS/CFT correspondence.
Theory Ten-dimensional supergravity admits both N = 1 {\displaystyle {\mathcal {N}}=1} and N = 2 {\displaystyle {\mathcal {N}}=2} supergravities, which differ by the number of the Majorana–Weyl spinor supercharges that they possess. The type IIB theory has two supercharges of the same chirality, equivalent to a single Weyl supercharge, with it sometimes denoted as the ten-dimensional N = ( 2 , 0 ) {\displaystyle {\mathcal {N}}=(2,0)} supergravity. The field content of this theory is given by the ten dimensional N = 2 {\displaystyle {\mathcal {N}}=2} chiral supermultiplet ( g μ ν , B , C 4 , C 2 , C 0 , ψ μ , λ , ϕ ) {\displaystyle (g_{\mu \nu },B,C_{4},C_{2},C_{0},\psi _{\mu },\lambda ,\phi )} . Here g μ ν {\displaystyle g_{\mu \nu }} is the metric corresponding to the graviton, while C p {\displaystyle C_{p}} are 4-form, 2-form, and 0-form gauge fields. Meanwhile, B {\displaystyle B} is the Kalb–Ramond field and ϕ {\displaystyle \phi } is the dilaton. There is also a single left-handed Weyl gravitino ψ μ {\displaystyle \psi _{\mu }} , equivalent to two left-handed Majorana–Weyl gravitinos, and a single right-handed Weyl fermion λ {\displaystyle \lambda } , also equivalent to two right-handed Majorana–Weyl fermions. The theory does admit a cosmological constant.
Algebra The superalgebra for ten-dimensional N = ( 2 , 0 ) {\displaystyle {\mathcal {N}}=(2,0)} supersymmetry is given by
{ Q α i , Q β j } = δ i j ( P γ μ C ) α β P μ + ( P γ μ C ) α β Z ~ μ i j + ϵ i j ( P γ μ ν ρ C ) α β Z μ ν ρ {\displaystyle \{Q_{\alpha }^{i},Q_{\beta }^{j}\}=\delta ^{ij}(P\gamma ^{\mu }C)_{\alpha \beta }P_{\mu }+(P\gamma ^{\mu }C)_{\alpha \beta }{\tilde {Z}}_{\mu }^{ij}+\epsilon ^{ij}(P\gamma ^{\mu \nu \rho }C)_{\alpha \beta }Z_{\mu \nu \rho }}
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