Ten dimensions admits both \mathcal N=1 and \mathcal N=2 supergravity, depending on whether there are one or two supercharges. Since the smallest
spinorial representations in ten dimensions are
Majorana–
Weyl spinors, the supercharges come in two types Q^\pm depending on their chirality, giving three possible supergravity theories. The \mathcal N=2 theory formed using two supercharges of opposite chiralities is denoted by \mathcal N=(1,1) and is known as type IIA supergravity. This theory contains a single
multiplet, known as the ten-dimensional \mathcal N=2 nonchiral multiplet. The fields in this multiplet are (g_{\mu\nu}, C_{\mu\nu\rho},B_{\mu\nu},C_\mu,\psi_\mu,\lambda,\phi), where g_{\mu\nu} is the
metric corresponding to the
graviton, while the next three fields are the 3-, 2-, and
1-form gauge fields, with the 2-form being the
Kalb–Ramond field. There is also a Majorana
gravitino \psi_\mu and a Majorana spinor \lambda, both of which decompose into a pair of Majorana–Weyl spinors of opposite chiralities \psi_\mu = \psi_\mu^++\psi_\mu^- and \lambda = \lambda^++\lambda^-. Lastly, there a
scalar field \phi. This nonchiral multiplet can be decomposed into the ten-dimensional \mathcal N=1 multiplet (g_{\mu\nu}, B_{\mu\nu}, \psi^+_\mu, \lambda^-, \phi), along with four additional fields (C_{\mu\nu\rho}, C_\mu, \psi_\mu^-, \lambda^+). In the context of string theory, the bosonic fields in the first multiplet consists of
NSNS fields while the bosonic fields are all
RR fields. The fermionic fields are meanwhile in the NSR sector.
Algebra The
superalgebra for \mathcal N=(1,1) supersymmetry is given by : \{Q_\alpha, Q_\beta\} = (\gamma^\mu C)_{\alpha \beta}P_\mu + (\gamma_* C)_{\alpha \beta}Z + (\gamma^\mu \gamma_* C)_{\alpha \beta}Z_\mu +(\gamma^{\mu\nu}C)_{\alpha \beta}Z_{\mu\nu} : \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + (\gamma^{\mu\nu\rho\sigma}\gamma_*C)_{\alpha \beta}Z_{\mu\nu\rho\sigma} + (\gamma^{\mu\nu\rho\sigma\delta}C)_{\alpha \beta}Z_{\mu\nu\rho\sigma \delta}, where all terms on the right-hand side besides the first one are the
central charges allowed by the theory. Here Q_\alpha are the spinor components of the Majorana supercharges while C is the
charge conjugation operator. Since the
anticommutator is symmetric, the only
matrices allowed on the right-hand side are ones that are symmetric in the spinor indices \alpha, \beta. In ten dimensions \gamma^{\mu_1\cdots \mu_p}C is symmetric only for p=1,2
modulo 4, with the
chirality matrix \gamma_* behaving as just another \gamma matrix, except with no index. : S_{IIA,\text{bosonic}} = \frac{1}{2\kappa^2} \int d^{10} x \sqrt{-g} e^{-2\phi}\bigg[R + 4 \partial_\mu \phi \partial^\mu \phi -\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho} - 2 \bar \psi_\mu \gamma^{\mu\nu\rho}D_\nu \psi_\rho + 2 \bar \lambda \gamma^\mu D_\mu \lambda \bigg] : \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{1}{4\kappa^2}\int d^{10}x \sqrt{-g}\big[\tfrac{1}{2}F_{2,\mu\nu}F^{\mu\nu}_2+\tfrac{1}{24}\tilde F_{4,\mu\nu\rho\sigma}\tilde F^{\mu\nu\rho\sigma}_4\big] -\frac{1}{4\kappa^2}\int B \wedge F_4 \wedge F_4 : \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{2\kappa^2}\int d^{10}x \sqrt{-g} \bigg[e^{-2\phi}(2 \chi^\mu_1 \partial_\mu \phi - \tfrac{1}{6} H_{\mu\nu\rho} \chi_3^{\mu\nu\rho} - 4 \bar \lambda \gamma^{\mu\nu}D_\mu \psi_\nu) - \tfrac{1}{2}F_{2,\mu\nu} \Psi_2^{\mu\nu} - \tfrac{1}{24}\tilde F_{4,\mu\nu\rho\sigma} \Psi_4^{\mu\nu\rho\sigma} \bigg]. Here H = dB and F_{p+1}=dC_p where p corresponds to a p-form
gauge field. {{refn|group=nb|Sometimes the notation |F_p|^2 = \tfrac{1}{p!} F_{\mu_1\cdots \mu_p}F^{\mu_1\cdots \mu_p} is used to write the canonically normalized
kinetic term for the gauge fields.}} The 3-form gauge field has a modified
field strength tensor \tilde F_4 = F_4 -A_1\wedge F_3 with this having a non-standard
Bianchi identity of d\tilde F_4 = -F_2\wedge F_3. Meanwhile, \chi_1^\mu, \chi_3^{\mu\nu\rho}, \Psi_2^{\mu\nu}, and \Psi_4^{\mu\nu\rho\sigma} are various fermion
bilinears given by{{refn|group=nb|The action and supersymmetry variations depend on the
metric signature used. Transforming from a mainily positive signature, denoted by primes, to a mainly negative one used in this article can be done through g'_{\mu\nu} = - g_{\mu\nu} implying that \gamma'^\mu = i\gamma^\mu, \gamma'_\mu = -i\gamma_\mu, and \gamma'_{*}=-\gamma_{*}. Additionally, the fields are often redefined as {e'}_\mu{}^a=e_\mu{}^a, {\psi'}_\mu=\psi_\mu, \lambda'=i\lambda, B'=-B, C_1'=-C_1, C'_3=C_3. }} : \delta e_\mu{}^a = \bar \epsilon \gamma^a \psi_\mu, : \delta \psi_\mu = (D_\mu + \tfrac{1}{8}H_{\alpha \beta\mu}\gamma^{\alpha \beta}\gamma_*)\epsilon + \tfrac{1}{16}e^\phi F_{\alpha \beta}\gamma^{\alpha \beta}\gamma_\mu \gamma_* \epsilon + \tfrac{1}{192}e^\phi F_{\alpha \beta \gamma \delta}\gamma^{\alpha \beta \gamma \delta}\gamma_\mu \epsilon, : \delta B_{\mu\nu} = 2\bar \epsilon \gamma_{*}\gamma_{[\mu}\psi_{\nu]}, : \delta C_\mu = -e^{-\phi}\bar \epsilon\gamma_* (\psi_\mu - \tfrac{1}{2}\gamma_\mu \lambda), : \delta C_{\mu\nu\rho} = -e^{-\phi}\bar \epsilon\gamma_{[\mu\nu}(3\psi_{\rho]}-\tfrac{1}{2}\gamma_{\rho]}\lambda) + 3 C_{[\mu}\delta B_{\nu \rho]}, : \delta \lambda = ({\partial\!\!\!/} \phi +\tfrac{1}{12}H_{\alpha \beta \gamma}\gamma^{\alpha \beta \gamma}\gamma_*)\epsilon + \tfrac{3}{8}e^\phi F_{\alpha \beta}\gamma^{\alpha \beta}\gamma_* \epsilon + \tfrac{1}{96}e^\phi F_{\alpha \beta \gamma \delta}\gamma^{\alpha \beta \gamma \delta}\epsilon, : \delta \phi = \tfrac{1}{2}\bar \epsilon \lambda. They are useful for constructing the
Killing spinor equations and finding the supersymmetric
ground states of the theory since these require that the fermionic variations vanish. == Related theories ==