Type IIA supergravity

After supergravity was discovered in 1976 with pure 4D ${\displaystyle {\mathcal {N}}=1}$ supergravity, significant effort was devoted to understanding other possible supergravities that can exist with various numbers of supercharges and in various dimensions. The discovery of eleven-dimensional supergravity in 1978 led to the derivation of many lower dimensional supergravities through dimensional reduction of this theory.^[5] Using this technique, type IIA supergravity was first constructed in 1984 by three different groups, by F. Giani and M. Pernici,^[1] by I.C.G. Campbell and P. West,^[2] and by M. Huq and M. A. Namazie.^[3] In 1986 it was noticed by L. Romans that there exists a massive deformation of the theory.^[4] Type IIA supergravity has since been extensively used to study the low-energy behaviour of type IIA string theory. The terminology of type IIA, type IIB, and type I was coined by J. Schwarz, originally to refer to the three string theories that were known of in 1982.^[6]

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Ten dimensions admits both ${\displaystyle {\mathcal {N}}=1}$ and ${\displaystyle {\mathcal {N}}=2}$ supergravity, depending on whether there are one or two supercharges.^{[nb 1]} Since the smallest spinorial representations in ten dimensions are Majorana–Weyl spinors, the supercharges come in two types ${\displaystyle Q^{\pm }}$ depending on their chirality, giving three possible supergravity theories.^[7]: 241 The ${\displaystyle {\mathcal {N}}=2}$ theory formed using two supercharges of opposite chiralities is denoted by ${\displaystyle {\mathcal {N}}=(1,1)}$ and is known as type IIA supergravity.

This theory contains a single multiplet, known as the ten-dimensional ${\displaystyle {\mathcal {N}}=2}$ nonchiral multiplet. The fields in this multiplet are ${\displaystyle (g_{\mu \nu },C_{\mu \nu \rho },B_{\mu \nu },C_{\mu },\psi _{\mu },\lambda ,\phi )}$, where ${\displaystyle g_{\mu \nu }}$ 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.^[8] There is also a Majorana gravitino ${\displaystyle \psi _{\mu }}$ and a Majorana spinor ${\displaystyle \lambda }$, both of which decompose into a pair of Majorana–Weyl spinors of opposite chiralities ${\displaystyle \psi _{\mu }=\psi _{\mu }^{+}+\psi _{\mu }^{-}}$ and ${\displaystyle \lambda =\lambda ^{+}+\lambda ^{-}}$. Lastly, there a scalar field ${\displaystyle \phi }$.

This nonchiral multiplet can be decomposed into the ten-dimensional ${\displaystyle {\mathcal {N}}=1}$ multiplet ${\displaystyle (g_{\mu \nu },B_{\mu \nu },\psi _{\mu }^{+},\lambda ^{-},\phi )}$, along with four additional fields ${\displaystyle (C_{\mu \nu \rho },C_{\mu },\psi _{\mu }^{-},\lambda ^{+})}$.^{[9]: 269 [nb 2]} 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.

The superalgebra for ${\displaystyle {\mathcal {N}}=(1,1)}$ supersymmetry is given by^[10]

${\displaystyle \{Q_{\alpha },Q_{\beta }\}=(\gamma ^{\mu }C)_{\alpha \beta }P_{\mu }+(\gamma _{*}C)_{\alpha \beta }Z+(\gamma ^{\mu }\gamma _{*}C)_{\alpha \beta }Z_{\mu }+(\gamma ^{\mu \nu }C)_{\alpha \beta }Z_{\mu \nu }}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +(\gamma ^{\mu \nu \rho \sigma }\gamma _{*}C)_{\alpha \beta }Z_{\mu \nu \rho \sigma }+(\gamma ^{\mu \nu \rho \sigma \delta }C)_{\alpha \beta }Z_{\mu \nu \rho \sigma \delta },}$

where all terms on the right-hand side besides the first one are the central charges allowed by the theory. Here ${\displaystyle Q_{\alpha }}$ are the spinor components of the Majorana supercharges^{[nb 3]} while ${\displaystyle C}$ 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 ${\displaystyle \alpha }$, ${\displaystyle \beta }$. In ten dimensions ${\displaystyle \gamma ^{\mu _{1}\cdots \mu _{p}}C}$ is symmetric only for ${\displaystyle p=1,2}$ modulo ${\displaystyle 4}$, with the chirality matrix ${\displaystyle \gamma _{*}}$ behaving as just another ${\displaystyle \gamma }$ matrix, except with no index.^{[7]: 47–48} Going only up to five-index matrices, since the rest are equivalent up to Poincare duality, yields the set of central charges described by the above algebra.

The various central charges in the algebra correspond to different BPS states allowed by the theory. In particular, the ${\displaystyle Z}$, ${\displaystyle Z_{\mu \nu }}$ and ${\displaystyle Z_{\mu \nu \rho \sigma }}$ correspond to the D0, D2, and D4 branes.^[10] The ${\displaystyle Z_{\mu }}$ corresponds to the NSNS 1-brane, which is equivalent to the fundamental string, while ${\displaystyle Z_{\mu \nu \rho \sigma \delta }}$ corresponds to the NS5-brane.

The type IIA supergravity action is given up to four-fermion terms by^[11]

${\displaystyle S_{IIA,{\text{bosonic}}}={\frac {1}{2\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}e^{-2\phi }{\bigg [}R+4\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{12}}H_{\mu \nu \rho }H^{\mu \nu \rho }-2{\bar {\psi }}_{\mu }\gamma ^{\mu \nu \rho }D_{\nu }\psi _{\rho }+2{\bar {\lambda }}\gamma ^{\mu }D_{\mu }\lambda {\bigg ]}}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -{\frac {1}{4\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}{\big [}{\tfrac {1}{2}}F_{2,\mu \nu }F_{2}^{\mu \nu }+{\tfrac {1}{24}}{\tilde {F}}_{4,\mu \nu \rho \sigma }{\tilde {F}}_{4}^{\mu \nu \rho \sigma }{\big ]}-{\frac {1}{4\kappa ^{2}}}\int B\wedge F_{4}\wedge F_{4}}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{\frac {1}{2\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}{\bigg [}e^{-2\phi }(2\chi _{1}^{\mu }\partial _{\mu }\phi -{\tfrac {1}{6}}H_{\mu \nu \rho }\chi _{3}^{\mu \nu \rho }-4{\bar {\lambda }}\gamma ^{\mu \nu }D_{\mu }\psi _{\nu })-{\tfrac {1}{2}}F_{2,\mu \nu }\Psi _{2}^{\mu \nu }-{\tfrac {1}{24}}{\tilde {F}}_{4,\mu \nu \rho \sigma }\Psi _{4}^{\mu \nu \rho \sigma }{\bigg ]}.}$

Here ${\displaystyle H=dB}$ and ${\displaystyle F_{p+1}=dC_{p}}$ where ${\displaystyle p}$ corresponds to a ${\displaystyle p}$-form gauge field. ^{[nb 4]} The 3-form gauge field has a modified field strength tensor ${\displaystyle {\tilde {F}}_{4}=F_{4}-A_{1}\wedge F_{3}}$ with this having a non-standard Bianchi identity of ${\displaystyle d{\tilde {F}}_{4}=-F_{2}\wedge F_{3}}$.^{[12]: 115 [nb 5]} Meanwhile, ${\displaystyle \chi _{1}^{\mu }}$, ${\displaystyle \chi _{3}^{\mu \nu \rho }}$, ${\displaystyle \Psi _{2}^{\mu \nu }}$, and ${\displaystyle \Psi _{4}^{\mu \nu \rho \sigma }}$ are various fermion bilinears given by^[11]

${\displaystyle \chi _{1}^{\mu }=-2{\bar {\psi }}_{\nu }\gamma ^{\nu }\psi ^{\mu }-2{\bar {\lambda }}\gamma ^{\nu }\gamma ^{\mu }\psi _{\nu },}$

${\displaystyle \chi _{3}^{\mu \nu \rho }={\tfrac {1}{2}}{\bar {\psi }}^{\alpha }\gamma _{[\alpha }\gamma ^{\mu \nu \rho }\gamma _{\beta ]}\gamma _{*}\psi ^{\beta }+{\bar {\lambda }}\gamma ^{\mu \nu \rho }{}_{\beta }\gamma _{*}\psi ^{\beta }-{\tfrac {1}{2}}{\bar {\lambda }}\gamma _{*}\gamma ^{\mu \nu \rho }\lambda ,}$

${\displaystyle \Psi _{2}^{\mu \nu }={\tfrac {1}{2}}e^{-\phi }{\bar {\psi }}^{\alpha }\gamma _{[\alpha }\gamma ^{\mu \nu }\gamma _{\beta ]}\gamma _{*}\psi ^{\beta }+{\tfrac {1}{2}}e^{-\phi }{\bar {\lambda }}\gamma ^{\mu \nu }\gamma _{\beta }\gamma _{*}\psi ^{\beta }+{\tfrac {1}{4}}e^{-\phi }{\bar {\lambda }}\gamma ^{\mu \nu }\gamma _{*}\lambda ,}$

${\displaystyle \Psi _{4}^{\mu \nu \rho \sigma }={\tfrac {1}{2}}e^{-\phi }{\bar {\psi }}^{\alpha }\gamma _{[\alpha }\gamma ^{\mu \nu \rho \sigma }\gamma _{\beta ]}\psi ^{\beta }+{\tfrac {1}{2}}e^{-\phi }{\bar {\lambda }}\gamma ^{\mu \nu \rho \sigma }\gamma _{\beta }\psi ^{\beta }-{\tfrac {1}{4}}e^{-\phi }{\bar {\lambda }}\gamma ^{\mu \nu \rho \sigma }\lambda .}$

The first line of the action has the Einstein–Hilbert action, the dilaton kinetic term^{[nb 6]}, the 2-form ${\displaystyle B_{\mu \nu }}$ field strength tensor. It also contains the kinetic terms for the gravitino ${\displaystyle \psi _{\mu }}$ and spinor ${\displaystyle \lambda }$, described by the Rarita–Schwinger action and Dirac action, respectively. The second line has the kinetic terms for the 1-form and 3-form gauge fields as well as a Chern–Simons term. The last line contains the cubic interaction terms between two fermions and a boson.

The supersymmetry variations that leave the action invariant are given up to three-fermion terms by^{[11][14]: 665 [nb 7]}

${\displaystyle \delta e_{\mu }{}^{a}={\bar {\epsilon }}\gamma ^{a}\psi _{\mu },}$

${\displaystyle \delta \psi _{\mu }=(D_{\mu }+{\tfrac {1}{8}}H_{\alpha \beta \mu }\gamma ^{\alpha \beta }\gamma _{*})\epsilon +{\tfrac {1}{16}}e^{\phi }F_{\alpha \beta }\gamma ^{\alpha \beta }\gamma _{\mu }\gamma _{*}\epsilon +{\tfrac {1}{192}}e^{\phi }F_{\alpha \beta \gamma \delta }\gamma ^{\alpha \beta \gamma \delta }\gamma _{\mu }\epsilon ,}$

${\displaystyle \delta B_{\mu \nu }=2{\bar {\epsilon }}\gamma _{*}\gamma _{[\mu }\psi _{\nu ]},}$

${\displaystyle \delta C_{\mu }=-e^{-\phi }{\bar {\epsilon }}\gamma _{*}(\psi _{\mu }-{\tfrac {1}{2}}\gamma _{\mu }\lambda ),}$

${\displaystyle \delta C_{\mu \nu \rho }=-e^{-\phi }{\bar {\epsilon }}\gamma _{[\mu \nu }(3\psi _{\rho ]}-{\tfrac {1}{2}}\gamma _{\rho ]}\lambda )+3C_{[\mu }\delta B_{\nu \rho ]},}$

${\displaystyle \delta \lambda =({\partial \!\!\!/}\phi +{\tfrac {1}{12}}H_{\alpha \beta \gamma }\gamma ^{\alpha \beta \gamma }\gamma _{*})\epsilon +{\tfrac {3}{8}}e^{\phi }F_{\alpha \beta }\gamma ^{\alpha \beta }\gamma _{*}\epsilon +{\tfrac {1}{96}}e^{\phi }F_{\alpha \beta \gamma \delta }\gamma ^{\alpha \beta \gamma \delta }\epsilon ,}$

${\displaystyle \delta \phi ={\tfrac {1}{2}}{\bar {\epsilon }}\lambda .}$

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.