In supersymmetry, type IIA supergravity is the unique supergravity in ten dimensions with two supercharges of opposite chirality. It was first constructed in 1984 by a dimensional reduction of eleven-dimensional supergravity on a circle.[1] [2] [3] The other supergravities in ten dimensions are type IIB supergravity, which has two supercharges of the same chirality, and type I supergravity, which has a single supercharge. In 1986 a deformation of the theory was discovered which gives mass to one of the fields and is known as massive type IIA supergravity.[4] Type IIA supergravity plays a very important role in string theory as it is the low-energy limit of type IIA string theory.
After supergravity was discovered in 1976 with pure 4D lN=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]
Ten dimensions admits both
lN=1
lN=2
Q\pm
lN=2
lN=(1,1)
This theory contains a single multiplet, known as the ten-dimensional
lN=2
(g\mu\nu,C\mu\nu\rho,B\mu\nu,C\mu,\psi\mu,λ,\phi)
g\mu\nu
\psi\mu
λ
\psi\mu=
| - | |
| \psi | |
| \mu |
λ=λ++λ-
\phi
This nonchiral multiplet can be decomposed into the ten-dimensional
lN=1
(g\mu\nu,B\mu\nu,
| + | |
| \psi | |
| \mu, |
λ-,\phi)
(C\mu\nu\rho,C\mu,
| -, | |
| \psi | |
| \mu |
λ+)
The superalgebra for
lN=(1,1)
\{Q\alpha,Q\beta\}=(\gamma\muC)\alphaP\mu+(\gamma*C)\alphaZ+(\gamma\mu\gamma*C)\alphaZ\mu+(\gamma\mu\nuC)\alphaZ\mu\nu
+(\gamma\mu\nu\rho\sigma\gamma*C)\alphaZ\mu\nu\rho\sigma+(\gamma\mu\nu\rho\sigma\deltaC)\alphaZ\mu\nu\rho\sigma,
where all terms on the right-hand side besides the first one are the central charges allowed by the theory. Here
Q\alpha
C
\alpha
\beta
| \mu1 … \mup | |
| \gamma |
C
p=1,2
4
\gamma*
\gamma
The various central charges in the algebra correspond to different BPS states allowed by the theory. In particular, the
Z
Z\mu\nu
Z\mu\nu\rho\sigma
Z\mu
Z\mu\nu\rho\sigma\delta
The type IIA supergravity action is given up to four-fermion terms by[11]
SIIA,bosonic=
| 1 | |
| 2\kappa2 |
\intd10x\sqrt{-g}e-2\phi[R+4\partial\mu \phi\partial\mu\phi-
| 1 | |
| 12 |
H\mu\nu\rhoH\mu\nu\rho-2\bar\psi\mu\gamma\mu\nu\rhoD\nu\psi\rho+2\barλ\gamma\muD\muλ]
| - | 1 |
| 4\kappa2 |
\intd10x\sqrt{-g}[\tfrac{1}{2}F2,\mu\nu
| \mu\nu | |
| F | |
| 2+\tfrac{1}{24}\tilde |
F4,\mu\nu\rho\sigma\tilde
| \mu\nu\rho\sigma | ||||
| F | ||||
|
\intB\wedgeF4\wedgeF4
| + | 1 |
| 2\kappa2 |
\intd10x\sqrt{-g}[e-2\phi(2
| \mu | |
| \chi | |
| 1 |
\partial\mu\phi-\tfrac{1}{6}H\mu\nu\rho
| \mu\nu\rho | |
| \chi | |
| 3 |
-4\barλ\gamma\mu\nuD\mu\psi\nu)-\tfrac{1}{2}F2,\mu\nu
| \mu\nu | |
| \Psi | |
| 2 |
-\tfrac{1}{24}\tildeF4,\mu\nu\rho\sigma
| \mu\nu\rho\sigma | |
| \Psi | |
| 4 |
].
H=dB
Fp+1=dCp
p
p
\tildeF4=F4-A1\wedgeF3
d\tildeF4=-F2\wedgeF3
| \mu | |
| \chi | |
| 1 |
| \mu\nu\rho | |
| \chi | |
| 3 |
| \mu\nu | |
| \Psi | |
| 2 |
| \mu\nu\rho\sigma | |
| \Psi | |
| 4 |
| \mu | |
| \chi | |
| 1 |
=-2\bar\psi\nu\gamma\nu\psi\mu-2\barλ\gamma\nu\gamma\mu\psi\nu,
| \mu\nu\rho | |
| \chi | |
| 3 |
=\tfrac{1}{2}\bar\psi\alpha\gamma[\alpha\gamma\mu\nu\rho\gamma\beta]\gamma*\psi\beta+\barλ\gamma\mu\nu\rho{}\beta\gamma*\psi\beta-\tfrac{1}{2}\barλ\gamma*\gamma\mu\nu\rhoλ,
| \mu\nu | |
| \Psi | |
| 2 |
=\tfrac{1}{2}e-\phi\bar
| \alpha\gamma | |
| \psi | |
| [\alpha |
\gamma\mu\nu\gamma\beta]
| \beta | |
| \gamma | |
| *\psi |
+\tfrac{1}{2}e-\phi\barλ\gamma\mu\nu\gamma\beta
| \beta | |
| \gamma | |
| *\psi |
+\tfrac{1}{4}e-\phi\barλ\gamma\mu\nu\gamma*λ,
| \mu\nu\rho\sigma | |
| \Psi | |
| 4 |
=\tfrac{1}{2}e-\phi\bar
| \alpha\gamma | |
| \psi | |
| [\alpha |
\gamma\mu\nu\rho\sigma\gamma\beta]\psi\beta+\tfrac{1}{2}e-\phi\barλ\gamma\mu\nu\rho\sigma
| \beta | |
| \gamma | |
| \beta\psi |
-\tfrac{1}{4}e-\phi\barλ\gamma\mu\nu\rho\sigmaλ.
The first line of the action has the Einstein–Hilbert action, the dilaton kinetic term, the 2-form
B\mu\nu
\psi\mu
λ
The supersymmetry variations that leave the action invariant are given up to three-fermion terms by[11] [13]
\delta
| a | |
| e | |
| \mu{} |
=\bar\epsilon\gammaa\psi\mu,
\delta\psi\mu=(D\mu+\tfrac{1}{8}H\alpha\gamma\alpha\gamma*)\epsilon+\tfrac{1}{16}e\phiF\alpha\gamma\alpha\gamma\mu\gamma*\epsilon+\tfrac{1}{192}e\phiF\alpha\gamma\alpha\gamma\mu\epsilon,
\deltaB\mu\nu=2\bar\epsilon\gamma*\gamma[\mu\psi\nu],
\deltaC\mu=-e-\phi\bar\epsilon\gamma*(\psi\mu-\tfrac{1}{2}\gamma\muλ),
\deltaC\mu\nu\rho=-e-\phi\bar\epsilon\gamma[\mu\nu(3\psi\rho]-\tfrac{1}{2}\gamma\rho]λ)+3C[\mu\deltaB\nu,
\deltaλ=({\partial/}\phi+\tfrac{1}{12}H\alpha\gamma\alpha\gamma*)\epsilon+\tfrac{3}{8}e\phiF\alpha\gamma\alpha\gamma*\epsilon+\tfrac{1}{96}e\phiF\alpha\gamma\alpha\epsilon,
\delta\phi=\tfrac{1}{2}\bar\epsilonλ.
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.
Since type IIA supergravity has p-form field strengths of even dimensions, it also admits a nine-form gauge field
F10=dC9
\starF10
d\starF10=0
SmassiveIIA=\tildeSIIA-
| 1 | |
| 4\kappa2 |
\intd10x\sqrt{-g}M2+
| 1 | |
| 2\kappa2 |
\intMF10,
where
\tildeSIIA
F2 → F2+MB
F4 → F4+\tfrac{1}{2}MB\wedgeB
M
F10
M
V(\phi)=0
lN=2
Compactification of eleven-dimensional supergravity on a circle and keeping only the zero Fourier modes that are independent of the compact coordinates results in type IIA supergravity. For eleven-dimensional supergravity with the graviton, gravitino, and a 3-form gauge field denoted by
(gMN',\psiM',AMNR')
g'MN=e-2\phi/3\begin{pmatrix}g\mu\nu+e2\phiC\muC\nu&-e2\phiC\mu\ -e2\phiC\nu&e2\phi\end{pmatrix}.
Meanwhile, the 11D 3-form decomposes into the 10D 3-form
A\mu\nu\rho' → C\mu\nu\rho
A\mu\nu11' → B\mu\nu
\tildeF4
F'\mu\nu\rho\sigma=e4\phi/3\tildeF\mu\nu\rho\sigma
Dimensional reduction of the fermions must generally be done in terms of the flat coordinates
\psiA'=
| M\psi | |
| e | |
| M |
| M | |
| {e'} | |
| A |
\psiA'\sim(\psia,λ)
\psia'=e\phi/6(2\psia-\tfrac{1}{3}\gammaaλ), \psi11'=\tfrac{2}{3}e\phi/6\gamma*λ,
where this is chosen to make the supersymmetry transformations simpler. The ten-dimensional supersymmetry variations can also be directly acquired from the eleven-dimensional ones by setting
\epsilon'=e-\phi/6\epsilon
The low-energy effective field theory of type IIA string theory is given by type IIA supergravity.[14] The fields correspond to the different massless excitations of the string, with the metric, 2-form
B
gs
\alpha'
gs
e\phi
ls=\sqrt{\alpha'}
2\kappa2=(2\pi)7{\alpha'}4
When string theory is compactified to acquire four-dimensional theories, this is often done at the level of the low-energy supergravity. Reduction of type IIA on a Calabi–Yau manifold yields an
lN=2
lN=1