Type IIB supergravity

In supersymmetry, type IIB supergravity is the unique theory of 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.^[1][2] It does not admit a fully covariant action due to the presence of a self-dual field although one can write down a pseudo-action which must be supplemented with the self-duality condition. 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.

After supergravity was discovered in 1976, there was a concentrated effort to construct the various possible supergravities which were classified in 1978 by Werner Nahm.^[3] This showed that there exist three types of supergravity in ten dimensions, later termed 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,^[1] and independently by Paul Howe and Peter West.^[2] 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.^[4] The main application of the theory is as the low-energy limit of type IIB strings, and so it plays in important role in string theory, type IIB moduli stabilisation, and the AdS/CFT correspondence.

The field content of type IIB supergravity is given by the ten dimensional ${\displaystyle {\mathcal {N}}=2}$ chiral supermultiplet ${\displaystyle (g_{\mu \nu },C_{4},B_{2},C_{2},C_{0},\psi _{\mu },\lambda ,\phi )}$.^[5] Here ${\displaystyle g_{\mu \nu }}$ is the metric corresponding to the graviton, while ${\displaystyle C_{p}}$ are 4-form, 2-form, and 0-form gauge fields. ${\displaystyle B_{2}}$ is the Kalb–Ramond field while ${\displaystyle \phi }$ is the dilaton.^[6]: 313 There is also a single left-handed Weyl gravitino ${\displaystyle \psi _{\mu }}$, equivalent to two left-handed Majorana-Weyl gravitinos, and a single right-handed Weyl fermion ${\displaystyle \lambda }$, also equivalent to two right-handed Majorana–Weyl fermions.^[7]: 271

The superalgebra for ten-dimensional ${\displaystyle {\mathcal {N}}=(2,0)}$ supersymmetry is given by^[8]

${\displaystyle \{Q_{\alpha }^{i},Q_{\beta }^{j}\}=\delta ^{ij}(P\gamma ^{\mu }C)_{\alpha \beta }P_{\mu }+(P\gamma ^{\mu }C)_{\alpha \beta }{\tilde {Z}}_{\mu }^{ij}+\epsilon ^{ij}(P\gamma ^{\mu \nu \rho }C)_{\alpha \beta }Z_{\mu \nu \rho }}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\delta ^{ij}(P\gamma ^{\mu \nu \rho \sigma \delta }C)_{\alpha \beta }Z_{\mu \nu \rho \sigma \delta }+(P\gamma ^{\mu \nu \rho \sigma \delta }C)_{\alpha \beta }({\tilde {Z}})_{\mu \nu \rho \sigma \delta }^{ij}.}$

Here ${\displaystyle Q_{\alpha }^{i}}$ with ${\displaystyle i=1,2}$ are the two Majorana–Weyl supercharges of the same chirality. Mathematically this is expressed by the condition that ${\displaystyle PQ_{\alpha }^{i}=Q_{\alpha }^{i}}$, where ${\displaystyle P={\tfrac {1}{2}}(1\pm \gamma _{*})}$ is the chirality projection operator, which can be either left or right, depending on the chirality of the supercharges.

The ${\displaystyle M^{ij}}$ matrices allowed are fixed by the fact that states must be representations of the ${\displaystyle {\text{SO}}(2)}$ R-symmetry group of the type IIB theory,^[9]: 240 which only allows for ${\displaystyle \delta ^{ij}}$, ${\displaystyle \epsilon ^{ij}}$ and trace-free symmetric ${\displaystyle Z^{ij}}$. Since the anticommutator is symmetric under an exchange of the spinor and ${\displaystyle i,j}$ indices, then the maximally extended superalgebra is constructed by adding to it all matrices of the same chirality with the same symmetry property. Therefore, it contains the matrices ${\displaystyle P\gamma ^{\mu _{1}\cdots \mu _{p}}C}$ that are symmetric if multiplied by ${\displaystyle {\tilde {Z}}^{ij}}$ or ${\displaystyle \delta ^{ij}}$, and antisymmetric if multiplied by ${\displaystyle \epsilon ^{ij}}$. In ten dimensions ${\displaystyle \gamma ^{\mu _{1}\cdots \mu _{p}}C}$ is symmetric for ${\displaystyle p=1,2}$ mod ${\displaystyle 4}$ and antisymmetric for ${\displaystyle p=3,0}$ mod ${\displaystyle 4}$.^{[9]: 47–48} Since the projection operator ${\displaystyle P}$ is a sum of the identity and a gamma matrix, this means that the symmetric combination works when ${\displaystyle p=1}$ mod ${\displaystyle 4}$ and the antisymmetric one whe ${\displaystyle p=3}$ mod ${\displaystyle 4}$. This yields all the central charges up to Hodge duals of the superalgeba.

The central charges are each associated to various BPS states that are found in the theory. The ${\displaystyle {\tilde {Z}}^{ij}}$ central charges correspond to the fundametnal string and the D1 brane, ${\displaystyle Z_{\mu \nu \rho }}$ is associated with the D3 brane, while ${\displaystyle Z_{\mu \nu \rho \sigma \delta }}$ and ${\displaystyle {\tilde {Z}}_{\mu \nu \rho \sigma \delta }^{ij}}$ give three 5-form charges.^[8] One is the D5-brane, another the NS5-brane, and the last is associated with the KK monopole.

For the supergravity multiplet to have an equal number of bosonic and fermionic degrees of freedom, the four-form ${\displaystyle C_{4}}$ has to have 35 degrees of freedom.^[7]: 271 This is achieved when the field strength tensor is self-dual ${\displaystyle \star {\tilde {F}}_{5}={\tilde {F}}_{5}}$, eliminating half of the degrees of freedom that would otherwise be found in a 4-form gauge field.

This presents a problem when constructing an action since the standard kinetic term for the 5-form self-dual field vanishes.^{[nb 1]} The original way around this was to only work at the level of the equations of motion where self-duality is just another equation of motion. While it is possible to formulate a covariant action with the correct degrees of freedom by introducing an auxiliary field and a compensating gauge symmetry,^[11] the more common approach is to instead work with a pseudo-action where self-duality is imposed as an additional constraint on the equations of motion.^[4] Such an action is not supersymmetric since it does not have an equal number of fermionic and bosonic degrees of freedom. Similarly, type IIB supergravity cannot be acquired by dimensional reduction of theory in higher dimensions.^[12]

The bosonic part of the pseudo-action for type IIB supergravity is given by^[13]: 114

${\displaystyle S_{IIB,{\text{bosonic}}}={\frac {1}{2\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}e^{-2\phi }{\bigg [}R+4\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}|H_{3}|^{2}{\bigg ]}}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -{\frac {1}{4\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}{\big [}|F_{1}|^{2}+|{\tilde {F}}_{3}|^{2}+{\tfrac {1}{2}}|{\tilde {F}}_{5}|^{2}{\big ]}-{\frac {1}{4\kappa ^{2}}}\int C_{4}\wedge H_{3}\wedge F_{3}.}$

Here ${\displaystyle {\tilde {F}}_{3}=F_{3}-C_{0}H_{3}}$ and ${\displaystyle {\tilde {F}}_{5}=F_{5}-{\tfrac {1}{2}}C_{2}\wedge H_{3}+{\tfrac {1}{2}}B_{2}\wedge F_{3}}$ are modified field-strength tensors for the 2-form and 4-form gauge fields.^[14] The resulting Bianchi identity for the 5-form is ${\displaystyle d{\tilde {F}}_{5}=H_{3}\wedge F_{3}}$. The notation employed for the kinetic terms is ${\displaystyle |F_{p}|^{2}={\tfrac {1}{p!}}F_{\mu _{1}\cdots \mu _{p}}F^{\mu _{1}\cdots \mu _{p}}}$. Self-duality ${\displaystyle {\tilde {F}}_{5}=\star {\tilde {F}}_{5}}$ has to be imposed by hand onto the equations of motion, making this a pseudo-action rather than a regular action.

The first line in the action contains the Einstein–Hilbert action, the dilaton kinetic term, and the ${\displaystyle B_{2}}$ field strength tensor. The first term in the second line has the appropriately modified field strength tensors for the three ${\displaystyle C_{p}}$ gauge fields, while the last term is a Chern–Simons term. The action is written in the string frame where the first line consists of the NSNS fields, with these terms being identical to those found in type IIA supergravity. The second integral meanwhile consists of the kinetic term for the RR fields.

Type IIB supergravity has a global noncompact ${\displaystyle SL(2,\mathbb {R} )}$ symmetry.^{[6]: 315–317} This can be made explicit by rewriting the action into the Einstein frame ${\displaystyle g_{\mu \nu }\rightarrow e^{\phi /2}g_{\mu \nu }}$ and defining the axio-dilaton complex scalar field ${\displaystyle \tau =C_{0}+ie^{-\phi }}$. Introducing the matrix

${\displaystyle M_{ij}={\frac {1}{{\text{Im}}\ \tau }}{\begin{pmatrix}|\tau |^{2}&-{\text{Re}}\ \tau \\-{\text{Re}}\ \tau &1\end{pmatrix}}}$

and combining the two 3-form field strength tensors into a doublet ${\displaystyle F_{3}^{i}=(H_{3},F_{3})}$, the action becomes^[15]: 91

${\displaystyle S_{IIB}={\frac {1}{2\kappa ^{2}}}\int d^{10}x{\sqrt {-g}}{\bigg [}R-{\frac {\partial _{\mu }\tau \partial ^{\mu }{\bar {\tau }}}{2({\text{Im}}\ \tau )^{2}}}-{\frac {1}{12}}M_{ij}F_{3,\mu \nu \rho }^{i}F_{3}^{j,\mu \nu \rho }-{\frac {1}{4}}|{\tilde {F}}_{5}|^{2}{\bigg ]}-{\frac {\epsilon _{ij}}{8\kappa ^{2}}}\int C_{4}\wedge F_{3}^{i}\wedge F_{3}^{j}.}$

This action is manifestly invariant under the transformation ${\displaystyle \Lambda \in SL(2,\mathbb {R} )}$ which transforms the 3-forms ${\displaystyle F_{3}^{i}\rightarrow \Lambda ^{i}{}_{j}F_{3}^{j}}$ and the axio-dilaton as

${\displaystyle \tau \rightarrow {\frac {a\tau +b}{c\tau +d}},\ \ \ \ {\text{where}}\ \ \ \Lambda ={\begin{pmatrix}d&c\\b&a\end{pmatrix}}.}$

Both the metric and the self-dual field strength tensor are invariant under these transformations. The invariance of the 3-form field strength tensors follows from the fact that ${\displaystyle M\rightarrow (\Lambda ^{-1})^{T}M\Lambda ^{-1}}$.

The equations of motion acquired from the supergravity action are invariant under the following supersymmetry transformations^[16]

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

${\displaystyle \delta \psi _{\mu }=(D_{\mu }\epsilon -{\tfrac {1}{8}}H_{\alpha \beta \mu }\gamma ^{\alpha \beta }\sigma ^{3})\epsilon +{\tfrac {1}{16}}e^{\phi }\sum _{n=1}^{5}{F\!\!\!/}^{(2n-1)}\gamma _{\mu }{\mathcal {P}}_{n}\epsilon ,}$

${\displaystyle \delta B_{\mu \nu }=2{\bar {\epsilon }}\sigma ^{3}\gamma _{[\mu }\psi _{\nu ]},}$

${\displaystyle \delta C_{\mu _{1},\cdots \mu _{2n-2}}^{(2n-2)}=-(2n-2)e^{-\phi }{\bar {\epsilon }}{\mathcal {P}}_{n}\gamma _{[\mu _{1}\cdots \mu _{2n-3}}(\psi _{\mu _{2n-2}]}-{\tfrac {1}{2(2n-2)}}\gamma _{\mu _{2n-2}]}\lambda )}$

${\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{\tfrac {1}{2}}(2n-2)(2n-3)C_{[\mu _{1}\cdots \mu _{2n-4}}^{(2n-4)}\delta B_{\mu _{2n-3}\mu _{2n-2}]},}$

${\displaystyle \delta \lambda =({\partial \!\!\!/}\phi -{\tfrac {1}{12}}H_{\mu \nu \rho }\gamma ^{\mu \nu \rho }\sigma ^{3})\epsilon +{\tfrac {1}{4}}e^{\phi }\sum _{n=1}^{5}{\frac {n-3}{(2n-1)!}}{F\!\!\!/}^{(2n-1)}{\mathcal {P}}_{n}\epsilon ,}$

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

Here ${\displaystyle F_{\mu _{1}\cdots \mu _{p}}}$ are the field strength tensors associated with the ${\displaystyle C^{(p-1)}}$ gauge fields, including all their magnetic duals for ${\displaystyle p>5}$ while ${\displaystyle {F\!\!\!/}^{(p)}=F_{\mu _{1}\cdots \mu _{p}}\gamma ^{\mu _{1}\cdots \mu _{p}}}$. Additionally, ${\displaystyle {\mathcal {P}}_{n}=\sigma ^{1}}$ when ${\displaystyle n}$ is even and ${\displaystyle {\mathcal {P}}_{n}=i\sigma ^{2}}$ when it is odd.

Type IIB supergravity is the low energy limit of type IIB string theory. The fields in the supergravity are directly related to the different massless states of the string theory. In particular, the metric, ${\displaystyle B_{2}}$ field, and dilaton are NSNS fields, while the three ${\displaystyle C_{p}}$ ${\displaystyle p}$-forms are RR fields. The gravitational coupling constant is related to the Regge slope through ${\displaystyle 2\kappa ^{2}=(2\pi )^{7}\alpha '^{4}}$.^[13]: 114

The global ${\displaystyle {\text{SL}}(2,\mathbb {R} )}$ symmetry of the supergravity is not a symmetry of the full type IIB string theory since it would for example mix ${\displaystyle B_{2}}$ and ${\displaystyle C_{2}}$ fields. This cannot happen since one of these is an NSNS state and the other an RR state, with these having different physics, such as the former coupling to strings while the latter does not.^[15]: 92 The symmetry is instead broken to a discrete subgroup ${\displaystyle {\text{SL}}(2,\mathbb {Z} )\subset {\text{SL}}(2,\mathbb {R} )}$ which is believed to be a symmetry of the full type IIB string theory.

The quantum theory is anomaly free, with the gravitational anomalies cancelling exactly.^[15]: 98 The pseudo-action receives much studied string corrections that are classified into two types. The first are quantum corrections in terms of the string coupling ${\displaystyle g_{s}}$ and the second in terms of the Regge slope ${\displaystyle \alpha '}$. These corrections play an important role in many moduli stabilisation scenarios.

Dimensional reduction of type IIA and type IIB supergravities necessarily results in the same nine-dimensional ${\displaystyle {\mathcal {N}}=2}$ theory since only one superalgebra of this type exists in this dimension.^[17] This is closely linked to the T-duality between the corresponding string theories.