Type IIB Supergravity

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 $\mathcal N=1$ and $\mathcal N=2$ 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 $\mathcal N=(2,0)$ supergravity. The field content of this theory is given by the ten dimensional $\mathcal N=2$ chiral supermultiplet $(g_, B, C_4, C_2, C_0, \psi_\mu, \lambda, \phi)$. Here $g_$ is the metric corresponding to the graviton, while $C_p$ are 4-form, 2-form, and 0-form gauge fields. Meanwhile, $B$ is the Kalb–Ramond field and $\phi$ is the dilaton. There is also a single left-handed Weyl gravitino $\psi_\mu$, equivalent to two left-handed Majorana–Weyl gravitinos, and a single right-handed Weyl fermion $\lambda$, also equivalent to two right-handed Majorana–Weyl fermions.

Algebra

The superalgebra for ten-dimensional $\mathcal N=(2,0)$ supersymmetry is given by

:$\ = \delta^(P\gamma^\mu C)_P_\mu + (P\gamma^\mu C)_\tilde Z_\mu^ + \epsilon^(P\gamma^C)_Z_$
:$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \delta^(P\gamma^C)_Z_ + (P\gamma^C)_(\tilde Z)^_.$
Here $Q^i_\alpha$ with $i=1,2$ are the two Majorana–Weyl supercharges of the same chirality. They therefore satisfy the projection relation $PQ^i_\alpha = Q^i_\alpha$ where $P = \tfrac(1 - \gamma_*)$ is the left-handed chirality projection operator and $\gamma_*$ is the ten-dimensional chirality matrix.

The $ij$ matrices allowed on the right-hand side are fixed by the fact that they must be representations of the $\text(2)$ R-symmetry group of the type IIB theory, which only allows for $\delta^$, $\epsilon^$ and trace-free symmetric matrices $Z^$. Since the anticommutator is symmetric under an exchange of the spinor and $i,j$ indices, the maximally extended superalgebra can only have terms with the same chirality and symmetry property as the anticommutator. The terms are therefore a product of one of the $ij$ matrices with $P\gamma^C$, where $C$ is the charge conjugation operator. In particular, when the spinor matrix is symmetric, it multiplies $\tilde Z^$ or $\delta^$ while when it is antisymmetric it multiplies $\epsilon^$. In ten dimensions $\gamma^C$ is symmetric for $p=1,2$ modulo $4$ and antisymmetric for $p=3,0$ modulo $4$. Since the projection operator $P$ is a sum of the identity and a gamma matrix, this means that the symmetric combination works when $p = 1$ modulo $4$ and the antisymmetric one when $p=3$ modulo $4$. This yields all the central charges found in the superalgebra up to Poincaré duality.

The central charges are each associated to various BPS states that are found in the theory. The $\tilde Z^$ central charges correspond to the fundamental string and the D1 brane, $Z_$ is associated with the D3 brane, while $Z_$ and $\tilde Z^_$ give three 5-form charges. One is the D5-brane, another the NS5-brane, and the last is associated with the KK monopole.

Self-dual field

For the supergravity multiplet to have an equal number of bosonic and fermionic degrees of freedom, the four-form $C_4$ has to have 35 degrees of freedom. This is achieved when the corresponding field strength tensor is self-dual $\star \tilde F_5= \tilde F_5$, 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 kinetic term for the self-dual 5-form field vanishes. 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, 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. Without this constraint the action cannot be supersymmetric since it does not have an equal number of fermionic and bosonic degrees of freedom. Unlike type IIA supergravity, type IIB supergravity cannot be acquired as a dimensional reduction of a theory in higher dimensions.

Pseudo-action

The bosonic part of the pseudo-action for type IIB supergravity is given by

:
S_ = \frac \int d^ x \sqrt e^\bigg H, ^2\bigg
:
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac\int d^x \sqrt\big Bianchi identity