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] 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.
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.
[5] 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.
, equivalent to two left-handed Majorana–Weyl gravitinos, and a single right-handed Weyl fermion
matrices allowed on the right-hand side are fixed by the fact that they must be representations of the
R-symmetry group of the type IIB theory,[10]: 240 which only allows for
indices, the maximally extended superalgebra can only have terms with the same chirality and symmetry property as the anticommutator.
is a sum of the identity and a gamma matrix, this means that the symmetric combination works when
This yields all the central charges found in the superalgebra up to Poincaré duality.
For the supergravity multiplet to have an equal number of bosonic and fermionic degrees of freedom, the four-form
, 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.
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,[12] 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.
[5] Without this constraint the action cannot be supersymmetric since it does not have an equal number of fermionic and bosonic degrees of freedom.
[13] The bosonic part of the pseudo-action for type IIB supergravity is given by[14]: 114 Here
has to be imposed by hand onto the equations of motion, making this a pseudo-action rather than a regular action.
The first term on the second line has the appropriately modified field strength tensors for the three
The first integral on the second line meanwhile consists of the kinetic term for the RR fields.
[7]: 315–317 This can be made explicit by rewriting the action into the Einstein frame
Introducing the matrix and combining the two 3-form field strength tensors into a doublet
and the axio-dilaton as Both the metric and the self-dual field strength tensor are invariant under these transformations.
The equations of motion acquired from the supergravity action are invariant under the following supersymmetry transformations[17] Here
The type IIB pseudo-action can also be reformulated in a way that treats all RR fluxes equally in the so-called democratic formulation.
, with a duality constraint imposed on all of them to get the correct number of degrees of freedom.
which is believed to be a symmetry of the full type IIB string theory.
[16]: 98 In string theory the pseudo-action receives much studied corrections that are classified into two types.
These corrections play an important role in many moduli stabilisation scenarios.
[19] This is closely linked to the T-duality between the corresponding string theories.