Type IIB supergravity Clause Samples
Type IIB supergravity. Type IIB string theory is a theory of supersymmetric closed oriented strings in 10d. It possesses N = (2, 0) supersymmetry, with two Majorana-Weyl supercharges with the same (positive) chirality. The associated spinors parametrising the supersymme- try variations are also Majorana-Weyl with negative chirality. The supersymmetry algebra admits a U (1)R R-symmetry rotating the two supercharges. The bosonic massless sector of the theory contains: the ▇▇▇▇▇-▇▇▇▇▇▇▇-▇▇▇▇▇- ▇▇▇▇▇▇▇ (NSNS) sector containing a metric g, a scalar field called the dilaton Φ and a two-form B; the ▇▇▇▇▇▇-▇▇▇▇▇▇ (RR) sector containing a scalar known as the axion C(0), a two form potential C(2) and a four form potential C(4) with self- dual field strength. The fermionic sector contains two gravitini, Ψ and a dilatini, λ. Due to the self-duality of the field strength of the four-form potential there is no canonical covariant Lagrangian formulation of the theory. The two scalars of the theory parametrise a scalar manifold, M which must have holonomy containing the U (1)R automorphism group, see for example [6]. Moreover the scalar manifold must be negatively curved and locally isometric to a symmetric manifold, this leaves a single choice for a simply connected manifold, namely the upper-half plane M = SL(2, R)/U (1)R (1.1) equipped with the SL(2, R) invariant metric ds2 = dτ dτ¯ (Im [τ ])2
Type IIB supergravity s The other decomposition of the E11 algebra, found by deleting node 10 as shown in figure 7, leads to an SL(10) gravity line that describes the IIB theory. One immediately sees that, in addition to yielding an A9 subalgebra, the deletion of node 10 in figure 7 gives an A1 subalgebra that corresponds to the expected SL(2, R) symmetry of type IIB supergravity. A derivation of the simple roots and fundamental weights relative to the A9 subalgebra found by deleting node 10 is given in appendix D.3. The low level generators of E11 decomposed with respect to the A9 and A1 subalgebras relevant to type IIB supergravity are Kab and the set of generators Rs, Ra1a2 , Ra1a2a3a4 , Ra1a2...a6 , Ra1a2...a8 , where s = 1, 2 is an SL(2) index and the A9 indices a, b, a1,... a8, 2 s s range from 1 to 10. Figure 7: The E11 Dynkin diagram appropriate to type IIB supergravity The E11 Chevalley generators with respect to the type IIB supergravity decomposition are [67] Ea = Kaa+1, for a = 1, 2, ..., 8, (4.55) 1 E9 = R9 10, (4.56) E10 = R2, (4.57) E11 = K910, (4.58) while the corresponding Cartan subalgebra is given by a Ha = Ka
a+1 K
Type IIB supergravity. Unlike the d = 10 type IIA supergravity theory, one cannot obtain d = 10 type IIB supergravity by dimensional reduction of eleven dimensional supergravity. However, in deriving the field content of type IIA supergravity, in the previous section, we constructed an N = 2 supermultiplet by taking a direct product of N = 1 supermultiplets containing spinor representations of opposite chirality, if we choose to construct the d = 10, N = 2 supergravity theory by taking the direct product of two N = 1 supermultiplets containing spinor representations of the same chirality we obtain type IIB supergravity. The direct product of 2 multiplets of the form 8v + 8c decomposes as (8v + 8c) × (8v + 8c) = bIIB + fIIB where bIIB = 1 + 1 + 28 + 28 + 35v + 35c (2.9) and fIIB = 8s + 8s + 56s + 56s. (2.10) 4!4! The bosonic type IIB supergravity fields lie in representations contained in bIIB while their fermionic counterparts lie in representations contained in fIIB. In the bosonic sector of the type IIB supergravity theory, one may identify the two fields lying in the trivial representations of SO(8) as the type IIB supergravity dilaton and axion, the propagating degrees of freedom of the graviton are again contained in the 35v, while the two 28 representations are attributable to two 2-form gauge fields. The bosonic field corresponding to the 35c are realised by a four-form gauge field, however, a four-form gauge field in ten dimensions has 8! = 70 independent components, therefore to give the correct propagating degrees of freedom a self duality condition must be imposed on the five-form field strength constructed out of the four-form gauge field. Moreover, the self duality condition must be imposed on the five-form field strength at the level of the equations of motion, rather than on the five-form field strength in the type IIB supergravity action. The fermionic type IIB supergravity fields lie in spinor representations of SO(8) contained in fIIA. For the type IIB supergravity theory there are two gravitinos with the same chirality, each transforming in the 56s of SO(8), and two dilatinos of the same chirality transforming in the 8s of The bosonic and fermionic sectors of type IIB supergravity both contain massless fields that transform under representations of SO(8) that are degenerate, for the bosonic sector we have two copies of the trivial representation and the 28 of SO(8), while in the fermionic sector we find two copies of the 56s and 8s spinor representations of ...