6.8 Type IIB Supergravity |
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D Λ+α |
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6.8.3The Bosonic Field Equations and the Standard Form of the Bosonic Action
Following Castellani and Pesando we write next the general form of the bosonic field equations and using the identifications of (6.8.25), (6.8.3), (6.8.7) we reduce them to those following from a standard supergravity action for p-branes. As discussed in the literature [24, 26, 27], the standard form of a supergravity action truncated to the graviton, the dilaton and the ni = pi + 2 field strengths that can couple to the world-volume actions of pi -branes is as follows:
Astandard = |
dD x det V −2R ω(V ) − |
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∂μϕ∂μϕ |
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where R = R |
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adopted in the rheonomic framework [17],13 and ai are characteristic exponents dictated by the structure of supergravity and playing an essential role in dictating the properties of p-brane solutions.14 Furthermore in (6.8.34) we have defined:
|F[n]|2 |
≡ gμ1ν1 · · · gμnνn Fμ1...μn Fν1...νn |
(6.8.35) |
F[n] |
= Fμ1...μn dxμ1 · · · dxμn |
(6.8.36) |
and we have not made explicit the Chern Simons couplings between field strengths that are on the other hand essential in the derivation of the exact field equations.
Introducing the definition of the dressed 3-form field strengths:
H=±|a1a2a3 = εαβ Λ±α Ha1a2a3 ; H7A|a1a2a3 = εΛΣ LΛAHaΣ1a2a3 |
(6.8.37) |
13Note that our R is equal to − 12 Rold , Rold being the normalization of the scalar curvature usually adopted in General Relativity textbooks. The difference arises because in the traditional literature the Riemann tensor is not defined as the components of the curvature 2-form Rab rather as −2 times such components.
14In the next Chap. 7 we will emphasize the role of the dilaton factors exp[−aϕ] in front of the p-form kinetic terms.
260 |
6 Supergravity: The Principles |
it was shown by Castellani and Pesando [34] that the exact bosonic field equations implied by the closure of the supersymmetry algebra have the following form:
R |
qr − 2 |
δq Rabab = −75 Fqa1−a4 F pa1−a4 − |
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δq Fa1−a5 F a1−a5 |
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16 H pa1a2 H−|qa1a2 |
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3 δqpH a1a2a3 H−|a1a2a3 |
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P p Pq + Pq P p − δqp P a Pa |
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D a Pa |
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H a1a2a3 H a a2a3 |
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H−|a1a2b |
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H+|a1a2b = −i20Fa1a2b1b2b3 H |
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εa1a2a3a4b1...b6 H=+ |
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At the purely bosonic level (i.e. disregarding all fermionic contributions), using the solvable parameterization (6.8.3) of the SL(2, R)/O(2) coset and inserting the identifications (6.8.25) we obtain the following expression for the dressed 3-forms in terms of string massless fields denoted NS or RR according to their origin in the Neveu Schwarz or Ramond Ramond sector:
H |
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2e−ϕ/2F NS |
i2eϕ/2F RR |
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F[RR5] |
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Using the Hodge dual of -forms in space-time dimensions D, the field equations (6.8.39)–(6.8.41) can be written in a more compact form. Let us begin with the scalar equation (6.8.39), it becomes:
d( P ) − 2iQ P + |
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(6.8.43) |