TK-NOTE/07-04 since: July 27, 2007 last update: 2014-06-21 13:31 Supergravities in ten and eleven dimensions — as a dictionary — Tetsuji KIMURA Yukawa Institute for Theoretical Physics, Kyoto University Sakyo-ku, Kyoto 606-8502, Japan [email protected] Abstract In this note we start eleven-dimensional supergravity which explicitly contains fermionic terms. We connect it to the type IIA supergravity action, and perform T-duality to obtain the type IIB supergravity. Further we also study the type I and heterotic supergravity actions. In this note we impose the self-dual condition on the Ramond-Ramond five-form field strength F5 by hand in type IIB supergravity. 1 Strategy We start from the eleven-dimensional supergravity. Along various kinds of duality transforma- tions, we connect five significant supergravity Lagrangian including fermions. We also inves- tigate local supersymmetry transformations in each theory. The reason is that, unfortunately, we only find the bosonic parts of supergravity theories while the local supersymmetry transformation rules are explicitly introduced. This situation is terrible to consider the Killing spinor equations and equations of motion, as well as the Bianchi identity in the presense of fermions. To comple- ment such a difficulty, we will derive the supergravity Lagrangians including fermions and their interactions, explicitly. It might be quite useful to fix the convention of fields and transformations for our future work on the investigation of duality transformations in terms of Killing spinors and spinorial geometry. 2 Contents 1 Strategy 2 2 Eleven-dimensional supergravity 5 3 Type IIA supergravity in ten dimensions 8 3.1 Duality transformation rules . 8 4 Type IIB supergravity in ten dimensions 9 4.1 Duality transformation rules . 9 5 Type I supergravity in ten dimensions 10 6 Heterotic supergravity in ten dimensions 11 7 Heterotic supergravity in ten dimensions, II 12 0 7.1 Lagrangian with higher-order corrections in α ...................... 12 7.2 Local supersymmetry variations . 13 A Convention 17 A.1 Contraction rule on antisymmetric tensors . 17 A.2 Antisymmetrized symbol . 17 A.3 Lorentz algebra . 17 A.4 Dirac conjugate and charge conjugate on spinors . 18 A.5 Covariant derivatives and curvature tensors . 19 A.6 Riemann tensor of Levi-Civita connection . 22 A.7 Contorsion . 22 A.8 Bismut torsion . 24 A.9 Differential forms . 25 A.10 Yang-Mills gauge fields: hermitian variables . 27 3 A.11 Yang-Mills gauge fields: anti-hermitian variables . 28 A.12 Chern-Simons three-forms . 28 B Conventions 30 C Geometries: complex, hermitian and K¨ahlermanifolds 31 C.1 Complex coordinates . 31 C.2 Metric on complex manifold: hermitian metric . 31 C.3 General coordinate transformations of connections . 33 C.4 Further analysis on hermitian manifold . 35 C.5 Additional constraint: Kahler¨ manifold . 40 D Introducing torsion 43 4 2 Eleven-dimensional supergravity Here let us summarize the convantion which appears in the note [4, 7]: Z S = d11x L ; (2.1a) p 1p 1 p 2κ2 L = β −g R(e; !) − −g Ψ ΓMNP D [ 1 (! + !b)]Ψ − −g F F MNPQ 11 0 11 2 11 M N 2 P 48 11 MNPQ 1 p + β −g Ψ ΓeMNP QRSΨ (F + Fb) 2 1 11 M N P QRS MNP QRSUV W XY + β2 "^ FMNPQ FRSUV CWXY ; (2.1b) b where !bMAB and FMNPQ are the supercovariantizations of !MAB and FMNPQ, respectively [5, 4]. The supercovariantization means that its supersymmetry variation does not contain derivatives of the supersymmetry parameter. However, such extension introduce fermion bilinear terms in it, which might be irrelevant in this note. Then we truncate such supercovariantized fields as b !bMAB = !MAB ; FMNPQ = FMNPQ ; (2.1c) MNP QRSUW XY where !MAB is a torsionless spin connection. We should also notice that "^ is not ··· the invariant tensor but the “antisymmetric symbol”, whose normalization is given as "^012 \ = 1. This is related to the tensor "MNP QRSUW XY as MNP QRSUV W XY MNP QRSUV W XY "^ = g11 " : (2.1d) The transformation rule under the local supersymmetry is given as 1 δe A = "ΓAΨ ; δC = α "Γ Ψ ; (2.1e) M 2 M MNP 2 [MN P ] NP QR δΨM = 2DM (!)" + 2TM "FNP QR : (2.1f) In the above formulation, various objects are defined in the following way: y i Ψ = iΨ Γ0 ;D (!) = @ − ! ΣAB ; (2.1g) M M M M 2 MAB eMNP QRS MNP QRS M[P QR S]N Γ = Γ + 12g Γ g ; (2.1h) ( ) α1 [N P QR] T NP QR = Γ NP QR − 8δ Γ : (2.1i) M 2 M M For later convenience, we introduced some unfixed coefficients β0, β1 and β2 in the Lagrangian, and α1 and α2 in the supersymmetry transformation rule. These are closely related to each other to preserve symmetry of the action: 2α1β1 2 1 1 β2 = − = β1 = − α1 ; α1β1 = − : (2.2a) α2 216 144 96 · 288β0 5 To arrange the usual form of the Einstein-Hilbert action, we fix the constant β0 to be unity: β0 = 1. In this setting we should choose the following explicit solution: 1 3 1 1 α = ; α = − ; β = − ; β = − : (2.2b) 1 144 2 2 1 192 2 (144)2 Let us rewrite the above forms (2.1) to connect other supergravity actions in ten-dimensional spacetime, which appear in the Polchinski’s book [15] and in the book written by Becker, Becker and Schwarz [1]. Let us rescale the gravitino ΨM ! 2ΨM : (2.3) Substituting (2.1c), (2.2b) and (2.3), we obtain the 11-dimensional supergravity action, which we will mainly use, instead of (2.1). First, the Lagrangian itself is p p 1 p 2κ2 L = −g R(e; !) − 2 −g Ψ ΓMNP D (!)Ψ − −g F F MNPQ 11 M N P 48 MNPQ 1 p − −g Ψ ΓeMNP QRSΨ F 48 M N P QRS 1 − "^MNP QRSUV W XY F F C : (2.4a) (144)2 MNPQ RSUV WXY The transformation rule under the local supersymmetry is given as A A δeM = "Γ ΨM ; δCMNP = −3"Γ[MN ΨP ] ; (2.5a) NP QR δΨM = DM (!)" + TM "FNP QR : (2.5b) NP QR In addition, the combination of the gamma matrices TM is rewritten in the following way: ( ) 1 [N P QR] T NP QR = Γ NP QR − 8δ Γ : (2.5c) M 288 M M Here it is worth describing the bosonic part of the action in terms of the differential form to com- pare the conventions in [15, 1]. Let us first extract the bosonic part of the action: Z ! 1 p 1 S = d11x −g R(e; !) − F F MNPQ boson 2κ2 48 MNPQ 11 Z − 1 11 MNP QRSUV W XY 2 2 d x "^ FMNPQFRST U CWXY : (2.6) 2κ11(144) Notice that the symbol "^MNP QRSUW XY does not depend on the curved space coordinates. It is just a number. Using the convention in appendix A, the last term is rewritten as Z − 1 11 MNP QRSUV W XY 2 d x "^ FMNPQFRSUV CWXY (144) Z 1 = − d11x g "MNP QRSUV W XY F F C (144)2 MNPQ RSUV WXY 6 Z − 1 M ^ N ^ · · · ^ Y = 2 dx dx dx FMNPQFRSUV CWXY (144) Z Z 4!4!3! 1 = − F ^ F ^ C = − F ^ F ^ C : (2.7) (144)2 4 4 3 6 4 4 3 Then, the bosonic action is simplified in the following way: ! Z n o 1 − 1j j2 − 1 ^ ^ Sboson = 2 (vol:) R(e; !) F4 F4 F4 C3 : (2.8) 2κ11 2 6 7 3 Type IIA supergravity in ten dimensions In this section let us derive the type IIA supergravity from the eleven-dimensional supergrav- ity. The derivation rule has already been investigated very well. Here we follow a convention introduced by [14], because the Lagrangians both in eleven- and in ten-dimensional spacetime are common forms in modern sense, while the fermion transformation rule itself is not explicitly discussed. 3.1 Duality transformation rules Here let us introduce the duality transformation rules from the eleven-dimensional supergravity to the type IIA supergravity, and vice versa. We only fucus on the bosonic parts. The former rule is given by − 2 Φ 4 Φ g^MN = e 3 gMN + e 3 AM AN ; C^MNP = CMNP ; (3.1a) ( ) 4 Φ 2 4 g^ = e 3 A ; C^ = B ; g^ = exp Φ : (3.1b) M\ M MN\ 3 MN \\ 3 Note that the objects with ^ denote the ones in eleven-dimensions, while the others in ten di- mensions. In addition, the symbol \ indicates the eleventh direction in the eleven-dimensional spacetime. In the same way, the latter rule is also given by ( ) ( ) 1 g^ g^ 2 M\ \N gMN = g^\\ g^MN − ;CMNP = C^MNP ; (3.2a) g^\\ ( ) g^M\ 3 3 AM = ;BMN = C^MN\ ; Φ = log g^\\ : (3.2b) g^\\ 2 4 In this note we omit the derivation of the above rules (see section 5.1 in [14]). 8 4 Type IIB supergravity in ten dimensions In this section let us derive the type IIB supergravity from the type IIA supergravity using the convention introduced by [14]. 4.1 Duality transformation rules Here let us introduce the duality transformation rules from the type IIA supergravity to the type IIB supergravity, and vice versa. We only fucus on the bosonic parts. The former rule is given by ( ) B(1) − 1 (1) (1) 9M 1 g^MN = gMN gM9g9N + BM9B9N ; g^9M = ; g^99 = ; (4.1a) g99 g99 g99 ( 2B(2) g ) ^ 2 (2) 9[M N]9 C9MN = BMN + ; (4.1b) 3 g99 "ijB(i) B(j) g ^ 8 ij (i) (j) 9[M j9jN P ]9 CMNP = C9MNP + " B9[M BNP ] + ; (4.1c) 3 g99 2B(1) g ^ (1) 9[M N]9 ^ g9M BMN = BMN + ; B9M = ; (4.1d) g99 g99 ( ) (2) (1) 1 A^ = −B + CB ; A^ = C; Φ^ = Φ − log g : (4.1e) M 9M 9M 9 2 99 Note that the objects with ^ denote the ones in type IIA, while the others in type IIB.