Spectrum Of Eleven-dimensional Supergravity On A PP-wave Background

Spectrum of Eleven-dimensional Supergravity on a PP-wave Background KEK/ hep-th/0307193, to appear in Phys. Rev. D in collaboration with Kentaroh Yoshida (KEK) . – p.1/36

Introduction 11-dimensional Supergravity: 32 Maximal supersymmetry, non-chiral theory . – p.2/36

Introduction 11-dimensional Supergravity: 32 Maximal supersymmetry, non-chiral theory First construction by Cremmer, Julia and Scherk (1978) 25th Anniversary! . – p.2/36

Introduction 11-dimensional Supergravity: 32 Maximal supersymmetry, non-chiral theory First construction by Cremmer, Julia and Scherk (1978) 25th Anniversary! Various dimensional supergravities appear Kaluza-Klein mechanism 10-dim. type IIA (non-chiral), type IIB (chiral), etc. AdS4 S 7 (with cosmological constant) singular G2 compactification 4-dimensional N 1 chiral supergravity . – p.2/36

Introduction 11-dimensional Supergravity: 32 Maximal supersymmetry, non-chiral theory First construction by Cremmer, Julia and Scherk (1978) 25th Anniversary! Various dimensional supergravities appear Kaluza-Klein mechanism 10-dim. type IIA (non-chiral), type IIB (chiral), etc. AdS4 S 7 (with cosmological constant) singular G2 compactification 4-dimensional N 1 chiral supergravity only a classical theory . – p.2/36

Towards Quantum Theory Supermembrane theory in 11-dimensions: Bergshoeff, Sezgin and Townsend (1987) . – p.3/36

Towards Quantum Theory Supermembrane theory in 11-dimensions: Bergshoeff, Sezgin and Townsend (1987) Quantum Mechanics of Supermembrane on FLAT background: de Wit, Hoppe and Nicolai (1988) 15th Anniversary! . – p.3/36

Towards Quantum Theory Supermembrane theory in 11-dimensions: Bergshoeff, Sezgin and Townsend (1987) Quantum Mechanics of Supermembrane on FLAT background: de Wit, Hoppe and Nicolai (1988) 15th Anniversary! Continuous spectrum, etc. UNSTABLE as a single object de Wit, Lüscher and Nicolai (1989) . – p.3/36

Towards Quantum Theory Supermembrane on PP-wave background Nakayama, Sugiyama and Yoshida (2002) discretized spectrum, supersymmetric, etc. STABLE as a single object . – p.4/36

Towards Quantum Theory Supermembrane on PP-wave background Nakayama, Sugiyama and Yoshida (2002) discretized spectrum, supersymmetric, etc. STABLE as a single object Does this theory contain supergravity? (Zero-mode spectrum should correspond to that of supergravity on PP-wave background.) . – p.4/36

Towards Quantum Theory Supermembrane on PP-wave background Nakayama, Sugiyama and Yoshida (2002) discretized spectrum, supersymmetric, etc. STABLE as a single object Does this theory contain supergravity? (Zero-mode spectrum should correspond to that of supergravity on PP-wave background.) We construct supergravity on PP-wave background and check the spectrum of fields. . – p.4/36

Field Contents eM A : vielbein ΨM : gravitino (Majorana spinor) CM N P : three-form gauge field . – p.5/36

Lagrangian of Eleven-dimensional Supergravity L eR 1 1 2 e ΨM 1 MNP b e FM N P Q F M N P Q Γ D N ΨP 48 e M N P QRS ΨN FP QRS e ΨM Γ 192 1 (144) M N P QRSU V W XY ε FM N P Q FRSU V CW XY 2 various conventions: D N ΨP N ΨP 1 4 b AB ΨP ωN AB Γ b N P QR M 8δ [N Γ e N P QR M Γ b P QR] Γ M e M N P QRS Γ b QR g S]N b M N P QRS 12g M [P Γ Γ ε012···10 1 weight 1 invariant tensor density . – p.6/36

Classical Field Equations 0 1 2 gM N R R M N 1 96 gM N FP QRS F P QRS 1 12 FM P QR FN P QR 1 e M N P QRS MNP b Γ 0 Γ D N ΨP ΨN FP QRS 96 ª 0 Q e FQM N P 18 (144)2 gM Z gN K gP L εZKLQRSU V W XY FQRSU FV W XY . – p.7/36

Spectrum on AdS4 S 7 hFµνρσ i 3m ²µνρσ X I Φµν···mn··· (x, y) φIµν··· (x) · Ymn··· (y) I d 11 AdS4 spin S 7 (SO(8)) number gM N (x, y) hµν (x) 2 1 1 Vµ[IJ] (x) 1 [IJ] Km (y) 28 S [IJKL] (x) 0 19 gmn Kp[IJ K KL]p 35 CM N P (x, y) P [IJKL] (x) 0 ΨM (x, y) I ψµ (x) 3/2 χ[IJK] (x) 1/2 [IJ] Km (y) : Killing vector [IJ KL] K(m Kn) [IJKL] K[mnp] (y) [IJK] ηm η I (y) b m η/ [IJK] 1Γ 9 35 8 56 η I (y) : Killing spinor (I 1, · · · , 8) . – p.8/36

Maximally Supersymmetric Spaces AdS4 S 7 Penrose Limit Flat Limit Kowalski-Glikman AdS7 S 4 Minkowski Penrose Limit . – p.9/36

Penrose Limit of AdS4 S 7 AdS4 S 7 coordinates: n o 2 ds2 RA cosh2 ρ · dt2 dρ2 sinh2 ρ · dΩ22 n o RS2 cos2 θ · dϕ2 dθ 2 sin2 θ · dΩ05 2 We choose a null geodesics of AdS4 S 7 as RS 2RA t 2ϕ ρ θ 0 . – p.10/36

Along this null direction we take RA : ds2 2dx dx G (dx )2 9 X (dxI )2 I 1 G 3 h³ µ 2 X 3 Ie 2 (x ) e I 1 9 ³ µ 2 X 6 I 0 4 I0 (x )2 i µ F 123 6 0 where x RA ρ x 1 2 (t 2ϕ) · 3 µ y 2RA θ x 2 RA (t 2ϕ) · µ 3 . – p.11/36

Fluctuation Fields gM N gM N hM N ΨM 0 ψ M CM N P CM N P C M N P gM N : pp-wave background 4 [ C123] F 123 µ . – p.12/36

Field Equations for Fluctuations From classical equation for gM N : 1 n o 0 g M N h P Q R P Q P Q h P Q P P h Q Q 2 o 1n P P P P M h N P N h M P M N h P P h M N 2 n o 1 gM N 2F P QRS P C QRS FP QRS FU QRS hP U 24 1 1 FM P QR [N C P QR] FN P QR [M C P QR] 3 3 1 FM P QR FN U QR hP U 4 (A-1) . – p.13/36

Field Equations for Fluctuations From classical equation for ΨM : 1 b M N 123 MNP b D N ψP µ Γ ψN 0 Γ 4 1 n M b 12 3N b 23 g 1N Γ b 31 g 2N ) µ g (Γ g Γ 4 b 23 g N Γ b 3 g 2N Γ b 2 g 3N ) g M 1 (Γ b 3 g 1N Γ b 1 g 3N Γ b 13 g N ) g M 2 (Γ o b 1 g 2N Γ b 12 g N Γ b 2 g 1N ) ψN g M 3 (Γ (A-2) . – p.14/36

Field Equations for Fluctuations From classical equation for CM N P : n S C Γ 0 4g QR R [Q C M N P ] ΓS RQ [S M N P ] RM [Q C SN P ] o S ΓS C Γ [Q M SP ] RN RP [Q C M N S] 1 QR n FSM N P ( R hQ S Q hR S S hRQ ) g 2 FQSN P ( R hM S M hR S S hRM ) FQM SP ( R hN S N hR S S hRN ) o FQM N S ( R hP S P hR S S hRP ) 1 144 gM Z gN K gP L εZKLQRSU V W XY FQRSU V C W XY (A-3) . – p.15/36

Gauge-fixing Conditions N h M h 0 ψ 0 C N P 0 light-cone gauge-fixing . – p.16/36

Hamiltonian We will encounter Klein-Gordon type equations of motion and have to evaluate its energy spectrum: ¡ α µ i φ(x , x , xI ) 0 x : evolution parameter α : arbitrary constant We express the Hamiltonian H i : H 1 3 µ X Ie Ie a a Ie 1 6 µ X I0 I0 I0 a a 1 2 µ (2 α) Last term zero-point energy E0 of the system (eigenvalue of H): E0 1 2 µ E 0 (φ) E 0 (φ) 2 α . – p.17/36

Bosonic Fields ( ) component of (A-1): 0 hII traceless condition . – p.18/36

Bosonic Fields ( ) component of (A-1): 0 hII traceless condition Substitute this into ( I) component of (A-1): hI 1 J hIJ non-dynamical . – p.18/36

Bosonic Fields ( ) component of (A-1): 0 hII traceless condition Substitute this into ( I) component of (A-1): hI 1 J hIJ non-dynamical ( IJ ) component of (A-3): C IJ 1 K C IJK non-dynamical . – p.18/36

Bosonic Fields ( ) component of (A-1): 0 hII traceless condition Substitute this into ( I) component of (A-1): 1 hI J hIJ non-dynamical ( IJ ) component of (A-3): C IJ 1 K C IJK non-dynamical Trace of (A-1) under the above conditions: h 1 ( )2 I J hIJ 1 3 µ C 123 non-dynamical . – p.18/36

Non-trivial Equations (IeJe) of (A-1) : e 0 ) of (A-1) : (IJ 0 0 (I J ) of (A-1) : f of (A-3) : (IeJeK) (IeJeK 0 ) of (A-3) : e 0 K 0 ) of (A-3) : (IJ (I 0 J 0 K 0 ) of (A-3) : 0 hIeJe 0 hIJ e 0 0 h I0J 0 2 µ δIeJe C 3 µ C IJ e 0 1 (1-a) (1-b) µ δ I 0 J 0 C (1-c) 3 0 C 2µ hIeIe (1-d) 0 C IJ e 0 µ hIJ e 0 (1-e) 0 C IJ e 0K0 (1-f) 0 CI0J 0K0 C IJ e 0 1 2 εIeK fL e CK fLJ e 0 1 6 µ εI 0 J 0K0W 0X0Y 0 C W 0 X 0 Y 0 (1-g) C 2C 123 . – p.19/36

(1-f): 0 CIeJeK 0 We find the zero-mode energy E0 (C IJ e 0 K 0 ) and degrees of freedom D(C IJ e 0 K 0 ): E0 (C IJ e 0K0 ) 2 D(C IJ e 0 K 0 ) 45 . – p.20/36

(1-b), (1-e): 0 hIJ e 0 µ CIJ e 0 CIJ e 0 µ hIJ e 0 Diagonalize hIJ e 0 and C IJ e 0: HIJ e 0 hIJ e 0 iC IJ e 0 Thus modified(1-b) and (1-e) are ¡ 0 µ i HIJ e 0 E 0 (HIJ e 0) 1 H IJ e 0 hIJ e 0 iC IJ e 0 0 ¡ µ i H IJ e 0 E 0 (H IJ e 0) 3 D(HIJ e 0 ) D(H IJ e 0 ) 18 . – p.21/36

(1-a), (1-c), (1-d): Apply similar consideration to (1-a), (1-c) and (1-d): h IeJe hIeJe 1 δIeJe hK fK f 3 h hIeIe iC h I0J 0 hI 0 J 0 1 6 h hIeIe iC δI 0 J 0 h K 0 K 0 Then we find ) E (h E 0 (h 0 e e I0J 0 ) 2 IJ E 0 (h) 0 D(h ) 5 IeJe E 0 (h) 4 D(h I 0 J 0 ) 20 D(h) D(h) 1 . – p.22/36

(1-g): Decomposing into self-dual part C I 0 J 0 K 0 and anti-self-dual part Cª I0J 0K0 : C I0J 0K0 Cª I0J 0K0 i 3! ε I0J 0K0W 0X0Y 0 i 3! ε C W 0X0Y 0 I0J 0K0W 0X0Y 0 Cª W 0X0Y 0 They satisfy the following equations: ¡ ª ¡ µ i C I 0 J 0 K 0 0 µ i C I 0 J 0 K 0 0 E 0 (C I0J 0K0 ) 3 E 0 (C ª I0J 0K0 ) 1 ª D(C ) D(C 0 0 0 I J K I 0 J 0 K 0 ) 10 . – p.23/36

Results We have fully solved the field equations for bosonic fluctuations and have derived the spectrum of graviton hM N and three-form gauge field CM N P . The resulting spectrum is splitting with a certain energy difference in contrast to the flat case. energy bosonic fields 4 h 3 H IJ e 0 C I0J 0K0 2 C IJ e 0K0 h ee 1 HIJ e 0 0 h degrees of freedom 1 IJ Cª I0J 0K0 18 10 h I0J 0 45 5 20 18 10 1 . – p.24/36

Fermionic Fields M component of (A-2): b P ψP 0 Γ Lorentz-type gauge-fixing condition . – p.25/36

Fermionic Fields M component of (A-2): b P ψP 0 Γ Lorentz-type gauge-fixing condition M component of (A-2): P ψP 0 ψ 1 I ψI non-dynamical . – p.25/36

M Ie component of (A-2): 0 n ³ 1 o b Γ b K K ψ e b G Γ Γ I 2 1 b 123 ³ b e ψe b eΓ δIeJe Γ µΓ I J J 4 We decompose ψI e 1 b b Γ Γ ψIe 2 ψIª e 1 b b Γ Γ ψIe 2 Then we obtain 1 b bK Γ Γ K ψI non-dynamical e 2 1 b 123 ¡ b b 0 ψIe µΓ δIeJe ΓIeΓJe ψJ e 2 ψIª e . – p.26/36

we shall introduce the following fields: ³ 1 b b Γ Γ δ ψ ψ e e IeJe Ie 3 I J Je k Ie b ψ Γ ψ 1 Ie b Ie or (δ f e 1 Γ b fΓ b e): Acting Γ I 3 K KI 0 k ψ1 k I k ψ1R b Γ-parallel mode 1 b 123 k 123 b µ Γ ψ1 0 ψ f µΓ ψ f K K 2 Decompose ψ and ψ1 e ψ e IR b Γ-transverse mode b 123 1 iΓ 2 b 123 1 iΓ 2 according to the chirality: ψ Ie ψ e IL k ψ1 k ψ1L b 123 1 iΓ 2 b 123 1 iΓ 2 ψ e I k ψ1 . – p.27/36

b 123 ): Multiplying chiral projection operator 12 (1 iΓ ³ ³ k k 0 µ i ψ1L 0 µ i ψ1R ³ ³ 1 1 ψ µ i 0 µ i ψIR 0 e e IL 2 2 Zero-mode energies and degrees of freedom of ψIR and ψ e e : IL E 0 (ψIR e ) 5 E 0 (ψIL e ) 3 2 2 ) D(ψ D(ψIR e e ) 8 (3 1) 16 IL . – p.28/36

M I 0 component of (A-2): o n ³ 1 b Γ b K K ψI 0 b G Γ 0 Γ 2 1 b 123 ³ bI0 Γ b J 0 ψJ 0 δI 0 J 0 Γ µΓ 4 b b The Γ-transverse mode and Γ-parallel mode are defined as ψI 0 ψI 0R k ψ2R b 123 1 iΓ 2 b 123 1 iΓ 2 1 b b Γ Γ ψI 0 2 ψI 0 k ψ2 ψI 0L k ψ2L b 123 1 iΓ 2 b 123 1 iΓ 2 ψI 0 k ψ2 . – p.29/36

b b Equations for the Γ-parallel mode and Γ-transverse mode: ³ ³ 5 5 k k 0 µ i ψ2R 0 µ i ψ2L 2 2 ³ ³ 1 1 ψ µ i 0 µ i ψI 0 0R I0L 2 2 We find that the zero-mode energiesand degrees of freedom: E 0 (ψI 0R ) 3 E 0 (ψI 0L ) 5 2 2 D(ψI 0 R ) D(ψI 0 L ) 8 (6 1) 40 . – p.30/36

b Linear combination of Γ-parallel modes: k ψR 2 5 k ψ1R k ψ2R k ψL 2 5 k k ψ1L ψ2L We can easily see that the re-defined fermions satisfy: ³ ³ 3 3 k k 0 µ i ψL 0 µ i ψR 2 2 k E 0 (ψR ) 1 k E 0 (ψL ) 2 k k D(ψR ) D(ψL ) 8 7 2 . – p.31/36

Results We have solved and derived the spectrum of gravitino ψM in the case of pp-wave background. As a result, we have found that the spectrum is splitting with a certain energy difference in the same manner with the spectrum of bosons. energy fermionic fields degrees of freedom k 7/2 ψL 5/2 ψIR e ψI 0L 16 40 3/2 ψIL e ψI 0R 16 40 1/2 ψR k 8 8 . – p.32/36

Results energy 4 bosons fermions h 1 k ψL 7/2 3 H IJ e 0 28 ψIR e C IJ e 0K0 h IeJe HIJ e 0 56 70 Cª I0J 0K0 ψI 0R 56 28 k ψR 1/2 0 ψI 0L h I0J 0 ψIL e 3/2 1 8 C I0J 0K0 5/2 2 D.O.F. h 8 1 . – p.33/36

Towards Quantum Theory, revisited Supermembrane as a matrix model: Nakayama, Sugiyama and Yoshida (2002), etc. discrete spectrum, STABLE as a single object . – p.34/36