New Massive Dual Gravity: beyond 3D

Transcription

1 New Massive Dual Gravity: beyond 3D Eric Bergshoeff Groningen University Work in progress together with Jose Juan Fernandez, Jan Rosseel and Paul Townsend Miami, December

2 Outline Introduction

3 Outline Introduction Massive Spin 1 in 2D

4 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D

5 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D?

6 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

7 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

8 Higher-derivative Gravity Consider Einstein gravity as a theory of interacting massless spin 2 particles around a Minkowski space-time background Problem: This theory is perturbative non-renormalizable ( ) 2 ab L R +a R µν +b(rµν ) 2 +cr 2 : renormalizable but not unitary Stelle (1977) massless spin 2 and massive spin 2 have opposite sign!

9 Special Case In three dimensions there is no massless spin 2! New Massive Gravity Can this be extended to four dimensions?

10 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

11 Proca ( m 2) A ν = 0, µ A µ = 0

12 Proca ( m 2) A ν = 0, µ A µ = 0 L = 1 4 Fµν F µν 1 2 m2 A µ A µ

13 Proca ( m 2) A ν = 0, µ A µ = 0 L = 1 4 Fµν F µν 1 2 m2 A µ A µ number of propagated modes is D 1 = { 3 for 4D 1 for 2D

14 Proca ( m 2) A ν = 0, µ A µ = 0 L = 1 4 Fµν F µν 1 2 m2 A µ A µ number of propagated modes is D 1 = { 3 for 4D 1 for 2D Note: the numbers become 2 (4D) and 0 (2D) for m = 0 use: F µν (A) ǫ µν F

15 Boosting up the Derivatives L = 1 2 Aµ G µ (A) 1 2 m2 A µ A µ with G µ (A) = λ F λµ (A) : Einstein tensor : µ G µ (A) = 0

16 Boosting up the Derivatives L = 1 2 Aµ G µ (A) 1 2 m2 A µ A µ with G µ (A) = λ F λµ (A) : Einstein tensor : µ G µ (A) = 0 µ A µ = 0 A µ = G µ (B) : gauge theory

17 Boosting up the Derivatives L = 1 2 Aµ G µ (A) 1 2 m2 A µ A µ with G µ (A) = λ F λµ (A) : Einstein tensor : µ G µ (A) = 0 µ A µ = 0 A µ = G µ (B) : gauge theory L 1 2 Bµ G µ (B)+ 1 m 2 G µ (B)G µ (B) higher-derivative Maxwell: unitary in 2D!

18 Mode Analysis lower number of derivatives from 4 to 2 by introducing an auxiliary field C µ : L 1 2 Bµ G µ (B)+ 1 2 Cµ G µ (B)+ m2 2 C µc µ after diagonalization the two fields describe a massless spin 1 and a massive spin 1 with relative minus sign only in 2D this is not a problem!

19 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

20 3D Einstein-Hilbert Gravity There are no massless gravitons Adding higher-derivative terms leads to massive gravitons

21 Fierz-Pauli ( m 2) h µν = 0, h µν = h νµ, η µν h µν = 0, µ h µν = 0 L FP = 1 2 hµν G lin µν(h)+ 1 2 m2( h µν h µν h 2), h η µν h µν no non-linear extension! number of propagating modes is 1 2 D(D +1) 1 D = { 5 for 4D 2 for 3D Note: the numbers become 2 (4D) and 0 (3D) for m = 0

22 Non-linear Extension in 3D µ hµν = 0 h µν = G lin µν (h) ( m 2 ) G lin µν(h) = 0, R lin (h) = 0 Non-linear generalization: g µν = η µν +h µν L = g [ R + 1m (R µν 2 R µν 38 )] R2 New Massive Gravity : unitary in 3D!

23 Mode Analysis Take NMG with metric g µν, cosmological constant Λ and coefficient σ = ±1 in front of R lower number of derivatives from 4 to 2 by introducing an auxiliary field f µν after linearization and diagonalization the two fields describe a massless spin 2 with coefficient σ = σ Λ and a massive 2m 2 spin 2 with mass M 2 = m 2 σ special cases: 3D NMG and D 3 critical gravity for special value of Λ

24 What did we learn? two theories can be equivalent at the linearized level (FP and boosted FP) but only one of them allows for a non-linear extension i.e. interactions! what about 4D? interacting spin 2 FP? we need massive spin 2 whose massless limit describes 0 d.o.f. Example : a b in 3D

25 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

26 Generalized spin-2 FP in 4D standard massive spin-2 : a b describes 5 d.o.f. m 0 2 d.o.f. m = 0 generalized massive spin-2 : a c b describes 5 d.o.f. m 0 0 d.o.f. m = 0 Curtright (1980)

27 Connection-metric Duality West (2001); Gonzalés, Khoudeir, Montemayor and Urrutia (2008) start with first-order form of EH with fields e a µ and ω ab µ linearize around Minkowski: e a µ = δ a µ +h a µ and add a FP mass term m 2 (h µν h νµ h 2 ) L h ω +ω 2 +h 2 solve for ω massive spin-2 FP with h (µν) and auxiliary h [µν] solve for h and write ω µ ab = 1 2 ǫabcd T µcd generalized spin-2 FP with T ab,c and auxiliary T [µcd]

28 Remark duality is singular for m 0 due to 1/m terms duality for massless case was discussed by West (2001)

29 4D New Massive Dual Gravity (NMDG) start with generalised spin-2 FP: T a,bc b c a T [a,bc] = 0, η bc Tb,ca = 0 T satisfies subsidiary condition c T a,bc = 0 solve for c Ta,bc = 0 T a,bc = G a,bc (T) L NMDG 1 2 Ta,bc G a,bc (T)+ 1 2m 2 T a,bc C a,bc (T) mode analysis L NMDG massless spin 2 plus massive spin 2

30 Outline Introduction Massive Spin 1 in 2D New Massive Gravity in 3D New Massive Dual Gravity in 4D? Conclusions

31 Conclusions can we introduce interactions? natural extension to general Young tableaux with 2 columns of height p and q in p+q+1 dimensions what about higher spin?