Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG)...

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Higher order curvature and Chern-Simons densities in GR Tigran Tchrakian Supersymmetry in Integrable Systems - SIS’16 International Workshop 28-30 December 2016 Hannover, Germany

Transcript of Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG)...

Page 1: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Higher order curvature and Chern-Simonsdensities in GR

Tigran Tchrakian

Supersymmetry in Integrable Systems - SIS’16International Workshop 28-30 December 2016

Hannover, Germany

Page 2: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Higher order ”Yang-Mills vs. gravitational” curvature terms

The p-Yang-Mills (p-YM) system is defined in terms of

F (2p) = F ∧ F ∧ · · · ∧ F , p times

p-fold antisymmetrised product of the curvature 2-form F (2) suchthat

L(p)YM = Tr |F (2p)|2 ,

I Can be defined in any dimensions d ≥ 2p

I Has dimensions L−4p (inverse length).

(On IR4p these systems support radial and axial instantons:Tch: Phys. Lett. B 150 (1985) 360 ; Chakrabarti, Tch. ibid. B162 (1985) 340)

Note: The p-YM terms scale as L−4p and can be employed tosatisfy the Derrick scaling requirement for finite energy .

Page 3: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

The p-Einstein-Hilbert (p-EH) systems are defined by p-foldantisymmetrised product of the Riemann curvature 2-form

Rabµν = ∂[µω

abν] + (ω[µων])

ab

and the (d − p)-fold antisymmetrised product of the Vielbein1-form eaµ such that

L(p,d)EH = εµ1µ2...µ2pν1ν2...νd−2peb1ν1 eb2ν2 . . . ebd−2pνd−2p εa1a2...a2pb1b2...bd−2p

··Ra1a2µ1µ2Ra3a4

µ33µ4. . .R

a2p−1a2pµ2p−1µ2p

I Can be defined in any dimensions d ≥ p Unlike its YMcounterpart, the definition depends on the dimension d

I Has dimensions L−2p (inverse length).

Note: The p-EH terms scale as L−2p, half as fast as p-YM terms,so are not in practice as useful for satisfying the Derrick scalingrequirement.

Page 4: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Effect of nA hair due to higher-order nA curvature terms:SO(D) in D + 1 dimensions

Brihaye, Chakrabarti, Tch. Class.Quant.Grav. 20 (2003)2765-2784; Breitenlohner, Maison, Tch. ibid. 22 (2005)5201-5222; Breitenlohner, Tch. ibid. 26 (2009) 145008

L =1

16πGR − 1

4TrFµνFµν+ τTrFµναβFµναβ , D = 4

I F 4: gauge fields counterpart of the Gauss-Bonnet term ingravity , and F (2p)2 of p-Einsten-Hilbert

I second order equations of motion

I such terms appear in d = 10 heterotic string action

I with an F 4 term: all known solutions are unstable!(Bartnik-McKinnon type), regular and black hole

Page 5: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Effect of non-Abelian (nA) hair due to CS terms:SO(D + 2) in D + 1 dimensions

Brihaye, Radu, Tch: Phys.Rev.Lett.106 (2011) 071101;Phys.Rev.D93 (2016) 124069)

d = 5 EYM model with a CS term

Ω(2)CS =

1

16πGR−κ ελµνρσ Tr Aλ

(FµνFρσ − FµνAρAσ +

2

5AµAνAρAσ

).

I truncation of N = 8, D = 5 gauged supergravity model

I gauge group SO(6) to enable supporting A0

I κ = 0: no finite mass solutions with non-Abelian hair (Volkov,Maeda)

I the Reissner-Nordstrom black hole becomes unstable anddevelops non-Abelian hair for κ 6= 0

Page 6: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

I the hairy solutions emerge as perturbations ofReissner-Nordstrom BH which develops an unstable mode fora critical value of the electric charge

I some hairy solutions are perturbatively stable

I the hairy solutions are thermodynamically favored over theReissner-Nordstrom Abelian configurations

I they have an electric charge

I Qualitatively the same effects occur when replacing theCS density with a |F (2p)|2 density with appropriatelylarge gauge group e.g . SO(N) , N = D + 2

Brihaye, Radu, Tch: Phys.Rev. D85 (2012) 084022, Radu, Tch,Yang: Phys.Rev.D93 (2016) 124069)

Page 7: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Some usual SO(D) CS densities: D = 2n = 4, 6 ond = 2n − 1 = 3, 5

For even D, in addition to the gamma matrices γa, a = 1, 2, . . . ,Dthere exists the chiral matrix γD+1 (γ2D+1 = 1).Thus we have two options in defining the SO(D) CS densities,

Ω(n)I and Ω

(n)II

Ω(2)I = ελµνTr Aλ

[Fµν −

2

3AµAν

]Ω(3)I = ελµνρσTr Aλ

[FµνFρσ − FµνAρAσ +

2

5AµAνAρAσ

]and

Ω(2)II = ελµνTr γ5Aλ

[Fµν −

2

3AµAν

]Ω(3)II = ελµνρσTr γ7Aλ

[FµνFρσ − FµνAρAσ +

2

5AµAνAρAσ

]

Page 8: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Passage from Yang-Mills–(Higgs) to Gravity

I Gauge group G of the (H)CS is the same as that of (H)CP

I For passage of YM → gravity, specialise G = SO(D) with

Ai = −1

2ωabi γab , i = 1, 2, . . .D

yielding gravitational (H)CP from non-Abelian (H)CP.

I The Higgs field Φ (if present) is the frame D-vector multiplet

Φ = 2φa γa,D+1 , (for later consideration)

Resulting gravitational CS has SO(D) frame vectors

I For gravitational CS in d = D − 1 dimensions to have SO(d)frame index, contract ωab = (ωαβ, ωα,D) by setting

ωα,Dµ = 0 ⇒ Rα,Dµν = 0 , α = 1, 2, . . . d .

Page 9: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Gravitational CS (GCS) densities Ω(n)GCS(I) and Ω

(n)GCS(II)

I 1st step: Evaluate the traces in Ω(n)I and Ω

(n)II with

Aµ = −1

2ωabµ γab , µ = 1, 2, . . . d , a = 1, 2, . . .D

I Ω(n)I 6= 0 for all even n

I Ω(n)I = 0 for all odd n

I Ω(n)II 6= 0 for all n

I 2nd step: Contract SO(D) to SO(d) by setting ωα,Dµ = 0

I Ω(n)GCS(I)=Ω

(n)I |SO(d) for all even n

I Ω(n)GCS(I)=0 for all odd n

I Ω(n)GCS(II) = 0 for all n, with contracted gauge group SO(d)

Conclusion:Gravitational CS densities exist only in d = 4p − 1dimensions, and are only of Type I.

Page 10: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

(Higgs)–Chern-Pontryagin → (Higgs)–Chern-Simonsn-th CP density in D = 2n dimensions is a total divergence

C(n)CP[F ] = ∇ ·Ω(n)[A,F ] , gauge invariant

CS density in d = 2n− 1 dimensions is defined as D-th componentof Ω(n)[A,F ]

Ω(n)CS[A,F ]

def.= Ωi=D [A,F ] , gauge variant

The HCP density in any dimension D < 2n is descended from CPin 2n dimensions and is a gauge invariant and total divergence

C(n,D)HCP [F ,DΦ, |Φ|2] = ∇ ·Ω(n,D)[F ,DΦ,DΦ, |Φ|2] , D even

= ∇ ·Ω(n,D)[A,F ,DΦ,DΦ, |Φ|2] , D odd

I Ω(n,d)HCS [F ,DΦ,DΦ, |Φ|2] is gauge invariant for even d = D − 1

I Ω(n,d)HCS [A,F ,DΦ,DΦ, |Φ|2] is gauge variant for odd d = D − 1

Page 11: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

SO(D) Higgs-CS densities (HCS) Ω(n,d)HCS : n = 3 on

d = D − 1 = 4, 3Tch. J.Phys. A44 (2011) 343001; ibid. A48 (2015) no.37, 375401

Ω(3,4)HCS = εµνρσ Tr Fµν Fρσ Φ

Ω(3,3)HCS = εµνλTr Γ5

[− 2η2Aλ

(Fµν −

2

3AµAν

)+

+ (Φ DλΦ− DλΦ Φ) Fµν

]Note that:

I In even dimensions d , the HCS density is gauge invariantI In odd dimensions d , the HCS density consists of two parts: A

Higgs independant gauge variant term being the usual CSdensity in the given dimensions, and, a gauge invariant Higgsdependant term.

This property persists for all HCS densities.

Page 12: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

SO(D) HCS densities Ω(n,d)HCS : n = 4 on d = 6, 5, 4, 3

Ω(4,6)HCS = εµνρστλ Tr Fµν Fρσ Fτλ Φ

Ω(4,5)HCS = εµνρσλ Tr γ7

[2η2Aλ

(FµνFρσFµνAρAσ +

2

5AµAνAρAσ

)+

+DλΦ (ΦFµνFρσ + FµνΦFρσ + FµνFρσΦ)

]Ω(4,4)HCS = εµνρσ Tr

(η2 FµνFkl +

2

9Φ2 FµνFkl +

1

9FµνΦ2Fkl

)−2

9(ΦDµΦDνΦ− DµΦΦDνΦ + DµΦDνΦΦ) Fρσ

]

Ω(4,3)HCS = εµνλ Tr Γ5

6η4 Aλ

(Fµν −

2

3Aµ Aν

)− 6 η2[Φ,DλΦ]Fµν

+([Φ,ΦDλΦΦ]− 2[Φ3,DλΦ]

)Fµν

Page 13: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Passage to gravity of SO(D) HCS

1st step: Evaluating the traces with,

For odd d = D − 1 (even D)

Aµ = −1

2ωabµ γab ⇒ Fµν = −1

2Rabµν γab

Φ = 2φa γa,D+1 ⇒ DµΦ = 2Dµφa γa,D+1 , Dµφ

a = ∂µφa + ωab

µ φb

For even d = D − 1 (odd D)

Aµ = −1

2ωabµ Σab ⇒ Fµν = −1

2Rabµν Σab

Φ = 2φa Σa,D+1 ⇒ DµΦ = 2Dµφa Σa,D+1 , Dµφ

a = ∂µφa + ωab

µ φb

(Σab spinor representations of SO(D) – no γ5 in even D.)Note: None of the Higgs dependant terms vanish when evaluatingthe traces.

Page 14: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

2nd step: In addition to the truncation ωα,Dµ = 0, the frame vectormultiplet φa = (φα, φD) as

I A SO(d) frame-vector field φα , α = 1, 2, . . . , d

I A scalar field, φ ≡ φD

such that the covariant derivative Dµφa = (Dµφ

α, φ) splits up into

Dµφα = ∂µφ

α + ωαβµ φβ

Dµφ = ∂µφ

The resulting gravitatioanl HCS (GHCS) densities can be added toa gravitaional Lagrangian, plus dynamical terms like

|∂µφ|2 , |Dµφα|2 , |D[µφ

αDν]φβ|2 , . . . etc .

as long as causality is preserved in the classical system.

Page 15: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

GHCS for (n, d) = (3, 3), (4, 5), (4, 3)

Ω(3,3)GHCS = Ω

(2,3)GCS(I) + 2 εµνλ εαβγ Rαβ

µν (φDλφγ − φγ∂λφ) .

Ω(4,5)GHCS = −1

2εµνρσλ εαβγδε Rαβ

µν Rγδρσ (φDλφ

ε − φε∂λφ)

Ω(4,3)GHCS = Ω

(2,3)GCS(I)−

−12 εµνλ εαβγ

(η2 − 1

2|φc |2

)Rαβµν (φDλφ

γ − φγ∂λφ) .

where Ω(2,3)GCS(I) is the usual gravitational CS density Ω

(n,d)GCS (I) with

n = 2, in d = 2n − 1 = 3.

Note1: that Ω(3)GCS(I) = 0, in d = 2n − 1 = 5,

Note2: the GHCS density Ω(n,d)GHCS reduces to the usual GCS density

Ω(n,d)GCS (I), if either φα or φ is suppressed.

Note3: Employing a GHCS density Ω(n,d)GHCS with n ≥ d + 2 results

in a symmetry-breaking type dynamics.

Page 16: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

GHCS for (n, d) = (3, 4), (4, 6), (4, 4)

Ω(3,4)GHCS = −1

4εµνρσ εαβγδ Rαβ

µν Rγδρσ φ

Ω(4,6)GHCS = −1

8εµνρστλ εαβγδεϕ Rαβ

µν Rγδρσ Rεϕ

τλ φ

Ω(4,4)GHCS = −εµνρσ εαβγδRαβ

µν

[1

8

(1− 1

3|φa|2

)Rγδρσ +

1

3φγδρσ

]φ+

+4

3φγDρφ

δ ∂σφ

.

where φγδρσ = D[ρφγDσ]φ

δ.

I In even d there are no frame-vector independant terms,I The GHCS density Ω

(n,d)GHCS vanishes if φ is suppressed,

I The GHCS density Ω(n,d)GHCS with n = d + 2 depends only on

the scalar φ, with no dynamical term for φ,I The GHCS density Ω

(n,d)GHCS with with n ≥ d + 2

I Displays no dynamical term for φ if φα is suppressed.I Displays symmetry-breaking if φα is not suppressed.

Page 17: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

CS Gravity (CSG) and Higgs-CS Gravity (HCSG)Chern-Simons gravities (in odd d) are constructed via the Ansatz

Aµ = −1

2ωαβµ γαβ + κ eαµ γαD ⇒

⇒ Fµν = −1

2

(Rαβµν − κ2 eα[µ eβν]

)γαβ + κTα

µνγαD

where Tαµν is the Torsion.

Note that this Ansatz effects the group contractionSO(D)→ SO(d) = SO(D − 1).

Substituting this in the SO(D) Chern-Simons density Ω(n)II yeilds a

superposition of a p-EH system with negative Λ, e.g . for d = 3

L(2,3)CSG = 2κ ελµνεαβγ

(eγλ Rαβ

µν −2

3κ2eαµ eβν eγλ

)

L(3)CSG = κ εµνρσλεabcde

(3

4eeλ Rab

µν Rcdρσ − ecρedσ eeλ Rab

µν +3

5κ4eaµebν ecρedσ eeλ

)

Page 18: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

An aspect of Chern-Simons gravity in higher dimensions

The Schwarzschild black hole solution:

ds2 =dr2

N(r)+ r2dΩ2

3 − N(r)dt2

EGB-Λ generic coefficients (Λ = −6/L2):

N(r) = 1 +r2

α

(1−

√1 +

r2(

M

r2− r2

L2)

)

CS theory of gravity : α =L2

2⇒ N(r) = 1− 2

√M

L+

2r2

L2

Exact solutions for some cases which are not known in genericEGB-Λ theory: black strings and spinning black holes,Brihaye, Radu: JHEP 1311 (2013) 049

Page 19: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Higgs-CS gravities (HCSG) in all d = D − 1These are constructed via the Ansatze

Aµ = −1

2ωαβµ γαβ + κ eαµ γαD , Φ = 2(φα γα,D+1 + φγD,D+1)

Aµ = −1

2ωαβµ Σαβ + κ eαµ ΣαD , Φ = 2(φαΣα,D+1 + φΣD,D+1)

Using the notation φαµ = Dµφα , φµ = ∂µφ ,

In d = 3, the HCSG models resulting from the n = 3, 4 HCS are

L(3,3)HCSG = 2η2κ ελµνεαβγ

(eγλ Rαβ

µν −2

3κ2eαµ eβν eλρ

)+2ελµνεγαβ

(−1

2(Rαβ

µν − κ2 eα[µeβν])(φγφλ − φφγλ) + κφαφβλT γµν

)L(4,3)HCSG = −3η2L(3,3)HCSG − 12ελµνεγαβ

[η2 − (|φδ|2)

·(−1

2(Rαβ

µν − κ2 eα[µeβν])(φγφλ − φφγλ) + κφαφβλT γµν

)

Page 20: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

In d = 4, the HCSG models resulting from the n = 3, 4 HCS are

L(3,4)HCSG = −εµνρσεαβγδ φ[Rαβµν Rγδ

ρσ − 4κ2 eγρ eδσRαβµν + 4κ4 eαµ eβν eγρ eδσ

]+2κεµνρσεαβγδ

(Rαβµν − κ2 eα[µ eβν]

)T γρσφ

δ

L(4,4)HCSG =

[η2 − 1

3(|φα|2 + φ2)

]L(3,4)HCSG −

2

3εµνρσεαβγδ ·

·[(

Rαβµν − κ2 eα[µ eβν]

)φγρ(φφδσ − 2φδφσ) +

2

3Tαµνφ

βφγρφδσ

]

I In odd dimensions, the leading terms in the HCSG models arethe CSG models.

I In even dimensions, the HCSG models consist exclusively of(φα, φ) dependant terms, and when φ = 0 they are pureTorsion models.

I In contrast to CSG models, the (φα, φ) dependant terms inHCSG models exhibit an explicit dependane on the Torsion.

I HCSG models with n ≥ d exhibit symmetry breaking.

Page 21: Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG) Chern-Simons gravities (in odd d) are constructed via the Ansatz A = 1 2! + e D))F

Acknowledgements

Thanks to Friedrich Hehl and Ruben Manvelyan