Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG)...
Transcript of Higher order curvature and Chern-Simons densities …CS Gravity (CSG) and Higgs-CS Gravity (HCSG)...
Higher order curvature and Chern-Simonsdensities in GR
Tigran Tchrakian
Supersymmetry in Integrable Systems - SIS’16International Workshop 28-30 December 2016
Hannover, Germany
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 .
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.
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
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
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)
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
]
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 .
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.
(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
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.
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
[Φ
(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
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.
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.
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.
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.
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
)
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 +
2α
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
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
)
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.
Acknowledgements
Thanks to Friedrich Hehl and Ruben Manvelyan