supersymmetry of classical solutions in chern-simons ...ism2012/talks/ism-shouvik.pdf ·...
Post on 21-May-2020
3 Views
Preview:
TRANSCRIPT
Supersymmetry of classical solutionsin Chern-Simons higher spin supergravity
Shouvik Datta
Centre for High Energy PhysicsIndian Institute of Science
[hep-th/1208.3921] with Justin R. David
Indian Strings Meeting, 2012
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 1/24
Outline
Introduction and motivationWhy higher spins?Chern-Simons higher spin supergravity
Supersymmetry of classical solutionsWhen are classical solutions supersymmetric?
Supersymmetry of solutions in the sl(3|2) theoryThe sl(3|2) superalgebraSupersymmetry of black holes and conical defects
Generalization of results for the sl(N |N − 1) theoryThe sl(N |N − 1) supergroupKilling spinors and holonomyHigher spin conical defectsSmoothness and supersymmetry
Outlook
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 2/24
Why higher spins?
� The AdS/CFT correspondence in its original form relates a string theoryon AdS5 × S5 to N = 4 super-Yang Mills in four dimensions.
� The full spectrum of string theory has an infinite tower of massive higherspin states, which in the tensionless limit become a theory of masslesshigher spins.This is hard! We use N →∞ and λ→∞ to truncate spins upto 2.
� If a tractable theory of truncated higher spins is known, they can serve astoy models to capture the complications of stringy gravity.
� Such a theory exists in d = 3 when higher spin gravity is formulated as aChern-Simons theory
� It’s also useful to make explorations to families of theories with extendedsymmetry groups. Why not higher spin symmetry?
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 3/24
Chern-Simons higher spin supergravity
� 2+1 dimensional gravity can be formulated as a Chern-Simons theory interms of one-forms (ω ± e) based on the algebra sl(2, R)⊕ sl(2, R).(Achucarro & Townsend ‘87, Witten ‘88)
� Higher spin gravity is based on the algebra sl(N,R)⊕ sl(N,R) withN > 2. These have spins 2, 3, 4 · · ·N . (Blencowe, ‘88, Vasiliev ‘92)
� It is natural to look for supersymmetric extensions of higher spin theoriesin AdS3. (Prokushkin & Vasiliev, ‘98)
� Given any supergroup G the Chern-Simons action is given by
S =k
2π
∫ [str
(ΓdΓ +
2
3Γ3
)− str
(ΓdΓ +
2
3Γ3
)]The equations of motion are
dΓ + Γ ∧ Γ = 0 , dΓ + Γ ∧ Γ = 0
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 4/24
Higher spins and holography
(2+1)-d Gravity
sl(2, R)⊕ sl(2, R)Virasoro algebra
CFT2
BTZ black hole,conical defects
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 5/24
Higher spins and holography
(2+1)-d Gravity (2+1)-d HS-Gravity
sl(2, R)⊕ sl(2, R) sl(N,R)⊕ sl(N,R)Virasoro algebra WN algebra
CFT2CFT2 with HS-currents
Minimal Models
BTZ black hole, Higher spin black holes,conical defects conical defects
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 5/24
Higher spins and holography
(2+1)-d Gravity (2+1)-d HS-Gravity (2+1)-d N=2 HS-SUGRA
sl(2, R)⊕ sl(2, R) sl(N,R)⊕ sl(N,R) sl(N |N − 1)⊕ sl(N |N − 1)
Virasoro algebra WN algebra super-WN algebra
CFT2CFT2 with HS-currents CFT2 with HS-supercurrents
Minimal Models Kazama-Suzuki Models
BTZ black hole, Higher spin black holes, ?conical defects conical defects
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 5/24
Higher spins and holography
(2+1)-d Gravity (2+1)-d HS-Gravity (2+1)-d N=2 HS-SUGRA
sl(2, R)⊕ sl(2, R) sl(N,R)⊕ sl(N,R) sl(N |N − 1)⊕ sl(N |N − 1)
Virasoro algebra WN algebra super-WN algebra
CFT2CFT2 with HS-currents CFT2 with HS-supercurrents
Minimal Models Kazama-Suzuki Models
BTZ black hole, Higher spin black holes, ?conical defects conical defects
� We shall be constructing and studying classical solutions of Chern-Simonstheories embedded in the superalgebra sl(N |N − 1).
� The study of supersymmetry in higher spin theories is a new subject and thereare no general results for when a classical solution is supersymmetric.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 5/24
Outline
Introduction and motivationWhy higher spins?Chern-Simons higher spin supergravity
Supersymmetry of classical solutionsWhen are classical solutions supersymmetric?
Supersymmetry of solutions in the sl(3|2) theoryThe sl(3|2) superalgebraSupersymmetry of black holes and conical defects
Generalization of results for the sl(N |N − 1) theoryThe sl(N |N − 1) supergroupKilling spinors and holonomyHigher spin conical defectsSmoothness and supersymmetry
Outlook
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 6/24
When are classical solutions supersymmetric?
� The classical solutions like higher spin black holes (Gutperle et al. ‘11 )and conical defects (Castro et al. ‘11 ) are given terms of 1-formsA = (ωa + ea)Ta and A = (ωa − ea)Ta.
� These obey the flatness condition, dA+A ∧A = 0.
� These solutions are supersymmetric when they are invariant underfermionic gauge transformations.
δεA ≡ ∂µεiGi +Aaµεi[Ta, Gi] = 0
εi is called the Killing spinor.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 7/24
When are classical solutions supersymmetric?
� The flatness conditions are the integrability constraints of the Killingspinor equation. Thus, the existence of Killing spinors is guaranteed forany flat connection.
� Additionally, we need to find under what conditions these spinors obeyproper periodicity requirements.
� We shall be interested in the class of N = 2 Chern-Simons theories basedon the supergroup sl(N |N − 1). The Cartan-Weyl representation of thissuperalgebra will be used for our analysis. (Frappat et al. ‘96)
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 7/24
Outline
Introduction and motivationWhy higher spins?Chern-Simons higher spin supergravity
Supersymmetry of classical solutionsWhen are classical solutions supersymmetric?
Supersymmetry of solutions in the sl(3|2) theoryThe sl(3|2) superalgebraSupersymmetry of black holes and conical defects
Generalization of results for the sl(N |N − 1) theoryThe sl(N |N − 1) supergroupKilling spinors and holonomyHigher spin conical defectsSmoothness and supersymmetry
Outlook
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 8/24
The sl(3|2) superalgebra
� Let’s now focus on the supersymmetry of classical solutions in thesimplest higher spin theory, sl(3|2).
� The sl(3|2) algebra can be obtained as the global part of the N = 2super-W3 algebra with large central charge (c→∞). (Romans ‘92)
This is often referred to as the semiclassical limit.
� These have the following generators
Bosonic : Jspin-1
, Lspin-2
, Vspin-2
, Wspin-3
Fermionic : G±spin-3/2
, U±spin-5/2
� The bosonic subalgebra is sl(3)W, T−
⊕ sl(2)T+
⊕ u(1)J
.
� We have explicitly verified the commutation relations by checking allJacobi identities.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 9/24
Classical solutions in the sl(3|2) theory
The gauge connections of the classical solutions (black holes and conicaldefects) embedded in sl(3|2) have the following generic form
A =( 1∑m=−1
(tmemρT−
m + smemρT+
m) +2∑
m=−2
(wmemρWm) + ξJ
)dx+
− ξJdx− + (T+0 + T−
0 )dρ
For exampleAdS3 :
A =
(eρ(T+
1 + T−1 ) +1
4e−ρ(T+
−1 + T−−1)
)dx+ + (T+
0 + T−0 )dρ+ 2ξJdφ
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 10/24
Classical solutions in the sl(3|2) theory
The gauge connections of the classical solutions (black holes and conicaldefects) embedded in sl(3|2) have the following generic form
A =( 1∑m=−1
(tmemρT−
m + smemρT+
m) +2∑
m=−2
(wmemρWm) + ξJ
)dx+
− ξJdx− + (T+0 + T−
0 )dρ
For exampleHigher spin black hole with W = µ = 0 :
A =
(eρT−1 −
2π
kLe−ρT−−1
)dx+ + 2ξJdφ+ (T+
0 + T−0 )dρ
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 10/24
Classical solutions in the sl(3|2) theory
The gauge connections of the classical solutions (black holes and conicaldefects) embedded in sl(3|2) have the following generic form
A =( 1∑m=−1
(tmemρT−
m + smemρT+
m) +2∑
m=−2
(wmemρWm) + ξJ
)dx+
− ξJdx− + (T+0 + T−
0 )dρ
For exampleHigher spin conical defect :
A =(e−ρδ−1T+−1 + eρδ1T
+1 + e−ρβ−1T
−−1
+ eρβ1T−1 + e−ρη−1W−1 + eρη1W1 + ξJ)dx+
− ξJdx− + (T−0 + T+0 )dρ
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 10/24
Supersymmetry of solutions in the sl(3|2) theory
� We have explicitly found Killing spinors for the previously known as wellas newly constructed black holes and conical defects (with sl(2)R-symmetry part) which can be embedded in the sl(3|2) theory.
� The general strategy involved the reduction of the Killing spinorequations to ODEs with constant coefficients.
� We have also carried out the analysis for the case of AdS3, spin-2 conicaldefects and the BTZ black hole. The supersymmetric conditions whichwe obtain for these match with those of the works done earlier.(Izquierdo & Townsend ‘95, Coussaert & Hennaux ‘93)
� For concreteness we shall be considering supersymmetry in one copy ofsl(3|2)⊕sl(3|2).
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 11/24
Supersymmetry of solutions in the sl(3|2) theoryThe general strategy
� The Killing spinor equation can be written in terms of the structureconstants of the superalgebra as
(∂µεr)Gr + εaAbµfbacGc = 0
� For connections of the form we are considering the above matrixequation is solved by
ε = R(ρ)eξx−∑i
cie−λix+zi
Here, Rab(ρ) = e(a)ρδab. λi and zi are eigenvalues and eigenvectorsof R−1(M+)R.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 12/24
Supersymmetry of solutions in the sl(3|2) theoryThe general strategy
� The Killing spinor equation can be written in terms of the structureconstants of the superalgebra as
∂µεc + (Mµ)caε
a = 0, with (Mµ)ac = Abµfbac
� For connections of the form we are considering the above matrixequation is solved by
ε = R(ρ)eξx−∑i
cie−λix+zi
Here, Rab(ρ) = e(a)ρδab. λi and zi are eigenvalues and eigenvectorsof R−1(M+)R.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 12/24
Supersymmetry of higher spin black holesSummary of results
BTZ black holein sl(2) with M = J Periodic Killing spinors
BTZ black holewith M = J Periodic Killing spinors
Higher spin black hole(Gutperle et al.)with W = µ = 0 ξ = ±
√2πLk or ξ = in2
A new R-chargedhigher spin black hole
with W = µ = 0
ξ = ±(√
2πL1
k ± 12
√2πL2
k
)or ξ = ± 1
2
√2πL2
k
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 13/24
Supersymmetry of higher spin conical defectsSummary of results
AdS3 in sl(2) Anti-periodic Killing spinors
AdS3 Anti-periodic Killing spinors
Higher spinconical defect
2ξ = ±i√αδ + in
2ξ = ±i√α(δ ± 2(β2 − ( 3
4η)2)1/2)
+ in
Conical defectsin sl(2) ξ = ±i
√γ
4 + in
Conical defectsin gravitational sl(2) ξ = ±i
√γ
4 + in or ξ = ±3i√γ
4 + in.
(cf. H. Tan ‘12)
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 14/24
Supersymmetry of higher spin conical defectsSummary of results
� In addition to the periodicity conditions we also need to check whetherthese conical defects are smooth i.e. whether the holonomy matrix istrivial.
� We need to impose energy bounds on these conical defects. These shouldhave energy higher than that of global AdS (−c/24) and the M = 0BTZ black hole (0).
� We have shown that smooth supersymmetric conical defects do not existin the sl(3|2) theory.
� However, on performing a similar analysis for the sl(4|3) case, we do findsmooth supersymmetric conical defects.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 14/24
Outline
Introduction and motivationWhy higher spins?Chern-Simons higher spin supergravity
Supersymmetry of classical solutionsWhen are classical solutions supersymmetric?
Supersymmetry of solutions in the sl(3|2) theoryThe sl(3|2) superalgebraSupersymmetry of black holes and conical defects
Generalization of results for the sl(N |N − 1) theoryThe sl(N |N − 1) supergroupKilling spinors and holonomyHigher spin conical defectsSmoothness and supersymmetry
Outlook
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 15/24
The sl(N |N − 1) supergroup� The supergroup for N = 2 Chern-Simons higher spin supergravity issl(N |N − 1).
� It has
Even part : sl(N)⊕ sl(N − 1)⊕ U(1)
Odd part : (N ,N − 1) + (N,N − 1)
� This supergroup has 2× 2N(N − 1) generators. Bosonic : Eij , Eı andfermionic : Ei, Eıj .
� The Cartan subalgebra is given by
HI = EII − EI+1,I+1 for I 6= N
HN = ENN − EN+1,N+1
� The commutation relations between each is these generators are explicitlyknown. In particular
[Hr, Gi] = αriGr, [J,Gi] = ±Gi� We shall be using the Cartan-Weyl basis for our analysis.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 16/24
Killing spinors and holonomyA generalized description
� We need to solve for Killing spinor equation
∂µεiGi +Aaµε
i[Ta, Gi] = 0
and analyse its periodicity so that the background corresponds toRamond or Neveu-Schwartz sectors.
� The holonomy matrix is given as
Holφ(A) = P exp
(∮Aµdx
µ
)� A formal solution to the Killing spinor equation can be written in terms
of the holonomy matrix as
ε(x) = P(e∫ xx0Aµdx
µ)ε(x0)P
(e−
∫ xx0Aµdx
µ)
� The periodicity can be determined by
ε(ρ, t, 2π) = b−1S−1e2π(λrHr+2ξJ)εi(ρ, t, 0)Gie−2π(λrHr+2ξJ)Sb
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 17/24
Killing spinors and holonomyA generalized description
� We need to solve for Killing spinor equation
∂µεiGi +Aaµε
i[Ta, Gi] = 0
and analyse its periodicity so that the background corresponds toRamond or Neveu-Schwartz sectors.
� The holonomy matrix is given as
Holφ(A) = b−1 exp
(∮aφdφ
)b where, b = eρL0
� A formal solution to the Killing spinor equation can be written in termsof the holonomy matrix as
ε(x) = P(e∫ xx0Aµdx
µ)ε(x0)P
(e−
∫ xx0Aµdx
µ)
� The periodicity can be determined by
ε(ρ, t, 2π) = b−1S−1e2π(λrHr+2ξJ)εi(ρ, t, 0)Gie−2π(λrHr+2ξJ)Sb
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 17/24
Killing spinors and holonomyA generalized description
� We need to solve for Killing spinor equation
∂µεiGi +Aaµε
i[Ta, Gi] = 0
and analyse its periodicity so that the background corresponds toRamond or Neveu-Schwartz sectors.
� The holonomy matrix is given as
Holφ(A) = b−1S−1 exp 2π(λrHr + 2ξJ)Sb
� A formal solution to the Killing spinor equation can be written in termsof the holonomy matrix as
ε(x) = P(e∫ xx0Aµdx
µ)ε(x0)P
(e−
∫ xx0Aµdx
µ)
� The periodicity can be determined by
ε(ρ, t, 2π) = b−1S−1e2π(λrHr+2ξJ)εi(ρ, t, 0)Gie−2π(λrHr+2ξJ)Sb
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 17/24
Killing spinors and holonomyA generalized description
� We need to solve for Killing spinor equation
∂µεiGi +Aaµε
i[Ta, Gi] = 0
and analyse its periodicity so that the background corresponds toRamond or Neveu-Schwartz sectors.
� The holonomy matrix is given as
Holφ(A) = b−1S−1 exp 2π(λrHr + 2ξJ)Sb
� A formal solution to the Killing spinor equation can be written in termsof the holonomy matrix as
ε(x) = P(e∫ xx0Aµdx
µ)ε(x0)P
(e−
∫ xx0Aµdx
µ)
� The periodicity can be determined by
ε(ρ, t, 2π) = e2π(λrαri±2ξ)b−1S−1εi(ρ, t, 0)GiSb
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 17/24
Killing spinors and holonomyA generalized description
� We need to solve for Killing spinor equation
∂µεiGi +Aaµε
i[Ta, Gi] = 0
and analyse its periodicity so that the background corresponds toRamond or Neveu-Schwartz sectors.
� The holonomy matrix is given as
Holφ(A) = b−1S−1 exp 2π(λrHr + 2ξJ)Sb
� A formal solution to the Killing spinor equation can be written in termsof the holonomy matrix as
ε(x) = P(e∫ xx0Aµdx
µ)ε(x0)P
(e−
∫ xx0Aµdx
µ)
� The periodicity can be determined by
ε(ρ, t, 2π) = e2π(λrαri±2ξ)εi(ρ, t, 0)Gi
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 17/24
Killing spinors and holonomyA generalized description
� The Killing spinor is therefore periodic when
λrαri ± 2ξ = ini
where, λr are eigenvalues of Holφ(A), αri s are the odd roots of thesuperalgebra, ξ is the U(1) field and ni ∈ Z.
� We have verified that this is the same relation for the periodicityconditions of black holes and conical defects in the sl(3|2) theory.
� This is important since the Chern-Simons action is independent of themetric on the manifold and the eigenvalues of the holonomy are the onlygauge invariant well-defined physical observables.
� The above result is true for any Chern-Simons supergravity theory.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 18/24
Higher spin conical defectsWe can write the gauge connections for conical defects in thesl(N |N − 1)× sl(N |N − 1) theory as (Castro et al. ‘11)
A = b−1ab+ b−1db , A = b−1ab+ b−1db
where b = exp(ρL0) and
a =
N−1∑k=1
Bk(ak, αak) +
2N−2∑k=N+1
Bk(ak, αak)
dx+ + 2ξJdφ
a = −
N−1∑k=1
Bk(γak,γαak) +
2N−2∑k=N+1
Bk(γak,γαak)
dx− − 2ξJdφ
One can write the holonomy matrix for the gauge connections, in terms oflinear combinations of the Cartan generators of the supergroup.
Upon imposing periodicity conditions (and using the ‘odd-root formula’) weget (for N ≥ 5) conditions like
i(ai − a) + 2ξ = ini (ni ∈ Z)
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 19/24
Smoothness and supersymmetry
� For the case of conical defects we also need to demand that the solutionsare smooth, i.e. the holonomy is trivial along the angular direction(Castro et al. ’11).� For even N , Holφ(A)= ±1N×N ⊕ 1(N−1)×(N−1)
� For odd N , Holφ(A)= 1N×N ⊕±1(N−1)×(N−1)
� ai, aı and ξ (eigenvalues of the holonomy matrix) are then restricted tointegers/half-integers.
� The Killing spinor periodicity conditions further constrain thesecombinations of integers.
� We now require to energy of the conical defect to obey the followingenergy bound (Izquierdo et al. ‘95, Castro et al. ‘11)
− c
24AdS3 (vacuum)
<c
24ε(N |N−1)
(str(a2
φ))< 0
BTZ with M=0
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 20/24
Are there supersymmetric conical defects which aresmooth?
� Periodicity requirements
mi
2− p + q = ni for even N, ri −
s2
+ t = ni for odd N.
� Energy bound conditions
For even N : 0 <1
4
N−1∑j=1
j odd
m2j −
2N−3∑=N+1
j odd
p2 −N(N − 1)q2 <
N(N − 1)
8
For odd N : 0 <
N−2∑j=1
j odd
r2j −
1
4
2N−2∑=N+1j even
s2 −N(N − 1)t2 <
N(N − 1)
8
� Smooth supersymmetric conical defects exist for N ≥ 5.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 21/24
Outline
Introduction and motivationWhy higher spins?Chern-Simons higher spin supergravity
Supersymmetry of classical solutionsWhen are classical solutions supersymmetric?
Supersymmetry of solutions in the sl(3|2) theoryThe sl(3|2) superalgebraSupersymmetry of black holes and conical defects
Generalization of results for the sl(N |N − 1) theoryThe sl(N |N − 1) supergroupKilling spinors and holonomyHigher spin conical defectsSmoothness and supersymmetry
Outlook
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 22/24
Outlook
� We have considered supersymmetry of classical solutions in thesl(N |N − 1) theory.
� Periodicity conditions of the Killing spinor can be formulated as productsof background holonomies with the odd roots of the superalgebra.
� We have arrived at the conditions which should hold good if smoothsupersymmetric defects are allowed in the theory.
� Smooth supersymmetric conical defects exist for N ≥ 4.
� The Killing spinor equations were explicitly solved for the sl(3|2) theory.It has been verified that the periodicity conditions obtained thereof agreewith the ‘odd-root formula’.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 23/24
Outlook
� How do we classify and compare the conical defects with the chiralprimaries of the Kazama-Suzuki model? (For the non-susy case, smoothconical defects correspond primaries of the minimal model.)
� What about conical surpluses?
� Thermodynamics of the newly constructed black holes.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 23/24
Thank you.
Shouvik Datta Supersymmetry of classical solutions in Chern-Simons higher spin supergravity 24/24
top related