supersymmetry of classical solutions in chern-simons ...ism2012/talks/ism-shouvik.pdf ·...

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


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?


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


Higher spins and holography

(2+1)-d Gravity

sl(2, R)⊕ sl(2, R)Virasoro algebra

CFT2

BTZ black hole,conical defects



(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



(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



(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.


Outline





Outlook


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.


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)


Outline





Outlook


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.


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




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




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


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).


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.


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.


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


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)


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.


Outline





Outlook


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.


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




∂µεiGi +Aaµε

i[Ta, Gi] = 0



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


µ)ε(x0)P

(e−

∫ xx0Aµdx

µ)






∂µεiGi +Aaµε

i[Ta, Gi] = 0



Holφ(A) = b−1S−1 exp 2π(λrHr + 2ξJ)Sb



µ)ε(x0)P

(e−

∫ xx0Aµdx

µ)






∂µεiGi +Aaµε

i[Ta, Gi] = 0






µ)ε(x0)P

(e−

∫ xx0Aµdx

µ)


ε(ρ, t, 2π) = e2π(λrαri±2ξ)b−1S−1εi(ρ, t, 0)GiSb




∂µεiGi +Aaµε

i[Ta, Gi] = 0






µ)ε(x0)P

(e−

∫ xx0Aµdx

µ)


ε(ρ, t, 2π) = e2π(λrαri±2ξ)εi(ρ, t, 0)Gi



� 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.


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)


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


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.


Outline





Outlook


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’.


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.


Thank you.


supersymmetry of classical solutions in chern-simons ...ism2012/talks/ism-shouvik.pdf ·...

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