SMOOTH SOLUTIONS IN VASILIEV THEORY Andrea Campoleoni Universit - - PowerPoint PPT Presentation

SMOOTH SOLUTIONS IN “VASILIEV THEORY”

A.C., T. Procházka, J. Raeymaekers, 1303.0880

Workshop on “Higher Spins, Strings and Duality”, Galileo Galilei Institute, Firenze, 7/5/2013

Andrea Campoleoni

Université Libre de Bruxelles & International Solvay Institutes

Einstein-Hilbert action

GRAVITY IN D = 2+1

2

← constant curvature!

Rab

l

≡ d⇤ab + ⇤ac ∧ ⇤c

b + 1

l2 ea ∧ eb = 0 T a ≡ dea + ⇤a

b ∧ eb = 0

I = 1 16⇥G

⁄

abc

3

ea ∧ Rbc + 1 3l2 ea ∧ eb ∧ ec

4

gµν = ⇥ab ea

µeb ν

⇒ I = 1 16⇤G

⁄

d3x√−g

3

R + 2 l2

4

Rewriting in terms of the metric Field equations

Einstein-Hilbert action

GRAVITY IN D = 2+1

2

I = 1 16⇥G

⁄

abc

3

ea ∧ Rbc + 1 3l2 ea ∧ eb ∧ ec

4

gµν = ⇥ab ea

µeb ν

⇒ I = 1 16⇤G

⁄

d3x√−g

3

R + 2 l2

4

!µ

a = 1

2 ✏a

bc !µ b,c

Rewriting in terms of the metric A couple of useful tricks...

. so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R)

Einstein-Hilbert action

GRAVITY IN D = 2+1

2

gµν = ⇥ab ea

µeb ν

⇒ I = 1 16⇤G

⁄

d3x√−g

3

R + 2 l2

4

!µ

a = 1

2 ✏a

bc !µ b,c

Rewriting in terms of the metric A couple of useful tricks...

. so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R)

• I =

1 8⇡G Z ✓ ea ∧ Ra + 1 6l2 ✏abc ea ∧ eb ∧ ec ◆

Einstein-Hilbert action

GRAVITY IN D = 2+1

2

gµν = ⇥ab ea

µeb ν

⇒ I = 1 16⇤G

⁄

d3x√−g

3

R + 2 l2

4

!µ

a = 1

2 ✏a

bc !µ b,c

Rewriting in terms of the metric A couple of useful tricks...

. so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R)

I = 1 16⇡G Z tr ✓ e ∧ R + 1 3l2 e ∧ e ∧ e ◆

with

( e = eaJa ! = !aJa

Achúcarro, Townsend (1986); Witten (1988)

Einstein-Hilbert action

GRAVITY IN D = 2+1

2

!µ

a = 1

2 ✏a

bc !µ b,c

A couple of useful tricks...

. so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R)

I = 1 16⇡G Z tr ✓ e ∧ R + 1 3l2 e ∧ e ∧ e ◆

with

( e = eaJa ! = !aJa

Achúcarro, Townsend (1986); Witten (1988)

Chern-Simons formulation on AdS3

e e e = l 2 ⇣ A − e A ⌘ ⇤ = 1 2 ⇣ A + e A ⌘

HIGHER SPINS IN D = 2+1

Natural generalization of the gravity frame action

3

For g = sl(N,R) describes fields of “spin” 2,3,...,N

e = eµ

A JA dxµ =

1

eµ

aJa + eµ ab Tab + · · ·

2

dxµ ⌅ = ⌅µ

A JA dxµ =

1

⌅µ

aJa + ⌅µ ab Tab + · · ·

2

dxµ

Blencowe (1989)

Example: the sl(3,R) algebra

[ Ja , Jb ] = abc Jc [ Ja , Tbc ] = m

a(bTc)m

[ Tab , Tcd ] = ⇤

• ⇥a(cd)bm + ⇥b(cd)am
• Jm

I = 1 16⇡G Z tr ✓ e ∧ R + 1 3l2 e ∧ e ∧ e ◆

with

R = d! + ! ∧ !

HIGHER SPINS IN D = 2+1

Natural generalization of the gravity frame action

3

For g = sl(N,R) describes fields of “spin” 2,3,...,N

e = eµ

A JA dxµ =

1

eµ

aJa + eµ ab Tab + · · ·

2

dxµ ⌅ = ⌅µ

A JA dxµ =

1

⌅µ

aJa + ⌅µ ab Tab + · · ·

2

dxµ

Blencowe (1989)

I = 1 16⇡G Z tr ✓ e ∧ R + 1 3l2 e ∧ e ∧ e ◆

with

R = d! + ! ∧ !

More in general: take any Lie algebra g with a non-degenerate Killing form and branch it under the adjoint action of sl(2,R)↪g

g = sl(2, R) ⊕

⇤ ⇧ ⌥

⌅ , a

g(⌅,a)

⌅ ⌃

2` + 1

dim =

HIGHER SPINS IN D = 2+1

Simple characterization in terms of Chern-Simons theories (for gauge fields) Field equations → flatness conditions Simple field equations, but rich space of solutions on AdS

Non-trivial topology → black holes Boundary conditions → boundary dynamics, AdS/CFT...

How to select “non-singular” solutions?

4

S = SCS[A] − SCS[ ⌃ A] e = l 2

• A −

A

• , ω = 1

2

• A +

A

• ⇒

F = dA + A ∧ A = 0 .

e F = d e A + e A ∧ e A

Gaberdiel, Gopakumar (2010) Gutperle, Kraus (2011) Gutperle, Kraus (2011) Castro, Gopakumar, Gutperle, Raeymaekers (2011)

OUTLINE

Coupling ∞ spins: hs[λ] Chern-Simons theories Smoothness criteria for asymptotically AdS solutions “Analytic continuation” of the sl(N) conical surpluses Conclusion

THE GAUGE SECTOR OF THE PROKUSHKIN-VASILIEV MODEL

FRAME-LIKE DESCRIPTION FOR HS

HS “vielbeins” and “spin connections”

7

eµa1... as−1 ωµb,a1... as−1

⇒ ⇒

Everything is traceless, then in D=2+1...

≈ (example: ) “Vielbeins” and “spin connections” have the same structure

Structure of the higher-spin generators:

eab... traceless ⇒ Tab... traceless in ab... eab... irreducible ⇒ !µ

a = 1

2 ✏a

bc !µ b,c

[ Ja , Tb1... bs−1 ] = m

a(b1Tb2... bs−1)m

Consider traceless and symmetric polynomials in

(+ traceless projection in the an indices) N-dim repr. for ⇒ N2

• 1ind. traceless matrices out of T’s with s<N

SL(N) HIGHER-SPIN THEORIES

For sl(3,R) the Jacobi identity fixes the algebra but...

⇒ 3-dim repr. for ⇒

8

General lesson to build higher-spin algebras: choose a representation of so(1,2) ≃ sl(2,R) and compute products of the representatives

[ Ja , Tbc ] = m

a(bTc)m ,

• [ Tab , Tcd ] = ⇤
• ⇥a(cd)bm + ⇥b(cd)am
• Jm

Tab = √ −σ ⇣ J(aJb) − 2 3 ηabJcJc ⌘

Ja Ja

Ta1... as ∼ J(a1 . . . Jas)

Ja

Hoppe (1982)

HIGHEST WEIGHT IRREPS OF SL(2,R)

λ≠N ⇒ two pairs of conjugate irreps

Realize the generators as e.g. and act on

9

sl(2,R) algebra:

[ J+ , J− ] = 2J0 , [ J± , J0 ] = ± J± C2 = J2

0 − 1

2(J+J− + J−J+) = 1 4(λ2 − 1)

Casimir:

J+ = y @ @x , J0 = 1 2 ✓ x @ @x − y @ @y ◆ , J− = − x @ @y .

vi = xiyλ−i−1 , ¯ vi = xλ−i−1yi , wi = xiy−(λ+i+1) , ¯ wi = x−(λ+i+1)yi ,

HIGHEST WEIGHT IRREPS OF SL(2,R)

λ≠N ⇒ two pairs of conjugate irreps

Realize the generators as e.g. and act on

9

sl(2,R) algebra:

[ J+ , J− ] = 2J0 , [ J± , J0 ] = ± J± C2 = J2

0 − 1

2(J+J− + J−J+) = 1 4(λ2 − 1)

Casimir:

J+ = y @ @x , J0 = 1 2 ✓ x @ @x − y @ @y ◆ , J− = − x @ @y .

vi = xiyλ−i−1 , ¯ vi = xλ−i−1yi , wi = xiy−(λ+i+1) , ¯ wi = x−(λ+i+1)yi ,

A FAITHFUL MATRIX REPR. OF HS[ ]

Irrep of sl(2,R) with highest weight : Building the hs[λ] generators: Explicit realization:

10

λ

(J+)jk = δj, k+1 , (J−)jk = j(j − λ) δj+1, k , (J0)jk = 1 2(λ + 1 − 2j) δj, k ,

t 1

2(λ − 1).

T ℓ

m = (−1)ℓ−m (ℓ + m)!

(2ℓ)!

• J−, . . . [ J−, [ J−
• ℓ − m terms

, (J+)ℓ ]]

• (T ℓ

m)jk = (−1)ℓ−m ℓ−m

• n = 0

ℓ − m n [ ℓ ]n [ 2ℓ ]n [ ℓ − λ ]n [ j − m − 1 ]ℓ−m−n δj, k+m ,

A.C., Procházka, Raeymaekers (2013) Pope, Romans, Shen (1990)

A HS[ ] CHERN-SIMONS THEORY

The satisfy Trace: Chern-Simons theory with hs[λ]⊕hs[λ] gauge algebra as a model for the interactions of spins 2,...,∞ Field equations:

11

λ

T ℓ

m

[ Ji, T `

m ] = (i` − m) T ` m+i

Bergshoeff, Blencowe, Stelle (1990); Vasiliev (1991)

tr v =

6 λ(λ2 − 1) lim

N→λ N

• j = 1

vjj ,

F = dA + A ∧ A = 0 . ¯ F = d ¯ A + ¯ A ∧ ¯ A = 0

A HS[ ] CHERN-SIMONS THEORY

The satisfy Trace: Chern-Simons theory with hs[λ]⊕hs[λ] gauge algebra as a model for the interactions of spins 2,...,∞ Field equations:

11

λ

T ℓ

m

What are the “admissible” connections?

[ Ji, T `

m ] = (i` − m) T ` m+i

Bergshoeff, Blencowe, Stelle (1990); Vasiliev (1991)

tr v =

6 λ(λ2 − 1) lim

N→λ N

• j = 1

vjj ,

F = dA + A ∧ A = 0 . ¯ F = d ¯ A + ¯ A ∧ ¯ A = 0

PROPERTIES OF THE HS[ ] MATRICES

12

λ

m

ts (T ℓ

m)j, j−m

The non-zero elements belong to the diagonal The are polynomials in j

Properties of the hs[λ] matrices:

m

ts (T ℓ

m)j, j−m

Is that enough? What kinds of linear combinations do we have to consider? What are their properties?

PROPERTIES OF THE HS[ ] MATRICES

12

λ

Properties of the hs[λ] matrices:

∃ N such that if j > k+N The matrix elements along a diagonal, for some fixed n, become polynomial in j for sufficiently large j

at vj, k = 0

l, vj, j+n

Khesin, Malikov (1996)

PROPERTIES OF THE HS[ ] MATRICES

12

λ

Properties of the hs[λ] matrices:

∃ N such that if j > k+N The matrix elements along a diagonal, for some fixed n, become polynomial in j for sufficiently large j

at vj, k = 0

l, vj, j+n

Khesin, Malikov (1996)

The trace is still well defined

tr v ∼ lim

N→λ N

X

j = 1

vjj

PROPERTIES OF THE HS[ ] MATRICES

12

λ

Properties of the hs[λ] matrices:

∃ N such that if j > k+N The matrix elements along a diagonal, for some fixed n, become polynomial in j for sufficiently large j

at vj, k = 0

l, vj, j+n

Khesin, Malikov (1996)

The trace is still well defined

tr v ∼ lim

N→λ N

X

j = 1

vjj

One can perform the substitution provided that is large enough

1) lim

N→λ

+ N.

SMOOTH ASYMPTOTICALLY ADS SOLUTIONS

ASYMPTOTICALLY ADS SOLUTIONS

Focus on a single chiral sector: Radial gauge fixing:

14

Boundary conditions → W∞[λ] algebra on the boundary

F = dA + A ∧ A = 0 .

g = eρ T 1

t

ρ Φ

as z ≡ φ+itE

a(z) = α T 1

1 + 12π

c

∞

• ℓ=1

α−ℓ Nℓ Wℓ+1(z) T ℓ

−ℓ ,

a(¯ z) = 0

Henneaux, Rey (2010) A.C., Fredenhagen, Pfenninger, Theisen (2010) Gaberdiel, Hartman (2011) A.C., Fredenhagen, Pfenninger (2011)

→

A = g−1a(z)g dz + g−1dg ,

→

SMOOTH SOLUTIONS?

How to identify smooth solutions?

The metric is no longer gauge invariant: the usual regularity conditions do not apply One can ask that the gauge field A is non singular

15

Gauge invariant characterization: holonomies

H(A) = P exp I A = g−1 exp(2πa) g

Smooth solutions → trivial holonomy around Φ H = e2πa = eiϕ0 T 0

0 ,

Gutperle, Kraus (2011) Castro, Gopakumar, Gutperle, Raeymaekers (2011) A.C., Procházka, Raeymaekers (2013)

HS[ ] PROJECTORS

How to control exponentials of hs[λ] elements? A simple class under control: the gl[λ] projectors

16

λ

ecP

⋆

= T 0

0 + (ec − 1)P.

g P ⋆ P = P,

⇒

HS[ ] PROJECTORS

How to control exponentials of hs[λ] elements? A simple class under control: the gl[λ] projectors

16

λ

General hs[λ] element in diagonal gauge

ecP

⋆

= T 0

0 + (ec − 1)P.

g P ⋆ P = P,

⇒

a = − i

• j

mjPj + λ2 − 1 6 i tr

j

mjPj

• T 0

(Pj)kl = δjkδjl

HS[ ] PROJECTORS

How to control exponentials of hs[λ] elements? A simple class under control: the gl[λ] projectors

16

λ

General hs[λ] element in diagonal gauge Holonomy:

ecP

⋆

= T 0

0 + (ec − 1)P.

g P ⋆ P = P,

⇒

a = − i

• j

mjPj + λ2 − 1 6 i tr

j

mjPj

• T 0

e2πa = e2π λ2−1

6

itr(

j mjPj)

• T 0

0 +

• j

(e−2πimj − 1)Pj

• .

HS[ ] PROJECTORS

How to control exponentials of hs[λ] elements? A simple class under control: the gl[λ] projectors

16

λ

General hs[λ] element in diagonal gauge Holonomy:

ecP

⋆

= T 0

0 + (ec − 1)P.

g P ⋆ P = P,

⇒

a = − i

• j

mjPj + λ2 − 1 6 i tr

j

mjPj

• T 0

e2πa = e2π λ2−1

6

itr(

j mjPj)

• T 0

0 +

• j

(e−2πimj − 1)Pj

• .

THE COUNTERPARTS OF THE SL(N) CONICAL SOLUTIONS

REWRITING OF THE SOLUTION

General gauge connection that exponentiate to 1

18

Order the integer mj and define sj = mj + j

solution specified by an ordered list of integers s1≥s2≥... the sj must become polynomials in j for large enough j

Looking at the matrix realization of J0 ~ T10 leads to

a = − i

• j

mjPj + λ2 − 1 6 i tr

j

mjPj

• T 0

a = − i

• j

sjPj + λ2 − 1 6 i tr

j

sjPj

• T 0

0 − i T 1

CONICAL SOLUTIONS

General gauge connection that exponentiate to 1

19

a = − i

• j

sjPj + λ2 − 1 6 i tr

j

sjPj

• T 0

0 − i T 1

The sj must become polynomials in j for j large Simplest solution: sj = S for j > N Smooth connections ↔ Young tableaux

length of the rows of Λ → rj = sj - S

CONICAL SOLUTIONS

General gauge connection that exponentiate to 1

19

a = − i

• j

sjPj + λ2 − 1 6 i tr

j

sjPj

• T 0

0 − i T 1

The sj must become polynomials in j for j large Simplest solution: sj = S for j > N Smooth connections ↔ Young tableaux

length of the rows of Λ → rj = sj - S

aΛ = − i

• j

rjPj + iB λ T 0

0 − i T 1

CONICAL SOLUTIONS

General gauge connection that exponentiate to 1

19

a = − i

• j

sjPj + λ2 − 1 6 i tr

j

sjPj

• T 0

0 − i T 1

The sj must become polynomials in j for j large Simplest solution: sj = S for j > N Smooth connections ↔ Young tableaux

length of the rows of Λ → rj = sj - S

aΛ = − i

• j

rjPj + iB λ T 0

0 − i T 1

λ = N ⇒ conical solutions of Castro, Gopakumar, Gutperle, Raeymaekers (2011)

CONCLUSIONS & OUTLOOK

hs[λ] can be conveniently treated as an algebra of infinite matrices

Simple discussion of smoothness conditions

Applications:

Drinfeld-Sokolov reduction i.e. asymptotics Conical solutions in Vasiliev theory

Perspectives:

Role on the CFT side of the smooth solution that are not classified by finite Young tableaux? (see talk by J. Raeymaekers) New thermodynamical branches built on these solutions?

20

Kraus, Perlmutter (2011) A.C., Procházka, Raeymaekers (2013) A.C., Procházka, Raeymaekers (2013) Khesin, Malikov (1996) A.C., Fredenhagen, Pfenninger (2011)