higher spins & strings - research school of physics...

Higher Spins & Strings

Matthias Gaberdiel

ETH Zürich

The Mathematics of CFT

ANU Canberra

13 July 2015

Based mainly on work with Rajesh Gopakumar,

1011.2986, 1106.1897, 1205.2472, 1305.4181,1406.6103, 1501.07236

AdS / CFT duality

Much recent progress in string theory has been related to AdS/CFT duality

AdS5 × S5

superstrings on SU(N) super Yang-Mills theory in 4 dimensions

=

4d non-abelian gauge theory similar to that appearing in the standard model of particle physics.

[Maldacena, ‘97, ...]

(

R

lPl

⇥4

= N

Relation of parameters

(

R

ls

⇥4

= g2

YMN = λgstring = g

2YM

The relation between the parameters of the two theories is

AdS radius in string units

‘t Hooft parameter

AdS radius in Planck units

Strong weak duality

For example, in the large N limit of gauge theory at large ‘t Hooft coupling

Supergravity (point particle) approximation is good for AdS description.

largelarge

(

R

lPl

⇥4

= N

(

R

ls

⇥4

= g2

YMN = λgstring = g

2YM

small

AdS/CFT duality

This is interesting since it gives insights into strongly coupled gauge theories using supergravity methods.

Many applications and insights:

‣ anomalous dimensions in N=4 SYM

‣ structural insights into amplitudes

‣ quark gluon plasma

‣ quantum critical systems

‣ ...

[Hartnoll,Herzog,Horowitz,Kachru,Sachdev,Son,...]

[Liu,Rajagopal,Wiedemann,Gubser,...]

[Witten,Cazacho,Arkani-Hamed,Alday,Maldacena, Korchemsky,Drummond,Sokatchev,...]

[Minahan,Zarembo,Beisert,Staudacher,...]

Conceptual understanding

However, at present, we are far from a conceptual understanding of why the duality works, and what ingredients are crucial for it, e.g. whether it requires

supersymmetry integrability ...

This is obviously an important question since in many applications these features are absent.

ls →∞

Weakly coupled gauge theory

In order to make progress in this direction analyse another corner of AdS/CFT: consider case where gauge theory is weakly coupled

small

`tensionless strings’[Sundborg, ‘01] [Witten, ‘01] [Sezgin,Sundell, ‘01]

(

R

lPl

⇥4

= N

(

R

ls

⇥4

= g2

YMN = λgstring = g

2YM

large

Higher spin theory

Resulting theory has an infinite number of massless higher spin fields, which generate a very large gauge symmetry.

maximally unbroken phase of string theory

effective description in terms of Vasiliev Higher Spin Theory.

Leading Regge trajectory

On the dual CFT side, the traces of bilinears of elementary Yang-Mills fields form closed subsector in free theory.

This subsector is believed to correspond to the leading Regge trajectory from the string point of view:

`vector-like’ HS -- CFT duality

[Chang, Minwalla, Sharma, Yin, ‘12] [MRG, Gopakumar, ‘14]

State of the Art

In the past this idea was taken as a general motivation to consider dualities relating

Vasiliev HS theory

on AdS

vector-like weakly coupled

CFT

However, recently interesting progress about how these dualities fit actually into stringy AdS/CFT correspondence has been made....

WN,k

λ =N

N + kand M

2 = −(1− λ2)

3d proposal

Concrete duality of this kind (somewhat similar to Klebanov & Polyakov proposal for AdS4/CFT3)

λ

AdS3: higher spin theory with a complex scalar of mass M

2d CFT: minimal models in large N ‘t Hooft limit with coupling

where

[MRG,Gopakumar, ‘10]

Outline

This version of the duality is bosonic, but can nevertheless be tested in quite some detail. In particular, we can match

‣ quantum symmetries

‣ spectrum

At the end I will also explain how this higher spin duality relates to stringy AdS3 -- CFT2 duality.

hs[λ] ⊕ C ⇠=U(sl(2))

hC2 −1

4(λ2 − 1)1i

sl(2,R) → hs[λ] ∼= sl(λ)

The HS theory on AdS3

Recall that pure gravity in AdS3: Chern-Simons theory based on [Achucarro, Townsend, ‘86]

[Witten, ‘88]

Higher spin description: replace [Prokushkin, Vasiliev, ‘98]

The AdS3 HS theory can be described very simply.

where [Bergshoeff et.al. ‘90]

[Pope, Romans, Shen, ‘90]

[Fradkin, Linetsky, ‘91]

[one spin field for each spin ]

sl(2, R)

s = 2, 3, . . .

V s

nwith |n| < s , s = 2, 3, . . .

hs[λ] :

Higher spin algebra

Generators of

`wedge algebra’

W∞[λ] algebra

For these higher spin theories asymptotic symmetry algebra can be determined following Brown & Henneaux, leading to classical

[Henneaux & Rey, ‘10] [Campoleoni et. al. ‘10] [MRG, Hartman, ‘11]

L0, L±1 → Ln , n ∈ Z

sl(2, R)

hs[λ] → W∞[λ]

s = 2, . . . ,∞, n ∈ Z

Asymptotic symmetry algebra

Asymptotic symmetry algebra extends hs algebra `beyond the wedge’:

pure gravity:

higher spin:

[Figueroa-O’Farrill et.al. ‘92]

(Virasoro)

generated by Vs

n

W∞[λ] = limN→∞

WN,k with λ =N

N + k.

Dual CFT

By the usual arguments, dual CFT should therefore have

Basic idea:

‘t Hooft limit of 2d CFT!

W∞[λ] symmetry.

WN,k :su(N)k ⊕ su(N)1

su(N)k+1

The minimal models

The minimal model CFTs are the cosets

General N: higher spin analogue of Virasoro minimal models. [Spin fields of spin s=2,3,..,N.]

e.g. Ising model (N=2, k=1) tricritical Ising (N=2, k=2) 3-state Potts (N=3,k=1),..

cN (k) = (N − 1)[

1−N(N + 1)

(N + k)(N + k + 1)

⇥

.

with central charge

[Bais et.al., ‘88] [Bouwknegt, Schoutens, ‘92]

W∞[λ]

Relation of symmetries

is a classical (commutative) Poisson algebra.

In order to understand relation to minimal model W-algebras, need to understand how to quantise it.

The asymptotic symmetry algebra

Quantisation

Quantisation is quite subtle since the Poisson algebra is non-linear --- cannot just replace Poisson brackets by commutators without violating Jacobi identities...

However, there seems to exist a unique way of defining a consistent quantum W-algebra (whose classical limit reduces to Poisson algebra).

[MRG, Gopakumar, ‘12]

[Blumenhagen, et.al. ‘94] [Hornfeck, ‘92-‘93]

[W 3m

, W 3n] = 2(m− n)W 4

m+n+

N3

12(m− n)(2m2 + 2n2

−mn− 8)Lm+n

+8N3

c(m− n) (LL)m+n +

N3c

144m(m2

− 1)(m2− 4)δm,−n

Quantum symmetry


There are two steps to this argument. To illustrate them consider an example. Naive quantisation of classical algebra leads to

spin-3 field

non-linear term

Jacobi identity

spin-3 field

non-linear term

[W 3m

, W 3n] = 2(m− n)W 4

m+n+

N3

12(m− n)(2m2 + 2n2

−mn− 8)Lm+n

+8N3

c(m− n) (LL)m+n +

N3c

144m(m2

− 1)(m2− 4)δm,−n

Λ(4)n

=X

p

: Ln−pLp : +15xnLn

[W 3m

, W 3n] = 2(m− n)W 4

m+n+

N3

12(m− n)(2m2 + 2n2

−mn− 8)Lm+n

+8N3

c + 225

(m− n) Λ(4)m+n

+N3c

144m(m2

− 1)(m2− 4)δm,−n

Jacobi identity

Jacobi identity determines quantum correction

where

Similar considerations apply for the other commutators.

[W 3m

, W 3n] = 2(m− n)W 4

m+n+

N3

12(m− n)(2m2 + 2n2

−mn− 8)Lm+n

+8N3

c(m− n) (LL)m+n +

N3c

144m(m2

− 1)(m2− 4)δm,−n

W(3)

· W(3)

∼

c

3· 1 + 2 · L +

32

(5c + 22)· Λ(4) + 4 · W

(4)

W(3)

· W(4)

∼ C433 · W

(3)+ · · ·

(

C4

33

)2=

64

5

λ2 − 9

λ2 − 4+ O( 1

c) .

Structure constants

The second step concerns structure constants. The fields can be rescaled so that

but then coupling constant

characterises algebra. Classical analysis determines

[MRG, Hartman, ‘11]

Structure constants

Classical analysis determines

(

C4

33

)2=

64

5

λ2 − 9

λ2 − 4+ O( 1

c) .

λ = N WN :

Structure constants

Classical analysis determines

Requirement that representation theory agrees for with

hs[λ]∣

∣

∣

λ=N

∼= sl(N, R)[Note: implies .] W∞[λ]|λ=N

= WN

(

C4

33

)2=

64

5

λ2 − 9

λ2 − 4+ O( 1

c) .

γ2≡

(

C4

33

)2=

64 (c+ 2) (λ− 3)(

c(λ+ 3) + 2(4λ+ 3)(λ− 1))

(5c+ 22) (λ− 2)(

c(λ+ 2) + (3λ+ 2)(λ− 1)) .

λ = 0 :

λ = 1 :

Explicit check

This formula has also been checked explicitly for the two special cases:

N complex bosons giving c=2N

N complex fermions with u(1) coset giving c = N-1

[MRG, Jin, Li, ‘13]

[Bakas, Kiritsis, ‘90]

[Bergshoeff et.al. ‘90]

Higher Structure Constants

Similarly, higher structure constants can be determined [Blumenhagen, et.al. ‘94 ]

[Hornfeck, ‘92-‘93]

C4

33C

4

44=

48(

c2(λ2− 19) + 3c(6λ3

− 25λ2 + 15) + 2(λ− 1)(6λ2− 41λ− 41)

)

(λ− 2)(5c+ 22)(

c(λ+ 2) + (3λ+ 2)(λ− 1))

(C5

34)2 =

25(5c+ 22)(λ− 4)(

c(λ+ 4) + 3(5λ+ 4)(λ− 1))

(7c+ 114)(λ− 2)(

c(λ+ 2) + (3λ+ 2)(λ− 1))

C5

45=

15

8(λ− 3)(c+ 2)(114 + 7c)(

c(λ+ 3) + 2(4λ+ 3)(λ− 1)) C

4

33

×

h

c3(3λ2

− 97) + c2(94λ3

− 467λ2− 483) + c(856λ3

− 5192λ2 + 4120)

+ 216λ3− 6972λ2 + 6756

i

.

C4

44=

9(c + 3)

4(c + 2)γ −

96(c + 10)

(5c + 22)γ−1

(C5

34)2 =

75(c + 7)(5c + 22)

16(c + 2)(7c + 114)γ2− 25

C5

45=

15 (17c + 126)(c + 7)

8 (7c + 114)(c + 2)γ − 240

(c + 10)

(5c + 22)γ−1

γ2≡

(

C4

33

)2

Higher Structure Constants

Actually, can rewrite all of them more simply as

where

These structure constants (and probably all) are actually determined in terms of by Jacobi identity.

[Candu, MRG, Kelm, Vollenweider, unpublished]


γ2

(C4

33)2 ≡ γ2 =

64(c + 2)(λ − 3)(

c(λ + 3) + 2(4λ + 3)(λ − 1))

(5c + 22)(λ − 2)(

c(λ + 2) + (3λ + 2)(λ − 1)) .

W∞[λ1] ∼= W∞[λ2] ∼= W∞[λ3] at fixed c

γ2 (rather than λ)

Quantum algebra

Thus full quantum algebra seems to be characterised by two free parameters: central charge c and

But

Thus there are three roots that lead to the same algebra:


`Triality’

W∞[N ] ∼= W∞[ N

N+k] ∼= W∞[− N

N+k+1] at c = cN (k)

Triality

In particular,

minimal model asymptotic symmetry algebra of hs theory

This is even true at finite N and k, not just in the ‘t Hooft limit!

This triality generalises level-rank duality of coset models of [Kuniba, Nakanishi, Suzuki, ‘91] and [Altschuler, Bauer, Saleur, ‘90].

hs[λ]

W∞[λ]

Symmetries

So the symmetries give strong evidence for the duality

HS on AdS3 2d CFT with

symmetry

=CS with

minimal models

=

(`, λ) (N, k)λ =

N

N + k

3`

4GN

= cN,ksize of AdS3 (sugra: c large)

Spectrum

Higher spin fields themselves correspond only to the vacuum representation of the W-algebra!

(Λ; 0)Contribution from all representations of the form is accounted for by adding to the hs theory a complex scalar field of mass

−1 ≤M2≤ 0 with M

2 = −(1− λ2) .

[Compatible with hs symmetry since hs theory has massive scalar multiplet with this mass.]

[Prokushkin, Vasiliev, ‘98]


Total 1-loop partition function

Z(1)pert =

∞∏

s=2

∞∏

n=s

1

|1− qn|2×

∞∏

l,l′=0

1

(1− qh+lq̄h+l′)2

The perturbative 1-loop partition function of the hs theory is then

higher spin fields

scalar fields

Total 1-loop partition function

Z(1)pert =

∞∏

s=2

∞∏

n=s

1

|1− qn|2×

∞∏

l,l′=0

1

(1− qh+lq̄h+l′)2

The perturbative 1-loop partition function of the hs theory is then

(f; 0)⊗r1 ⊗ (̄f; 0)⊗r2modes

We have shown analytically that this agrees exactly with CFT partition function of representations in ‘t Hooft limit! [MRG,Gopakumar, ‘10]

[MRG,Gopakumar,Hartman,Raju, ‘11]

(Λ; 0)

WN

Zpert =

X

Λ

|χ(Λ,0)|2 .

Spectrum

(Λ; ν) with ν 6= 0

The remaining states, i.e. those of the form

seem to correspond to conical defect solutions (possibly dressed with perturbative excitations).

[Castro, Gopakumar, Gutperle, Raeymaekers, ‘11] [MRG, Gopakumar, ‘12] [Perlmutter, Prochazka, Raeymaekers, ‘12]

Thus

N = 4

λ =N

N + k + 2

Recent developments

In order to understand relation to string theory have studied supersymmetric version of duality, in particular, case with large superconformal symmetry.

hs theory based on

shs2[λ]Wolf space cosets

su(N + 2)k ⊕ so(4N + 4)1su(N)k+2 ⊕ u(1)κ

⊕ u(1)κ .

in ‘t Hooft limit with .


AdS3 × S3× S

3× S

1

Vir⊕ su(2)⊕ su(2)⊕ u(1)N = 4

Symmetries

The Wolf space coset CFTs have same symmetry as dual CFT of string theory on

with 4 supercharges

Large

[Boonstra, Peeters, Skenderis, ‘98; Elitzur, Feinerman, Giveon, Tsabar, ‘99;

de Boer, Pasquinucci, Skenderis, ‘99; Gukov, Martinec, Moore, Strominger, ‘04; ...]

N = 4

(

T4)⊗(N+1)

/SN+1

Relation to string theory

AdS3 × S3× T

4AdS3 × S

3× S

3× S

1

hs theory based on

Wolf space cosets

shs2[λ]string theory

symmetric orbifold

large N = 4small

λ =N

N + k + 2→ 0

Contraction

While we cannot compare these two dualities directly, the large superconformal symmetry contracts to the small superconformal symmetry in the limit in which the level k goes to infinity

Indeed, this just describes the case where the radius of one of the two 3-spheres goes to infinity, and hence the sphere approximates flat space.

⊂

hs theory in string theory

N = 4

AdS3 × S3× T

4AdS3 × S

3× S

3× S

1

hs theory based on

Wolf space cosets

shs2[λ]string theory

symmetric orbifold

large N = 4small

λ→0−→

SymN+1(T

4) ≡ (T4)⊗(N+1)/SN+1

su(N + 2)k ⊕ so(4N + 4)1su(N)k+2 ⊕ u(1)κ

⊕ u(1)κ .

Wolf space cosets

What happens to the Wolf space cosets in this limit?

[MRG, Suchanek, ‘11]

As in bosonic case, the `perturbative’ part of the spectrum can be identified with the U(N)-singlet sector

Hpert =M

Λ

(0; Λ)⊗ (0; Λ∗) =⇣

4(N + 1) free bosons4(N + 1) free fermions

⌘

/U(N)


bosons: 2 · (N,1)⊕ 2 · (N̄,1)⊕ 4 · (1,1)

fermions: (N,2)⊕ (N̄,2)⊕ 2 · (1,2)

Untwisted sector

Here the free bosons and fermions transform as

U(N) su(2)

The other coset representations can be interpreted

as twisted sectors (and descendants) of this continuous

orbifold — actually, can give very concrete identification....

[MRG, Gopakumar, ‘14] [MRG, Kelm, ‘14]

SN+1

bosons: 4 · (N+ 1,1) = 4 · (N,1)⊕ 4 · (1,1)

fermions: 2 · (N+ 1,2) = 2 · (N,2)⊕ 2 · (1,2)

Comparison

This now looks very similar to the untwisted sector of the symmetric orbifold

Indeed, this sector is generated by free bosons and fermions in

su(2)

SymN+1(T

4) ≡⇣

T4(N+1)

⌘

/SN+1

SN+1 ⊂ U(N)

Branching rule

In fact

and under this embedding, we have the branching rules

NU(N) → NSN+1N̄U(N) → NSN+1

bosons: 4 · (N,1)⊕ 4 · (1,1)

fermions: 2 · (N,2)⊕ 2 · (1,2)

Comparison

Wolf coset:

bosons: 2 · (N,1)⊕ 2 · (N̄,1)⊕ 4 · (1,1)

fermions: (N,2)⊕ (N̄,2)⊕ 2 · (1,2)

Symmetric orbifold:

Thus the action of the permutation group on the free bosons and fermions is induced from the U(N) action!

⊂

λ → 0

Subtheory

U(N) invariant states of free theory

untwisted sector of sym. orbifold

i.e. invariant states of free theory

perturbative part of CFT dual of hs theory for

(part of)

CFT dual of string theory in this limit

It therefore follows that

hs theory is closed subsector of string theory!


SN+1

SN+1

Stringy symmetry

From the hs point of view, the symmetric orbifold (i.e. the stringy CFT dual) is characterised by an extended chiral algebra.

multiplicity of singlet representation of

The character of this stringy algebra equals

Zstringy(q, y) =X

Λ

D(Λ)χ(0;Λ)(q, y)

Stringy algebra

In terms of the representations of the Wolf space coset algebra, the extended stringy symmetry algebra is generated by

symmetric orbifold generators with free fields

Asym.orb. = W(N=4)1 [0] ⊕

0M

n,n̄

(

0; [n, 0, . . . , 0, n̄])

n+ n̄

Natural interpretation in terms of Regge trajectories, with labelling the different trajectories.m = n+ n̄


Tension perturbation

Study perturbation by exactly marginal operator that corresponds to switching on string tension — SO(4) inv. combination of moduli from 2-cycle twisted sector.

[MRG, Peng, Zadeh, ‘15]

Anomalous dimension from diagonalisation of mixing matrix

where

N (W (s)) =

bs+hΦc−1X

l=0

(−1)l

l!(L−1)

lW

(s)−s+1+l

Φperturbing

field

[Note: ] ∂z̄W(s) = g πN (W (s)) .

γij = hN (W i) , N (W j)i

cf. also [Hikida, Roenne, ‘15] [Creutzig, Hikida, ‘15]

Explicit results

spin

M2

spincubic terms

quadratic terms

Explicitly we find that

quartic terms

quintic terms


◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆××

××

××××

××

▲▲ ▲▲▲▲

××

×× ××××

■■ ■■ ■■

××

××××

▲▲××

1 2 3 4 5 6 7 8 9 11 13 15

1

2

3

4

56

8

10

Mixing and log behaviour

Generically, fields of the same unperturbed spin (incl. those from different columns) will mix, and the full diagonalisation problem is complicated.


However, have good estimate for diagonal entries for the original higher spin fields,

and hence for associated dispersion relation

E(s) ' s+ a log s

γ(s)

' a log s

RR background

E(s) ' s+ a log s+ 0 · (log s)2

Since this is of the form

suggests that AdS3 background has pure RR flux.

[Loewy, Oz, ‘03] [David, Sadhukhan, ‘14]

This is also in line with situation for AdS5….

[Gubser, Klebanov, Polyakov, ‘02], [Frolov, Tseytlin, ‘02] [Roiban, Tirziu, Tseytlin, ‘07], [Roiban, Tseytlin, ‘07],…

λ → 0

Stringy completion

At stringy description characterised by extended chiral algebra (which could be directly obtained from symmetric orbifold).

It is natural to believe that the same idea should also work away from this special point --- this suggests a new avenue for how to find the CFT dual of string theory on

AdS3 × S3× S

3× S

1

Conclusions

‣ Explained evidence for bosonic minimal model holography and sketched large generalisation.

‣ In the supersymmetric case found natural embedding of CFT dual of hs theory into CFT dual of string theory.

‣ Gives first concrete realisation of idea that hs theory emerges as a subsector of string theory in tensionless limit.

N = 4

Open problems & future directions

HS viewpoint: new perspective on stringy CFT

‣ Find Lie algebra structure of stringy symmetry

‣ Find other stringy modular invariants

‣ higher dimensional analogue?

cf. [Beisert,Bianchi,Morales,Samtleben, ‘04]

‣ Interpretation from D1-D5 viewpoint

‣ Relation to spin chain picture

Open problems:

cf. [Chang, Minwalla, Sharma, Yin, ‘12]

[Borsato, Ohlsson Sax, Sfondrini, Stefanski, ‘14] [Babichenko, Stefanski, Zarembo, ‘09]

Interesting CFT problems

Prove that is consistent W-algebra for

all values of and c. W[λ]λ

Show that every W-algebra with this spin content

(one simple field for each spin s>1) is of this form.

Representation theory

Unitarity?

Supersymmetric generalisations, N=1, N=2, N=4.

Interesting Lie problems

Give good characterisation of stringy symmetry

algebra = wedge algebra of symmetric orbifold.

Relation to Yangian symmetry?

Representation theory

Constraints for perturbation theory via Wigner-Eckert

type argument.

Stringy chiral algebra

Explicitly, we find

This reproduces precisely vacuum character of symmetric orbifold (from DMVV).

Zstringy(q, y) = χ(0;0)(q, y) + χ(0;[2,0,...,0])(q, y) + χ(0;[0,0,...,0,2])(q, y)

+ χ(0;[3,0,...,0,0])(q, y) + χ(0;[0,0,0,...,0,3])(q, y)

+ χ(0;[2,0,...,0,1])(q, y) + χ(0;[1,0,0,...,0,2])(q, y)

+ 2 · χ(0;[4,0,...,0,0])(q, y) + 2 · χ(0;[0,0,0,...,0,4])(q, y)

+ χ(0;[0,2,0,...0,0])(q, y) + χ(0;[0,0,...0,2,0])(q, y)

+ χ(0;[3,0,...,0,1])(q, y) + χ(0;[1,0,0,...,0,3])(q, y)

+ 2 · χ(0;[2,0,0,...,0,2])(q, y) + · · ·

Light States

The stringy extension does not contain any of the `light’ states any longer since they do not give rise to allowed (untwisted) representations of the extended chiral algebra.

Spacetime intepretation: classical hs solutions do not all lift to string theory.

higher spins & strings - research school of physics...

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