String Theory as a Higher Spin Theory

String Theory
as a Higher Spin Theory
(Or “A Song For David”)
Rajesh Gopakumar
ICTS-TIFR, Bengaluru,
Jerusalem, Mar. 3rd, 2016
Based on: M. R. Gaberdiel and R. G.
(arXiv:1512.07237 & 1501.07236)
Multiply Connected, Multiply Indebted
For being my Guru.
For being the chair of the International
Advisory Board at ICTS-TIFR.
For being the inspiration for the theme
of this talk (unbroken stringy symmetries).
For being the inspiration for the title of
this talk…..
As vs. Is
Large N 2d CFTs ….
Often know the full spectrum explicitly.
Z(q, q̄) = |Zvac(q)|2 +
!
|Zh(q)|2
h
Chiral Sector
non-trivial primaries
Simplification when there is a large N limit.
ZCF T = |Zvac
ZCF T
(q)|2[1+
!
wedge
(q)|2]
|Zh
! h " (n)
χY | 2
|
= |Zvac|2
n
Single trace
primaries
Y
Young Tableau
Fock Space
Character
….And Their AdS3 Duals
ZCF T
Massless
gauge fields
! " (n)
2
2
χY |
|
= |Zvac|
n
Y
(bulk) (bulk)
= Ztree Z1−loop
Massive fields
(labelled by ’n’)
Thus in Vasiliev theory on AdS3, a tower of massless
gauge fields with spin s=2,3…
And one massive scalar field in a special irrep
(minimal rep.) of HS symmetry algebra.
The mass and couplings to higher spin fields
completely fixed by the symmetry algebra.
Minimal Model Holography
Z (vas) =
c
−
(q q̄) 24 |
Tree
!!
s=2
!
1
2×
h+j q̄ h+k )−1
|
(1
−
q
n)
(1
−
q
n=s
j,k=0
1-loop det. of gauge fields
Matches with
pert
ZCF T =
1-loop det. of scalar
! (wedge)
c
(q q̄)− 24 |Zvac(q)|2(
|b
(q)|2)
(Λ;0)
Λ
is the vacuum character of W1 .
b(Λ;0)(q) are
Fock space characters built from the
qh
single particle one b(f ;0)(q) =
- minimal rep.
(1 − q)
Minimal rep. one with smallest no. of states.
Stringy CFTs
Only knowing the CFT, could have deduced the
Vasiliev gauge symmetries and irrep of the matter.
Use this as a model to study a 2d CFT dual to string
theory at a tensionless point - maximally symmetric.
Candidate: symmetric product orbifold CFT for D13
4
AdS
⇥
S
⇥
T
D5 system dual to string theory on
.
3
Try to understand the unbroken symmetries from the
viewpoint of the Vasiliev symmetry.
And how these symmetries organise the matter.
Symmetric Product CFTs
General structure of symmetric orbifold
orb = |Z̃ (q)|2 +
ZCF
vac
T
!
|ZU (q)|2 +
!
4 N
(T
CFT ) /SN .
|ZT (q)|2
T
U
Focus first on untwisted sector in large N limit.
|Z̃vac
(q)|2 +
!
c
−
2
|ZU (q)| = (q q̄) 24
where
n=1
yq n
(1−q r q̄ r̄ y ℓȳ ℓ̄)−d(r,ℓ)d(r̄,ℓ̄)
r,r̄=0 ℓ,ℓ̄
U
1
Y
1
! !
1/2 2
(1
1 y
q n )4
1 n 1/2 2
q
=
X
d(r, l) q r y l
r,l
(four bosons and fermions).
Can be expressed as a fock space character
|Z̃vac(q)|2[
!
(wedge)
|ΦY
(q)|2]
Y
Built from a single 1-particle character
(wedge)
Φf
(q) .
A Gigantic H-Spin Theory
Structure very parallel to Vasiliev H-spin
theory except on a much larger scale.
Chiral sector:
c
−
Z̃vac(q) = q 24
Much bigger than
Note d(r, ℓ) ∝ exp(a
Similarly
√
.
(1−(−1)2r q r y ℓ)−d(r,ℓ)
r,ℓ
!
c
(1 − q r )−(r−1)
Zvac = q − 24
r=2
r) exponentially
∞
!
"
1 + yq
(wedge)
=
Φf
(q)
n=1
In contrast to
!
string
b(f ;0)(q) =
# "
n−1/2 2
#
−1 n−1/2 2
1+y q
(1 − q n)4
qh
.
(1 − q)
Vasiliev
Vasiliev
.
larger -HSS.
−1
.
string
Oscillators for HS
To further develop the parallel:
Vasiliev H-spin symmetry algebra built from 2
oscillators ŷα . (α = 1, 2)
[ŷα, ŷβ ] = 2iϵαβ (1 − (2λ − 1)k))
Arbitrary monomials in ŷα give (wedge)
(s)
generators Wn (|n| < s).
SL(2,R) algebra (L0, L±1) built from bilinears in ŷα .
More Oscillators for HSS
4 N
(T
) /SN in 1-1
Z̃
From vac , single particle chiral sector of
4
T
correspondence with the chiral sector of at large N.
J (s)
N
1 X (s)
$p
Ji
N i=1
These are all independent generators at large N. E.g.
N
X
i=1
(@ i )2,
N
X
i=1
(@ i )4
.
This HSS algebra can be generated by monomials built from
usual free boson (and fermion) oscillators αm (m ∈ Z).
[αm, αn] = mδm+n,0
Vasiliev HS subalgebra generated by bilinears in these
oscillators - usual bosonic W1 construction.
Minimal Reps
The parallel continues to the minimal reps.
For the HS case the minimal rep. is one
generated by lowest weight state s.t. y1|0 >= 0 .
So states are (y2) |0 > and is a rep. of HS.
n
Similarly, the minimal rep. of the HSS
generated by lowest weight state s.t. α |0 >= 0 (m ≥ 0) .
m
Again has the smallest no. of states of all HSS
reps. at any given level - many null states.
Through the Lens of HS
In fact, minimal rep of HSS decomposes into an
infinite number of HS representations (incl. min. rep.)
(wedge)
Φf
(q, y)
=
!
(wedge)
b(0;[m,0...,0,n])(q, y)
m,n(̸=(0,0)))
Most of the reps. on the RHS appear as multiparticles in Vasiliev holography but here are singleparticle reps. in the string theory.
Their coupling to Vasiliev fields determined by that of
minimal HS rep.
The full HSS+min. rep. is a huge Vasiliev theory.
The Vertical Higher Spin Algebra
•
•
Can view the HSS algebra in terms of reps. of HS as well.
Decomposes into an infinite number of reps. e.g. in the
bosonic case
can group
into
monomials of oscillators.
Higher
Spin
Square
[1,0,0,..]
[0,1,0,0,..]
[0,0,1,0,0,..]
[0,0,0,1,0,0,..]
[0,0,0,0,1,0,0,..]
(ev)
W1
[1]
(ev)
W1
[1]
descendants
@l
i
@l
i k i
@
……..
@l i@k
i m i
@
quartic
quintic
The Horizontal H-Spin Algebra
•
Novel observation: Alternatively organise the generators in a
horizontal way starting with top row of each column.
•
By fermionisation,
the topSquare
row are bilinear of fermions which
Higher Spin
generates a different H-spin algebra - W1+1 [ = 0] .
W1+1 [0]
W1+1 [0]
(@
descendants
a m
)
(@ 2
a
(@ 3
a
)(@
a m 1
(@ 4
a
)(@
a m 1
)(@
a m 1
)
)
)
Two Decompositions
•
•
Vertical Decomposition: monomials can be assembled into
specific reps. of W1 [ = 1] - ⇤+ = [0n 1 , 1, 0 . . . 0].
Corresponds to character decomposition:
1
Y
k=1
1
(1
1
X
qn
Qn
=1+
k
q )
j=1 (1
n=1
qj )
•
Horizontal Decomposition: Various fermion composites are
different reps. of W1+1 [ = 0] ;⇤+ = 0 and ⇤ = [m, 0 . . . 0, m] .
•
It corresponds to another slicing of the generators as:
1
Y
k=1
1
(1
1
X
m2
2
q
Qm
=1+
k
q )
(1
j=1
m=1
q j )2
The Higher Spin Square
•
The vertical and horizontal algebras generate the structure of
the H-Spin square for the full unbroken stringy symmetry.
•
All commutators given in terms of H-spin commutators.
(@ i )l+1
……..
(@ 4 i )(@ i )l
@l
i
@l i@k
i
@l i@k
i m i
@
quartic
quintic
……..
descendants
(vert)
W1
(@ 3 i )(@ i )l
(vert)
W1
(@ 2 i )(@ i )l
(hor)
W1
As vs. Is (again!)
•
3
4
AdS
⇥
S
⇥
T
Evidence for string theory on
having a large
3
unbroken stringy symmetry that is parallel to H-spin symm.
•
•
•
HSS algebra exponentially larger but determined by HS.
•
Vasiliev theory a model for how constraining extended gauge
invariance can be. Essentially fixes the dynamics.
•
Would hope that one can uncover a similar rigidity for string
theory (interactions!) - why string theory is unique.
Matter sector (untwisted) also parallels HS theory.
Matter sector (twisted) seems to have a nice structure in terms
of “near-minimal” reps of HSS. Need to understand better.
Thanks, David,
For Being There.
Happy B’day!