Star-product:Higher-Spin Theory vs. String Field Theory
String Field Theory and Related Aspects VI, Trieste
Evgeny Skvortsov(based on 1210.7963, 1301.4166 with V. Didenko, Jianwei Meiand some papers to appear with K. Alkalaev and M. Grigoriev)
AEI and Lebedev Institute
29 July 2014
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Outline
Star-product, Gaussians, group structure,projectors, sewing Projectors (D-branes)
Higher-Spin Amplitudes without Higher-Spintheory
Uniformization of Vasiliev theory
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
HST vs. SFT
There are two general conjectures relating higher-spin theoryand string theory:(i) Higher-Spin theory is a tensionless limit of String theory(ii) String theory is a broken phase of Higher-Spin theoryThere are more concrete conjectures by Chang Minwalla,Sharma, Yin, 2012 and Gaberidiel and Gopakumar, 2014 andnon-stringy conjecture by Klebanov-Polyakov; Sezgin-Sundellthat relates Vasiliev theory on AdS4 to Free/CriticalO(N)-model.
(i) and (ii) are not well-understood at present, but somecomputations in HST and SFT are identical thanks to thestar-product
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Unification by Star-product
Star-product is a tool to handle computations with canonicaloperators in QFT. It appears naturally in SFT (Bars).Vasiliev HST is a classical theory. Nevertheless thestar-product is its essential ingredient. Moreover, it seems thatSFT and HST share many interesting (simple) solutions andobservables.
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Star-product
Let Y A be some canonical operators, e.g. Y A = qm, pn
[Y A, Y B ] = 2iAAB
Replace noncommutative algebra of Y A with the algebra ofcommuting generating elements Y A while deforming the usualdot-product into non-commutative star-product, whichcontains information about operator product and someadditional information, which specifies the ordering ofoperators
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Star-product
[Y A, Y B ] = 2iAAB
exp-formula
f (Y ) ? g(Y ) = f (Y ) exp i(←−∂A ΩAB−→∂B
)g(Y )
∫-formula
f (Y ) ? g(Y ) =
∫dU dV f (Y +U)g(Y +V ) exp i(ΩABU
AV B)
Symplectic metric : AAB =1
2(ΩAB − ΩBA)
Ordering : SAB =1
2(ΩAB + ΩBA)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Relation to sp(2N)
Oscillators provide a realization of sp(2N)
TAB = − i
4YA,YB?
In addition we have
[TAB ,YC ] = YAABC + YBAAC
sp(2N) `Heisenberg algebra
TAB , Y A, 1 are treated as even elements
Ortho-symplectic algebra, osp(1|2N)
TAB are even and Y A are odd, but bosonic
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Trace vs. super-trace
Y = qm, pn is Z2-graded so we can think of it either as ofalgebra or as of super-algebra, while all elements are bosonic
Trace
tr(f ) =
∫d2NY f (Y )
needs f to be integrable
Super-trace
str(f ) = f (0)
works nice for many reasonable functions, reduces to a tracefor even functions
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussian solutions of interest
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)SFT
There are quite a fewsolutions known
Vacuum
Perturbative states
Sliver
Wedge
Butterfly
D-branes
...
HST
There are few solutions known atpresent
Didenko-Vasiliev Black hole(Generalized by Sundell andIazeolla)
Boundary-to-bulk propagators
Cosmological (Sezgin,Sundell, Iazeolla)
that’s all
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Heisenberg group of plane waves
Φ(~ξ, q) ? Φ(~η, p) = Φ(~ξ + ~η, q + p + ~ξ · ~η)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Heisenberg group of plane waves
Φ(~ξ, q) ? Φ(~η, p) = Φ(~ξ + ~η, q + p + ~ξ · ~η)
Trace vs. super-trace
Φ(~ξ1) ? ... ? Φ(~ξn) = Φ(∑i
~ξi ,∑i<j
~ξi · ~ξj)
str = exp i(∑i<j
~ξi · ~ξj)
tr = str × δ(∑i
~ξi)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Hidden Symplectic group
Φ(A) ? Φ(B) = N(A,B)Φ(f (A,B))
f (A,B) = 11 + BA(B − I ) + 1
1 + AB (I + A)
N(A,B) = det−12 |1 + AB |
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Hidden Symplectic group
Φ(A) ? Φ(B) = N(A,B)Φ(f (A,B))
f (A,B) = 11 + BA(B − I ) + 1
1 + AB (I + A)
N(A,B) = det−12 |1 + AB |
Cayley Map C (a) = 1−a1+a , a ∈ Sp(2N)
The group structure is manifest nowf (C (a),C (b)) = C (ab)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Symplectic group
Group element is
G (a) = 2N
det12 |1+a|
exp i(
12YC (a)Y
)G (a) ? G (b) = G (ab)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)SpH(2N) = Sp(2N) n HN
(a, ~u, x) (b, ~v , y) = (ab, ~u + a~v , x + y + ~ua~v)
Generalized Cayley map (A, ~ξ, q)⇐⇒ (a, ~u, q)
A =1− a
1 + a
~ξ = ±2
(1
1 + a
)· ~u
q = x + ~u · 1
2
(1− a
1 + a
)· ~u .
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)SpH(2N) = Sp(2N) n HN
(a, ~u, x) (b, ~v , y) = (ab, ~u + a~v , x + y + ~ua~v)
The group law of SpH(2N) is respected now by thestar-product
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Gaussians and star-product
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Projectors Φ ? Φ = Φ, i.e. A3 = A or A2 = I
Cayley map fails to be invertible for projectorsC−1(A) = 1−A
1+A does not exist, formally we areapproaching the boundary at infinity of SpH .The star-product is still well-defined.A gives rise to two matrix projectors
(I + A)(I − A) = 0
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
D-brane surgery
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Sewing rules
one remove ± eigen spaces from A(B) and gluethem together into a new projectorA B = (A + B)−1(2I + B − A)
Still a√I : (A B)2 = I
Associativity : A (B C ) = (A B) CForgetful : A B C = A C
(1 + A)Y ? Φ(A B) = Φ(A B) ? (I − B)Y = 0
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Amplitude for A2 = I projectors
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)Tr(Φ1?...?Φn) =
∏i
1
|Ai + Ai+1|1/4exp i
∑j
(Qj+Pj)
Qi =1
8ξi (Ai+1 Ai + Ai Ai−1) ξi 〈OOJ〉
Pi =1
4ξi (I + Ai+1 Ai ) ξi+1 〈JJ〉
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
HST Amplitudes
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)=
In 4d Vasiliev HST it turned out that boundary-to-bulkpropagators are Gaussians, where q = logK , ξ encodes thespin degrees of freedom of boundary single-trace operators,A = D logK is a vector pointing from the bulk point to theboundary where the operator is inserted. Sundell and Colombosuggested that
〈J ...J〉 = str(Φ ? ...Φ)
should compute the correlation function of an infinite multipletof conserved currents Js = φ∂sφ + ... one can built of a freescalar/fermion.
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Large transformations
Φ(A, ~ξ, q) = exp i(
12YAY + Y ξ + q
)δΦ = [Φ, ξ]? ξ ∈ so(3, 2) ∈ HS
G−1? ?G =⇒ OR
A→ (αA + β)(γA + δ)−1[α βγ δ
]∈ Sp(4)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Diagrammatics: two-point
Vasiliev eq. CFT Amplitude
=
propagator 〈JsJs〉 str(Φ ? Φ)
〈JJ〉 =1
|x12|2exp(P12)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Diagrammatics: three-point
Vasiliev eq. CFT Amplitude
=
2nd-order,Yin, Giombi
〈JsJsJs〉str(Φ ? Φ ? Φ),
Colombo, Sundell
〈JJJ〉 =1
|x12||x23||x31|cos(Q2
13+Q321+Q1
32) cos(P12) cos(P23) cos(P31)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Diagrammatics: four-point
Vasiliev eq. CFT Amplitude
+ =
3nd-order 〈JsJsJsJs〉 str(Φ ? Φ ? Φ ? Φ)
〈JJJJ〉 = 1
|x12||x23||x34||x41|cos(Q2
13 + Q324 + Q4
31 + Q143) cos(P12) cos(P23) cos(P34) cos(P41)
+ permutations
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Vasiliev HS theory vs. Amplitudes
Amplitudes: unintegrated
all HS algebra invariants are traces tr(Φ1 ? ... ? Φn) ordecorated Wilson loops, which are in fact the same.These are essentially nonlocal since they are unintegrated anddo not depend on the interaction point.
this should match
Vasiliev HS theory: integrated
The HS algebra is deformed, structure constants go over intostructure functions. These vertices are integrated and hencelocal
S =
∫AdS
φG−1φ + φ3 + ...
δφ = dξ + [φ, ξ] + φ2ξ + ...
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
AdS/CFT dictionary
CFT AdS
〈J ...J〉 Tr(Φ ? ... ? Φ) correlators
[Q, J] δΦ = [Φ, ξ] symmetries
Q〈J ...J〉 = 0 δ Tr(Φ ? ... ? Φ) ≡ 0 Ward iden.
Effective action for Vasiliev theory
S =∑N
aNTr(ΦN)
Uniformization of Vasiliev Higher-Spin Theories
Higher-spin theory is based on formal consistency and gaugesymmetry rather than any simple geometric/algebraicprinciple. It relies on a very subtle effect of star-productalgebra — nontriviality of the theory depends on class offunctions. At present it is difficult to detach the star-productrealization from the theory to reveal its invariant meaning.The general structure is that of A∞.
dΨ = V2(Ψ,Ψ) + V3(Ψ,Ψ,Ψ) + ...
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Uniformization of Vasiliev Higher-Spin Theories
Flat connection : dW + W ?W = 0
Compatibility : dTa + [W ,Ta]? = 0
(Super)-algebra : [Ta,Tb]?,± = f cabTc
δW = dξ + [W , ξ]?
δTa = [Ta, ξ]?
Know-how
The algebra that Ta form, osp(1|2)
The associative algebra W and Ta take values in
The vacuum to expand over (AdS)
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Just osp(1|2)
Definitions : e, e = −2E f , f = 2F e, f = H
Relations : [H , e] = e [H , f ] = −f [E , f ] = e [F , e] = f
Consequences : [H ,E ] = +2E [H ,F ] = −2F [E ,F ] = H
Minimal set of relations
[e, f , e] = e [e, f , f ] = −f[e, e, f ] = −2e [f , f , e] = 2f
Four relations for two generators, could be better
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Magic of osp(1|2)
Casimir has a square root
Υ = [e, f ] +1
2
Υ, e = 0 Υ, f = 0
[Υ,E ] = 0 [Υ,F ] = 0 [Υ,H] = 0
Υ2 is a Casimir operator
Truly minimal set of relations
Υ, Sα = 0 Sα = (e, f )
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Vasiliev equations
Essential part of any Vasiliev system
dW + W ?W = 0
dSα + [W , Sα]? = 0
Sν ? Sα ? Sν = Sα
usually Υ = Sα ? Sα + 1 called B ? κ
Action for osp(1|2) : str(S4 + S2)
suggested by Prokushkin, Segal, Vasiliev
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory
Conclusions
1 SFT and HST share many interesting solutions together, all ofthem being related to Gaussians in star-product algebra
2 Generic Gaussians form a group, SpH(2N), which leads toexplicit formulas for Amplitudes
3 Projectors are at infinity of SpH(2N), the star-productsimulates Wick theorem
4 All HS amplitudes can be computed explicitly
5 Vasiliev theory has a simple algebraic meaning
Evgeny Skvortsov Star-product:Higher-Spin Theory vs. String Field Theory