star-product: higher-spin theory vs. string field theory · star-product: higher-spin theory vs....

33
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 Mei and 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

Upload: others

Post on 21-Jul-2020

8 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 2: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 3: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 4: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 5: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 6: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 7: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 8: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 9: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 10: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 11: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 12: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 13: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 14: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 15: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 16: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 17: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 18: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 19: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 20: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 21: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 22: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 23: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 24: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 25: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 26: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 27: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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)

Page 28: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 29: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 30: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 31: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 32: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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

Page 33: Star-product: Higher-Spin Theory vs. String Field Theory · Star-product: Higher-Spin Theory vs. String Field Theory String Field Theory and Related Aspects VI, Trieste Evgeny Skvortsov

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