Twistors and Conformal Higher-Spin Theorytristan/talks/Moscow_2016.pdf · Twistors and Conformal...

Twistors and Conformal Higher-Spin Theory

Tristan Mc Loughlin Trinity College Dublin

Based on work with Philipp Hähnel & Tim Adamo 1604.08209, 1611.06200 .

http://arxiv.org/abs/arXiv:1604.08209




Given the deep connections between twistors, the conformal group and conformal geometry it natural to

ask if there is a twistor description of conformal higher-spin theories.

Main idea of twistor theory is to relate differential field equations in space-time to holomorphic structures on a

complex manifold

Starting point is the twistor equation

• A, A’ = 0,1 are two-component spinor indices.

• is the covariant derivative where we make use of the bi-spinor notation for vectors. The metric is:

• Equation is conformally invariant. Underwe have so that

• Only consistent for: where

• The space of solutions form a four dimensional vector space over the complex numbers.

Twistor Theory

rAA0 = ra

r(AA0!

B) = 0

gab = ✏AB✏A0B0

gab 7! gab = ⌦2gab!A = !A

r(AA0 !

B) = ⌦�1r(AA0!

B)

Cabcd = ABCD✏A0B0✏C0D0 + A0B0C0D0✏AB✏CD

ABCD!D = 0

In Minkowski space the twistor equation has non-trivial solutions:

• are constant spinor fields defining a four complex- dimensional vector space of solutions called twistor space, .

• We denote elements of this solution space and we can represent them in a non-conformally invariant fashion by the pair of fields

• We can similarly define dual twistors as solutions of

Flat Twistor Space

!

A =�!

A + ix

AA0 �⇡A0

⇡A0 =�⇡A0

�!A,

�⇡A0

W↵

W↵ = (⇡A0 ,!A)

T↵ ' C4

Z↵ = (µA0,�A)

r(A0

A µB0) = 0

We identify points in with complex projective lines in via the incidence relation

Twistor Theory

CMPT

CM PT

µ

A0= ix

A0A�A

Lx

⇠= CP1

x

Projective Twistor space corresponds to the identification

Z↵ ⇠ tZ↵ , t 2 C⇤

PT ⇠= CP3

One application of twistor theory gives an identification between complex data on Twistor space and solutions of linear massless equations [Hitchen ‘80, Eastwood et al ’81]

via the Penrose transform

where is the volume form on . Can prove [Hitchen ‘80, Atiyah ’79]

Twistor Theory

�A1...A2h =1

2⇡i

Z

Lx

�A1 . . .�A2h↵ ^D�

D� = ✏AB�Ad�B ⌘ h�d�i CP1

@↵ = 0

↵ is a (0,1)-form with values in . O(�2h� 2)

�A1...A2h is a space-time field of helicity 2h.

rA01A1�A1...A2h = 0

space of solutions on CM of

H0,1(PT;O(�2h� 2)) ⇠= rA01A1�A1...A2h = 0

Self-dual ActionsGeneralized Weyl curvature tensors can be decomposed as:

For YM and Weyl Gravity (and linearized HS)

Schematically we can write this as

or introducing Lagrange multiplier

which has E.o.M.:

Provides an expansion about self-dual sector:

Csµ(s) ⌫(s)

S

s[�] =

1

"

2

Zd

4x + Boundary term

" = 0

S

s[�] =1

2"2

Zd

4x

⇣ AIBI

AIBI + e A0IB

0I

e A0IB

0I

⌘

S

s[�, G] =

Zd

4xGAIBI

AIBI � "

2

2

Zd

4xGAIBI G

AIBI.

C(s)A(s)A0(s)B(s)B0(s) = ✏AIBI

e A0IB

0I+ ✏A0

IB0I AIBI

AIBI = "2 GAIBI , rAIA0IGAIBI = 0 .

Form curved twistor spaces by deforming the complex structure

where the deformation is a (1,0)-vector valued (0,1)-form

To be well-defined on projective space, , must have homogeneity 1 under coordinate rescalings. Also has the invariance

The condition for the deformation to give rise to an integrable complex structures is

T@f = @ + f

f = f↵↵ dZ↵ ⌦ @↵

f↵ ⇠= f↵ + Z↵⇤

f↵

PT

Complex curved projective twistor

spaces

Self-dual space-times:

PT

Penrose/Ward

ABCD = 0

@ = dZ↵ @

@Z↵

@2f = (@f↵ + f� ^ @�f

↵)@↵ = 0

Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field

@2f = (@f↵ + f� ^ @�f

↵)@↵ = 0


SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)


SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)

Holomorphic (3,0) volumeform of homogeneity 4. In terms of homogeneous coordinates

D3Z = ✏↵1↵2↵3↵4Z↵1Z↵2Z↵3dZ↵4


(0,1)-form with homogeneity -5satisfying constraint g↵Z

↵ = 0

SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)


Equation of motion for tensor field

Describes a helicity-(-2) particle moving in a self-dual background. Corresponds to self-dual sector of Weyl gravity

@fg↵ = 0P.T. rC

A0rDB0GABCD = 0

SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)

SS.D. =

Zd

4x

p�g G

ABCD ABCD




To include the anti-self-dual interactions we add term

This is equivalent to usual Weyl action up to topological terms.

@fg↵ = 0P.T. rC

A0rDB0GABCD = 0

SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)

SS.D. =

Zd

4x

p�g G

ABCD ABCD

SA.S.D. = �✏

Zd

4x

p�g G

ABCDGABCD




To include the anti-self-dual interactions we add twistor term constructed by using the Penrose transform involving the deformation which gives rise to a MHV vertex like expansion.

Can be used to very efficiently calculate all Einstein gravity MHV scattering amplitude.

@fg↵ = 0P.T. rC

A0rDB0GABCD = 0

SS.D. =

Z

PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f

↵)

SS.D. =

Zd

4x

p�g G

ABCD ABCD

Following Maldacena’s argument for truncating conformal gravity to Einstein gravity with a non-zero cosmological constant, Adamo and Mason gave a twistor prescription for extracting EG amplitudes by restricting the fields to a “Unitary” sector :

Use infinity twistor for AdS space,

plane-wave wave-functions for and and produces AdS analogue of scattering amplitudes which in the flat limit reproduce Hodges formula for MHV amplitudes after accounting for overall powers of .

Can we generalise this to Higher Spin Fields?

f↵ = I�↵@�h

g↵ = I↵�Z� h

h, ˜h describe ±2 helicity gravitons}

h h

⇤

I↵� =

✓⇤✏AB 00 ✏A

0B0

◆

Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time

Higher Spin Deformations

@f = @ + f@ = dZ↵ @

@Z↵


using a multi-index notation.


@f = @ + f↵I ⌦ @↵I

f↵I is a rank-n symmetric tensor valued (0,1)-form of homogeneity n on . In order to be defined on it must have the invariance

T PT

f↵1...↵n ! f↵1...↵n + Z(↵1⇤↵2...↵n)

Spin-(|I|+1)




@f = @ + f↵I ⌦ @↵I

f↵I is a rank-n symmetric tensor valued (0,1)-form of homogeneity n on . In order to be defined on it must have the invariance

T PT

f↵1...↵n ! f↵1...↵n + Z(↵1⇤↵2...↵n)

⇤(↵1...↵n�1)

Spin-(|I|+1)

symmetric tensor of homogeneity n-1



We impose the equation of motion

and can write an action function by introducing a Lagrange multiplier field


@f = @ + f↵I ⌦ @↵I Spin-(|I|+1)

(@f↵I + f�I ^ @�If↵I ) = 0 6= @2 = 0

SS.D. =

Z

PT⌦ ^ g↵I ^ (@f↵I + f�I ^ @�If

↵I )



We impose the equation of motion

and can write an action function by introducing a Lagrange multiplier field


@f = @ + f↵I ⌦ @↵I Spin-(|I|+1)

(@f↵I + f�I ^ @�If↵I ) = 0 6= @2 = 0

g↵I = g(↵1...↵n) is a (0,1)-form of homogeneity -n-4 s.t. g↵1...↵nZ

↵1 = 0

SS.D. =

Z

PT⌦ ^ g↵I ^ (@f↵I + f�I ^ @�If

↵I )

At a linearized level the deformation and the Lagrange multiplier define elements of cohomology groups

Via the Penrose transform for homogeneous tensors [Eastwood, Mason] we have

Linearized Deformations

H0,1(PT ;O(�n� 4)) H0,1(PT ;O(n))&

rA01A1 . . .rA0

nAnGA1...AnB1...Bn = 0@g↵1...↵n = 0P.T.

@f↵1...↵n = 0 rA01

(A1. . .rA0

nAn

�B1...Bn)A01...A

0n= 0

At a linearized level the deformation and the Lagrange multiplier define elements of cohomology groups

Straightforward to calculate flat-space on-shell spectrum á la Witten & Berkovits : e.g. spin-3

Linearized Deformations

H0,1(PT ;O(�n� 4)) H0,1(PT ;O(n))&

-3

+2

-2

+2

-2

+1

-1

= 12 on-shell states

-3

+3

-3

+3+3

There is a nice trick for calculating conserved charges in a linearised spin-n/2 massless theory. Given a field satisfying

and a solution of the twistor equation

we can form a solution to Maxwell equation

and so a complex charge

Hence for every solution of the twistor equation we find two conserved charges (half are actually zero).

Linearized Conserved Charges

rA1A01�A1...An = 0

rB01(B1�B1...Bn�2) = 0

�A1A2 = �A1...An�A3...An

µ+ iq1

4⇡

I

S�AB✏A0B0

dx

AA0^ dx

BB0=S

⇠ iF + ⇤F

For conformal higher-spin theory we can do a similar construction.

Given a conformal higher-spin field we can use a symmetric trace-free -twistor

to form a solution of the Maxwell equation

Each charge corresponds to a solution of the twistor equation. Easy to count in flat-space that there are

This is essentially the number of conformal Killing tensors and matches with the calculation of Eastwood. Here it is easy to generalise to arbitrary self-dual backgrounds.

Linearized Conserved Charges

G↵1...↵s�1B1...Bs+1

[ss]

W↵1...↵s�1�1...�s�1

�B1B2 = G↵1...↵s�1B1B2...Bs+1W↵1...↵s�1

B3...Bs+1

112s

2(s+ 1)2(2s+ 1)2

At linearized level twistor action appears to describe conformal higher-spin theory.

What about non-linear terms?

We define a deformed Dolbeault operator involving all spins

then demand that each term vanishes

Lower-spin fields source higher-spin fields and can’t truncate to just e.g. spin-3 fields. Self-dual action is simply the sum of constraints.

Defines a holomorphic structure on the infinite jet bundle of symmetric product of dual tangent bundles of twistor space.

Can straightforwardly write an action by introducing Lagrange multiplier fields.

Infinite-Spin Deformations

@f = @ +1X

|J|=0

f�J@�J

@f↵I +

|I|X

|J|=0

1X

|K|=0

C|K||J|f(↵J�K ^ @�Kf↵I�J ) = 0@2

f = 0 ,

On-shell flat space spectrum forms a non-diagonalizable representation of Poincaré algebra. We can identify a “unitary” sub-sector analogous to EG in CG

Choose AdS background, evaluate action on plane-wave solutions to find the AdS analogue of 3-pt MHV-bar “amplitudes”

Thus accounting for powers of we reproduce the unique flat space answer consistent with Poincaré symmetry and helicity constraints.

Unitary Self-Interactions

g↵1...↵n = I↵1�1 . . . I↵n�nZ�1 . . . Z�n hs

f↵1...↵n = I�1↵1 . . . I�n↵n@�1 . . . @�nhs hs & hs

describe helicity states.±s

⇤

fM3,�1(�s1,+s2,+s3) = ⇤s1�1eN (s1,s2,s3) [2 3]s1+s2+s3

[1 2]s1+s3�s2 [3 1]s1+s2�s3(1 + ⇤⇤P )

s23|1 �4(P )

ASD interactionsSo far we have only considered the self-dual sector to motivate the interactions consider the quadratic case

Sspin�n =

Zd

4xG

AI�AI � �

Zd

4xG

AIGAI

�1 �2

x

AA0

P.T.

P.T.

M

It is convenient to picture real, Euclidean space-times. In this case is fibered over the space-time thus we see the action corresponds to a fibre wise product of twistor spaces

MPT

PT⇥M PT

ASD interactions

We can use the twistor formulation to propose a full ASD action but it’s rather involved. The interaction term for arbitrary negative helicity CHS fields on a conformal gravity background is relatively pleasing

and can be expanded to compute arbitrary (-s, -s, 2, 2, 2, ...) MHV amplitudes in a flat-space limit:

where

However here it is defined for arbitrary self-dual backgrounds and so provides a general quadratic coupling of CHS fields to conformal gravity.

S(2)int [f

(2), g] =

Z

PT ⇥MPT⌦ ^ ⌦0

1X

I=0

Z↵I Z 0�I g�I (Z) g↵I (Z0)

lim⇤!0

fMn,0

⇤/ �4(P )

h12i2s+2

h1ii2 h2ii2��12i

12i

�� ,

�ij =[i j]

hi ji , for i 6= j

�ii = �X

j 6=i

[i j]

hi jih⇠ ji2

h⇠ ii2 .

Conclusions/Outlook• Described a twistor action to describe higher-spin fields on arbitrary

self-dual backgrounds.

• Flat space-time action/spectrum is that of conformal higher-spin theory and produces linearised charges.

• Interactions involve one copy of all spins s>0. Have interpretation of holomorphic structure for infinite jet bundle of homogeneous tensors.

• Identified a “unitary” sub-sector, analogues of EG inside CG, which can be used to reproduce MHV and MHV-bar 3-pt amplitudes up powers of cosmological constant. Can be used to calculate n-point (-s, -s, 2, 2, 2, ...) MHV amplitude.

• What is the full non-linear interacting theory. CHS theory of Segal? What is the “unitary” subsector?

• Can we calculate something interesting? All MHV “amplitudes”? Correlation functions? Partition functions?

• Classical solutions? Instantons? Black Holes?

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