twistors and conformal higher-spin theorytristan/talks/moscow_2016.pdfcorresponds to self-dual...
TRANSCRIPT
Twistors and Conformal Higher-Spin Theory
Tristan Mc Loughlin Trinity College Dublin
Based on work with Philipp Hähnel & Tim Adamo 1604.08209, 1611.06200 .
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
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
SS.D. =
Z
PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f
↵)
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
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
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
(0,1)-form with homogeneity -5satisfying constraint g↵Z
↵ = 0
SS.D. =
Z
PT⌦ ^ g↵ ^ (@f↵ + f� ^ @�f
↵)
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
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
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
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
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
Write an action, [Berkovits/Witten ’04], by introducing a Lagrange multiplier field
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
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↵
Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time
using a multi-index notation.
Higher Spin Deformations
@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)
Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time
using a multi-index notation.
Higher Spin Deformations
@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
Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time
using a multi-index notation.
We impose the equation of motion
and can write an action function by introducing a Lagrange multiplier field
Higher Spin Deformations
@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 )
Consider deformations of Dolbeault operator for a curved twistor space corresponding to an arbitrary self-dual space time
using a multi-index notation.
We impose the equation of motion
and can write an action function by introducing a Lagrange multiplier field
Higher Spin Deformations
@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?