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A menagerie of ambitwistor string theories
Lionel Mason
The Mathematical Institute, Oxfordlmason@maths.ox.ac.uk
Eurostrings, Cambridge, 24/3/2015
With David Skinner. arxiv:1311.2564, and variouscollaborations with Tim Adamo, Eduardo Casali, Yvonne Geyer,Arthur Lipstein, Ricardo Monteiro & Kai Roehrig, 1312.3828,1404.6219, 1405.5122, 1406.1462, CGMMRS 150?.????.
[Cf. also Cachazo, He, Yuan arxiv:1306.2962, 1306.6575,1307.2199, 1309.0885, 1412.3479]
Amplitude formulae for massless theories.
Theorem (Cachazo, He, Yuan 2013,2014)Tree-level massless amplitudes in d-dims are integrals/sums
M(1, . . . , n) = �d
X
i
ki
!Z
CP1n
Il(✏li , kj)Ir (✏r
i , kj)
Vol SL(2,C)Y
i
0�(ki ·P(�i))d�i
where Il(✏li , ki ,�i) and Ir (✏r
i , ki ,�i) depend on the theory.
• polarizations ✏li for spin 1, ✏l
i ⌦ ✏ri for spin-2 (ki · ✏i = 0 . . . ).
• Introduce skew 2n ⇥ 2n matrices M =
✓A C
�Ct B
◆,
Aij =ki · kj
�i � �j, , Bij =
✏i · ✏j
�i � �j, Cij =
ki · ✏j
�i � �j, for i 6= j
and Aii = Bii = 0, Cii = ✏i · P(�i).• For YM, Il = Pf 0(M), Ir =
Qi
1�i��i�1
.• For GR Il = Pf 0(Ml), Ir = Pf 0(Mr )
Gravity
EM
EYM YM
YMS
gen.YMS
BI
DBI
�4
NLSM
compactify
generalize
compactify
generalize
“compactify”
“compactify”compactify
single trace
corollary
“compactify”
squeeze
squeeze
Figure: Theories studied by CHY and operations relating them.
Geometry of ambitwistor space A
Complexify real space-time MR ; M, and null covectors P.A := space of complex null geodesics with scale of P.
• A = T ⇤M|P2=0/{D0} where D0 := P ·r = geodesic spray.• D0 has Hamiltonian P2 wrt symplectic form ! = dPµ ^ dxµ.• ✓ = Pµdxµ symplectic potential, ! = d✓, descends to A.
Theorem (LeBrun 1983)The complex structure on A/scalings determines M andconformal metric g. Correspondence is stable underdeformations of the complex structure of PA that preserve ✓.
From deformations of A to the scattering equations
• ✓ determines complex structure on PA via ✓ ^ d✓d�2. So:• Deformations of complex structure $ [�✓] 2 H1
@(PA, L) .
PropositionFor �gµ⌫ = e
ik ·x✏µ✏⌫ on flat space-time
�✓ = �(k · P)�gµ⌫PµP⌫ = �(k · P)eik ·X (✏ · P)2 .
Ambitwistor repn ) �(k · P) ) scattering equs.
Chiral strings in ambitwistor space
⌃ = Riemann surface, and complexified (M, g).Bosonic ambitwistor string action:
• X : ⌃! M, P 2 X ⇤T ⇤M ⌦ K
SB =
Z(P · @X � e P2/2) .
with e 2 ⌦0,1 ⌦ T , where K = ⌦1,0⌃ and T = T 1,0⌃.
• e enforces P2 = 0,• P2 generates gauge freedom: �(X ,P, e) = (↵P, 0, 2@↵).
So target is:Ambitwistor space: A = T ⇤M|P2=0/{gauge}.
Quantization of bosonic ambitwistor stringGauge fix e = 0 ; ghosts (b, c) 2 (K 2,T ) plus usual(b, c) 2 (K 2,T ) for diffeos
Sghost
=
Zb@c + b@c , Q =
ZcT + cP2 .
• Integrated vertex ops = perturbations of action $ �g.• Action is
R✓ =
RP · @X so integrated vertex operator is
Vi =
Z
⌃�✓(�i) =
Z
⌃�(ki · P(�i)) e
ik ·X(�i )(✏i · P(�i))2 .
• Quantum consistency implies field equations:
{Q,Vi} = 0 , k2 = 0, kµ✏µ⌫ = 0 .
Fixed vertex ops $ quotient by G = SL(2,C)⇥ C3 ; amplitudeas path-integral
M(1, . . . , n) =Z
D[X ,P, . . .]
Vol Ge
iSnY
i=1
Vi .
Evaluation of amplitude
• Take e
iki ·X(�i ) factors into action to give
S =1
2⇡
Z
⌃P · @X + 2⇡
X
i
ik · X (�i) .
• Gives field equations @X = 0 and,
@P = 2⇡X
i
ik�2(� � �i) .
• Solutions X (�) = X = const. , P(�) =P
iki
���id� .
Thus path-integral reduces to
M(1, . . . , n) = �d(X
i
ki)
Z
(CP1)n�3
Qi0�(ki · P) (✏i · P(�i))
2
Vol G
We see P(�) appearing and scattering equations.
Unfortunately: amplitudes for S ⇠R
M R + R3.
Evaluation of amplitude
• Take e
iki ·X(�i ) factors into action to give
S =1
2⇡
Z
⌃P · @X + 2⇡
X
i
ik · X (�i) .
• Gives field equations @X = 0 and,
@P = 2⇡X
i
ik�2(� � �i) .
• Solutions X (�) = X = const. , P(�) =P
iki
���id� .
Thus path-integral reduces to
M(1, . . . , n) = �d(X
i
ki)
Z
(CP1)n�3
Qi0�(ki · P) (✏i · P(�i))
2
Vol G
We see P(�) appearing and scattering equations.
Unfortunately: amplitudes for S ⇠R
M R + R3.
Worldsheet matter
• Decorate null geodesics with spin vectors, vectors forinternal degrees of freedom & other holmorphic CFTs.
• TakeS = SB + Sl + Sr
where Sl , Sr are some worldsheet matter CFTs.• Total vertex operators given by
v lv r �(k · P) e
ik ·X
with v l , vr worldsheet currents from Sl , Sr resp..• Amplitudes become
M(1, . . . , n) = �d(X
i
ki)
Z
(CP1)n�3Il IrQ
i0�(ki · P)
Vol G
where Il , Ir are worldsheet correlators of v ls, vr s resp..• In good situations, Q-invariance and discrete symmetries
(GSO) rule out unwanted vertex operators.
Spinning light rays and worldsheet SUSY, S Let µ be spin 1/2, fermion on ⌃, RNS spin vector:
S =
Zgµ⌫
µ@ ⌫ � �Pµ µ
� ; constraints P · = 0 ; worldline susy
D = · @
@X+ P · @
@ , {D,D} = P · @
@X.
• Gauge fix � = 0 ; bosonic ghosts (�, �) 2 (K 3/2,T 1/2).• Extend BRST operator with Q =
R�P · .
• For v l or vr , must replace ✏ · P by
u = �(�)✏ · (fixed), v = ✏ · P + ✏ · k · .
• Worldsheet correlator hu1u2v3 . . . vni = Pf 0(M) for Il or Ir .• GSO symmetry ( , �,�) ! (� ,��,��) ) need u or v .
Free Fermions and current algebras
• Free ‘real’ Fermions ⇢a 2 Cm ⌦ K 1/2
S⇢ =
Z
⌃�ab⇢
a@⇢b , a = 1, . . . n,
• These generate current algebra e.g., if g is Lie algebra withstructure constants f abc , m = dim g then
ja = f abc⇢b⇢c , ja(�)jb(0) =k�ab
�2 +f abcjc
�+ . . . .
Here level k = C.• Current algebra gives Vertex ops
v = t · j .
• Correlators give ‘Parke-Taylor’ factors + unwantedmulti-trace terms
hv1 . . . vni =tr(t1 . . . tn)
�12�23 . . .�n1+ . . .
where �ij = �i � �j .
Comb system [Casali-Skinner]
• To avoid multitrace terms, take current algebra level k = 0.• But, then there is no trace hv1 . . . vni = 0.• Gauge fermionic spin 3/2 current to give two fixed vertex
operators to end chain of structure contants ‘comb’.• Use fermions ⇢a, ⇢a 2 g⌦ K 1/2, bosons qa, ya 2 g⌦ K 1/2
SCS =
Z
⌃⇢a@⇢
a + qa@ya + � tr⇢
✓[⇢, ⇢]
2+ [q, y ]
◆.
• Gauge fix � = 0 ; ghosts (�, �)
• Gives vertex operators
u = �(�)t ·⇢ u = �(�)t ·⇢ , v = t ·[⇢, ⇢] , v = [⇢, ⇢]+[q, y ] .
• To be nontrivial, correlator must have just one untilded VO
hu1u2v3 . . . vni = C(1, . . . , n) := tr(t1[t2, [t3, . . . [tn�1, tn] . . .])�12 . . .�n1
.
The 2013 CHY formulae & ambitwistor models
Above lead essentially to original models:• (Sl ,Sr ) = (S ,S ) ; type II gravity,• (Sl ,Sr ) = (SCS,S ) ; heterotic with YM,• (Sl ,Sr ) = (SCS,SCS) ; bi-adjoint scalar.
The first two are critical in d = 10 (with appropriate gaugegroup in heterotic case).The latter two come with unphysical gravity.SCS improves on current algebras in avoiding multi-trace terms.
Combined matter systems
S 1, 2 = S 1 + S 2 two worldsheet susy’s for Sl or Sr . This ismaximum. It gives VO currents
u = �(�1)k · 2 , v = k · 1k · 2 .
S ,⇢ = S + S⇢ combines ‘real’ Fermions with susy, ; VOcurrents as usual for S and
ut = �(�)t · ⇢ , vt = k · t · ⇢ .
S ,CS =R⌃ ·@ +⇢a@⇢a+qa@ya+�
⇣P · + tr⇢
⇣[⇢,⇢]
2 + [q, y ]⌘⌘
.
With ghosts etc., the VO currents are those for S and
ut = �(�)t · ⇢ , ut = �(�)t · ⇢ ,vt = k · t · ⇢+ t · ([⇢, ⇢] + [q, y ]) , vt = k · t · ⇢+ t · [⇢, ⇢] .
GSO now reverses signs of all fields in matter system.
Einstein-T ⇤Yang-MillsWith (Sl ,Sr ) = (S ,CS,S ) we obtain Einstein-T ⇤YM.
• Graviton vertex operators now come from
ulur = �(�)✏· �(�)✏· , v lv r = ✏·(P+ k ) ✏·(P+ k · ) .
Worldsheet correlators give Pf 0(M)Pf 0( eM).• Two types of gluon VOs from ul
tur , v l
t vr and ul
tur , v l
t vr .
• With gluons, worldheet correlator gives sum of
C(T1) . . . C(Tr )Pf 0(⇧)Pf 0(M)
where (T1, . . . ,Tr ) is a partition of gluons, ⇧ from CHY.
Theory is linear YM on full YM background i.e., T ⇤YM + gravityZ
MRNS dvol + tr(A ^ DA
⇤FA)
with A $ ultu
r , v lt v
r , A $ ultu
r , v lt v
r .Can replace SCS with anomalous SYM with single gluon type.
Ambitwistor strings with combined matter
Sl
Sr
S S 1, 2 S(m)⇢, S(N)
CS, S(N)CS
S E
S 1, 2 BI Galileon
S(m)⇢, EM
U(1)mDBI EMS
U(1)m⇥U(1)m
S(N)CS, EYM ext. DBI EYMS
SU(N)⇥U(1)mEYMS
SU(N)⇥SU(N)
S(N)CS YM Nonlinear � EYMS
SU(N)⇥U(1)mgen. YMSSU(N)⇥SU(N)
Biadjoint ScalarSU(N)⇥SU(N)
Table: Theories arising from the different choices of matter models.
Models from different geometric realizations of AWe can start with other formulations of null superparticles
• Pure spinor version (Berkovits) S =R
P · @X + p↵@✓↵ + . . ..• Green-Schwarz version:
S =
ZP · @X + Pµ�
µ↵�✓
↵@✓� .
• d = 4, Twistor-strings of Witten, Berkovits & Skinner
A = {(Z ,W ) 2 T⇥ T⇤|Z · W = 0}/{Z · @Z � W · @W}
S =
Z
⌃W · @Z + a Z · W
• In 4d have full ambitwistor representation [Geyer, Lipstein, M. 1404.6219]
S =
Z
⌃Z · @W � W · @Z + aZ · W
1 Not same as twistor string, (Z ,W ) spinors on world sheet.2 Valid for any amount of supersymmetry.3 New simpler 4d formulae with reduced moduli.
Relation to null infinity, BMS and soft gravitonsGeyer,Lipstein & M. 1406.1462 (cf. Adamo, Casali & Skinner 1405.5122).
All real null geodesics intersect I ± so AR = T ⇤I ±; so
A = T ⇤I +C [ T ⇤I �
C glued over AR .
and
Vertex op = �✓ = @(infinitesimal diffeo T ⇤ I � ! T ⇤I +) .
The diffeos intertwine with BMS transformations.Soft gravitons: k ! 0 then diffeo ! supertranslation.Gives Strominger/Weinberg soft graviton thm as Ward identity.Theorem (Strominger/Weinberg)Weinberg soft theorem: as kn ! 0
M(1, . . . , n) ! M(1, . . . , n � 1)n�1X
i=1
(✏n · ki)2
kn · ki,
follows from supertranslation equivariance.
Summary & Outlook
Chiral ↵0 = 0 ambitwistor strings, use LeBrun’s correspondence& give theories underlying CHY formulae new & old.
• NS sector of type II sugra extends to Ramond as in RNSstring via spin-operator from bosonizing s. [Adamo, Casali, Skinner]
How do these work in new theories?• Incorporates colour/kinematics Yang-Mills/gravity ideas.• Many new critical models; criticality gives extension to
loops. [Adamo, Casali, Skinner]
• At genus g, P is a 1-form and acquires dg zero-modes.• These are the loop momenta for g-loops.• Very likely gives correct answer for loop processes
[Casali-Tourkine].
• Quantization ties scattering of null geodesics into that forgravitational waves.
The end
Thank You
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