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