[PPT] - Ambitwistor Strings for Four Dimensions Yvonne Geyer Mathematical PowerPoint Presentation

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Ambitwistor Strings for Four Dimensions

Yvonne Geyer

Mathematical Institute, Oxford

New Geometric Structures in Scattering Amplitudes September 23, 2014 - Oxford

Based on YG, Arthur Lipstein and Lionel Mason arXiv: 1404.6219, 1406.1462

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 1 / 38

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Motivation

Since the formulation of the twistor string theories [Witten, Berkovits, Skinner], many remarkable formulae for tree-level scattering amplitudes have been developed [RSVW, ACCK], [Hodges, Cachazo-YG, Cachazo-Skinner, Cachazo-He-Yuan]. This inevitably raises questions regarding the underlying theories: What is the origin of these representations of Yang-Mills and gravity scattering amplitudes? Recent work has focussed on answering this question, and beyond providing a geometric explanation of the formulae, it also facilitated extensions in various directions. In particular, the CHY [Cachazo-He-Yuan] representation has been understood as arising from string theories in ambitwistor space, the space of null geodesics.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 2 / 38

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Motivation Scattering equations and CHY formulae

Scattering equations and CHY formulae

[Cachazo-He-Yuan]

P(σ) holomorphic map from Riemann sphere into momentum space, P : CP1 → CPd, P(σ) =

j kj σ−σj .

Scattering equations ki · P(σi) =

ji

ki · kj

σi − σj = 0

Representation of YM and gravity scattering amplitudes

A =

n

i=1 dσi

Vol SL(2;C) 1 n

i=1 σi,i+1

′

i ¯

δ(ki · P(σi)) Pf′(Ψ) M =

n

i=1 dσi

Vol SL(2;C)

′

i ¯

δ(ki · P(σi)) Pf′(Ψ) Pf′(˜ Ψ)

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 3 / 38

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Motivation The Ambitwistor String in d=10

The Ambitwistor String in d=10

Ambitwistor space A

Ambitwistor space = space of complex null geodesics in MC Symplectic quotient of cotangent bundle of (supersymmetric) spacetime (X, P, Ψ) ∈ T∗M by constraints P2 = 0 and Ψr · P = 0

A :=

(Xµ, Pµ, Ψµ

r ) ∈ T∗M

P2 = 0, Ψr · P = 0

{D0, Dr}

with Hamiltonian vector fields D0 = P · ∇, Dr = Ψr · ∇ + P · ∂Ψr

A is a symplectic holomorphic manifold, with symplectic potential Θ = P · ¯ ∂X + 1

2

r

Ψr · dΨr

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 4 / 38

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Motivation The Ambitwistor String in d=10

The Ambitwistor String in d=10

RNS ambitwistor string

[Mason-Skinner] (see also [Adamo-Casali-Skinner, Berkovits])

Complexify action of massless spinning particle S =

1 2π

P · ¯

∂X + 1

2

r Ψr · ¯

∂Ψr − e

2P2 − χrP · Ψr

Geometrically, the action is obtained from the symplectic potential Θ, and the gauge fields e and χr impose the constraints. This reduces the phase space to A. BRST operator Q =

cT + ˜

c 2P2 + r γrP · Ψr + ˜ b 2γrγr,

nilpotent in d = 10 as in the usual superstring In particular, the correlation functions of appropriate VO can be shown to yield the CHY formulae.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 5 / 38

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Motivation Four dimensions

Four dimensions

In four dimensions, the space of null geodesics has an alternative spinorial representation in addition to the vector representation used in the formulation of the RNS ambitwistor string. This suggests that the ambitwistor string ideas can be implemented naturally to construct models for Yang-Mills and gravity. These models allow for any amount of supersymmetry, and the correlation functions lead to new, remarkably simple formulae for tree-level scattering amplitudes which are supported on the scattering equations parity invariant.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 6 / 38

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Motivation Four dimensions

Outline

1

Ambitwistor strings in d=4 Ambitwistor space Worldsheet Theory

2

Yang-Mills

3

Gravity

4

Ambitwistor Strings at Null Infinity Geometry and Symmetries Worldsheet theory Soft limits

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Ambitwistor strings in d=4 Ambitwistor space

Ambitwistor strings in d=4

Ambitwistor space A4

Alternative twistorial representation: Z = (λα, µ ˙

α, χr) ∈ T = C4|N

W = (˜

µα, ˜ λ ˙

α, ˜

χr) ∈ T∗ = C4|N

Ambitwistor space is the quadric Z · W = 0 inside T × T∗

A :=

(ZI, WI) ∈ T × T∗ | Z · W = 0

Z ∂

∂Z − W ∂ ∂W

,

which can be seen as the symplectic quotient of T × T∗ by the Hamiltonian Z · W. A is thus a symplectic manifold with the potential

Θ = i

2(W · dZ − Z · dW)

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Ambitwistor strings in d=4 Ambitwistor space

Ambitwistor strings in d=4

Ambitwistor space A4

Comments: The incidence relations

µ ˙

α = i(xα ˙ α + iθaα˜

θ ˙

α a)λα ,

χa = θaαλα ˜ µα = −i(xα ˙

α − iθaα˜

θ ˙

α a)˜

λ ˙

α ,

˜ χa = ˜ θ ˙

α a˜

λ ˙

α

realize a point in chiral Minkowski space as a quadric, CP1 × CP1. Define Pα ˙

α = λα˜

λ ˙

α, then the null geodesic constraint P2 = 0

(appearing in the vectorial representation) is explicitly solved.

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Ambitwistor strings in d=4 Worldsheet Theory

Worldsheet Theory

Motivation: In analogy to the ambitwistor string in d = 10, we will complexify the action of a massless spinning particle, the Ferber superparticle. Again, the action S is determined by the symplectic potential Θ, and the constraint Z · W = 0 is imposed by introducing a gauge field a; S = 1 2π

Σ

W · ¯

∂Z − Z · ¯ ∂W + a Z · W .

Here, (Z, W) are spinors on the worldsheet,

(Z, W) ∈ Ω0(Σ, (T × K 1/2) × (T∗ × K 1/2))

Adding worldsheet gravity and gauge-fixing yields the BRST operator Q =

c(W · ∂Z − Z · ∂W) + uZ · W

Note: In general anomalous!

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 10 / 38

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Yang-Mills Vertex operators

Yang-Mills Amplitudes

Vertex Operators

Introduce integrated and unintegrated vertex operators for self-dual and anti self-dual fields V′

a =

dsa

sa

¯ δ2(λa − saλ)eisa([µ ˜

λa]+χr˜ ηar)j · ta

V′

a =

dσaV′

a

Va =
dsa

sa

¯ δ2(˜ λa − sa˜ λ)eisa(˜

µ λa+˜ χrηr

a)j · ta

Va =
dσa

Va where j denotes a current algebra, and ta are Lie algebra elements. More convenient representation of the supersymmetry: Va =

dsa

sa

¯ δ2|N(λa − saλ|ηa − saχ)eisa[µ ˜

λa]j · ta ,

Va =

dσaVa

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Yang-Mills Amplitudes

Worldsheet correlation function

NkMHV amplitudes as correlation function

A =

V1 . . .

VkVk+1 . . . Vn

.

Take exponential factors appearing in the vertex operators into the action to obtain the effective field equations

¯ ∂σZ = ¯ ∂ (λ, µ, χ) =

k

i=1

si (λi, 0, ηi) ¯

δ (σ − σi) , ¯ ∂σW = ¯ ∂

˜

µ, ˜ λ, ˜ χ

=

n

p=k+1

sp

0, ˜

λp, 0 ¯ δ(σ − σp). (Z, W) are worldsheet spinors, thus unique solution

Z(σ) =

k

i=1

1 (σ σi) (λi, 0, ηi) ,

W(σ) =

n

p=k+1

1 (σ, σp)

0, ˜

λp, 0

.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 12 / 38

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Yang-Mills Amplitudes

A =

V1 . . .

VkVk+1 . . . Vn

Yang-Mills amplitudes

A =

n

a=1 d2σa

Vol GL(2,C) 1 n

a=1(a a+1)

k

i=1 ¯

δ2(˜ λi − ˜ λ(σi)) n

p=k+1 ¯

δ2|N(λp − λ(σp)) .

where

λ(σ) =

k

i=1

λi (σ, σi) , ˜ λ(σ) =

n

p=k+1

˜ λp (σ, σp) .

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 13 / 38

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Yang-Mills Amplitudes

A =

V1 . . .

VkVk+1 . . . Vn

Yang-Mills amplitudes

A =

n

a=1 d2σa

Vol GL(2,C) 1 n

a=1(a a+1)

k

i=1 ¯

δ2(˜ λi − ˜ λ(σi)) n

p=k+1 ¯

δ2|N(λp − λ(σp)) .

where

λ(σ) =

k

i=1

λi (σ, σi) , ˜ λ(σ) =

n

p=k+1

˜ λp (σ, σp) .

In particular, these tree-level scattering amplitudes localize fully on the support of the scattering equations contain only σ moduli, no additional moduli from the degree d of a line bundle are manifestly parity invariant.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 13 / 38

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Yang-Mills Amplitudes

The Scattering Equations in d = 4

Momentum conservation On support of the scattering equations

n

p=k+1 λp˜

λp = n

p=k+1

k

j=1 ˜

λp

λj (σp σj) = − k j=1 λj˜

λj,

Scattering Equations In twistorial representation: Twistorial Scattering Equations 0 = [˜

λi, ˜ λ(σi)],

i = 1, . . . , k

˜ λ(σ) = n

p=k+1 ˜ λp (σ,σp)

0 = λp, λ(σp), p = k + 1, . . . , n

λ(σ) = k

i=1 λi (σ, σi)

Define Pα ˙

α(σ) = λα(σ)˜

λ ˙

α(σ), then the twistorial scattering equations

imply the (usual) scattering equations

λα

a˜

λ ˙

α a · Pα ˙ α(σa) = 0

Note: Twistorial scattering equations refined by MHV degree.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 14 / 38

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Yang-Mills Amplitudes

Proof of the new formula

Comparison to the RSVW formula

These new representations for tree-level scattering amplitudes can be proven by mapping them onto the well-known RSVW formula for N = 4 SYM. Recall the RSVW formula

ARSVW =

d

r=0 d4|4Zr

Vol GL(2,C)

n

a=1 dσa (a a+1)

n

a=1 Aa(Z)

with momentum eigenstates Aa(Z) and map moduli Zr(σ) = (λ, µ, χ). The equality ARSVW = A is established by integrating out Zr(σ) and a change of variables si =

1 k

l=1,li(i l) ti

i = 1, . . . k sp = k

l=1(p l) tp

p = k + 1, . . . , n.

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Gravity Gravity as an ambitwistor string

Gravity as an ambitwistor string

Worldsheet Theory

In analogy to the twistor string proposed in [Skinner], we can construct an ambitwistor string theory for Einstein gravity. Field content: worldsheet spinors

(Z, W) ∈ Ω0(Σ, (T × K 1/2) × (T∗ × K 1/2)) (ρ, ˜ ρ) ∈ ΠΩ0(Σ, (T × K 1/2) × (T∗ × K 1/2))

Breaking conformal invariance: Introduce the infinity twistors IIJ, IIJ, which determine a preferred metric on spacetime and encode a cosmological constant.

IIJIJK = ΛδI

K,

IIJ = ǫαβ Λǫ ˙

α˙ β

,

IIJ = Λǫαβ ǫ ˙

α˙ β

.

In particular, rank(I) = 2 for Λ = 0.

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Gravity Gravity as an ambitwistor string

Gravity as an ambitwistor string

Worldsheet Theory

We can now formulate the gravitational action S = 1 2π

Z · ¯

∂W − W · ¯ ∂Z + ˜ ρ · ¯ ∂ρ − ρ · ¯ ∂˜ ρ .

For Einstein gravity, we furthermore have the current algebra Ka = (Z · W, ρ · ˜

ρ, Z · ˜ ρ, W · ρ, Zρ, [W ˜ ρ], ρ ρ, [˜ ρ ˜ ρ]) .

Gauging all the currents yields the BRST operator Q, with ghosts

(βa, γa) and structure constants Ca

bc of the current algebra Ka.

QBRST =

cT + γaKa − i

2βaγbγcCa

bc ,

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Gravity Vertex operators and amplitudes

Gravity Amplitudes

Vertex Operators

As in YM, we obtain integrated and unintegrated vertex operators for sd and asd fields, corresponding to the on-shell pull-back from T or T∗, Vh =

Σ

δ2(γ)h ,

V˜

h =

Σ

δ2(ν)˜

h ,

Vp =

Σ

(1 + ρ · ∂Z˜ ρ · ∂W) dtp

t3

p

¯ δ2(λp − tpλ(σp)) [˜ λ(σp) ˜ λp] eitp[µ(σp)˜

λp] ,

Vi =
Σ

(1 + ρ · ∂Z˜ ρ · ∂W) dti

t3

i

¯ δ2(˜ λi − ti˜ λ(σi)) λ(σi) λi eiti˜

µ(σi)λi .

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Gravity Vertex operators and amplitudes

Gravity Amplitudes

Correlation function

Amplitudes are now given by the worldsheet correlation function

M =

V˜

h1 k

i=2
V˜

hi n−1

p=k+1

VhpVhn

.

(1) As in YM, solve the equations of motion of the effective action; Z(σ) = k

i=1 1 (σ σi) (λi, 0, ηi) ,

W(σ) = n

p=k+1 1 (σ,σp)

0, ˜

λp, 0

.

For Λ = 0, no contractions between Vp and

Vi.

To perform the calculation, note that the correlator of the fermionic

(ρ, ˜ ρ) system is the determinant of the matrix of possible contractions.

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Gravity Vertex operators and amplitudes

Gravity Amplitudes

Correlation function M =

V˜

h1

k

i=2

V˜

hi

n−1

p=k+1 VhpVhn

Gravity Amplitudes

M =

n

a=1 d2σa

Vol GL(2,C) det′(H) k i=1 ¯

δ2(˜ λi − ˜ λ(σi)) n

p=k+1 ¯

δ2|N(λp − λ(σp))

where

H = H

H
,

and for i, j ∈ {1, ..., k} and p, q ∈ {k + 1, ..., n},

Hij = i j

(i j),

i j,

Hii = − k

j=1,ji Hij

Hpq = [p q]

(p q),

p q,

Hpp = − n

q=k+1,qp

Hpq.

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Gravity Vertex operators and amplitudes

Proof of the new formula

Comparison to the Cachazo-Skinner formula

Outline of the proof As in Yang-Mills, this new representation for tree-level scattering amplitudes can be proven by establishing a correspondence to the Cachazo-Skinner formula for supergravity. The proof follows along the same idea as in YM; integrating out moduli and redefining variables. Link/Grassmannian-like representation Along similar lines, we can prove the equality of this new representation to the Link /Grassmannian-like representations found in [Cachazo-Mason-Skinner]: substitute the momentum eigenstates in the vertex operators by elemental states. fZi(Z) =

ds

s2h−1 δ4|N(Zi − sZ) ,

fWi(Z) =

ds

s2h−1 exp(sWi · Z) .

These are wave functions that are supported at points or planes in twistor space.

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Summary

We have defined new ambitwistor string theories in d = 4, leading to simple representations of tree-level scattering amplitudes in YM and gravity with arbitrary degree of supersymmetry,

A =

n

i=1 d2σi Vol GL(2,C) 1 n i=1(i i+1) k

i=1

¯ δ2|N(λi−λ(σi))

n

a=k+1

¯ δ2(˜ λa−˜ λ(σa)) M =

n

i=1 d2σi Vol GL(2,C) det′(H) det′(

H)

k

i=1

¯ δ2|N(λi−λ(σi))

n

a=k+1

¯ δ2(˜ λa−˜ λ(σa)).

These formulae for A and M localize on support of scattering equations depend on very few moduli are ambidextrous, and manifestly parity invariant.

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Ambitwistor Strings at Null Infinity

Motivation

Recall that ambitwistor space is defined as the phase space of complex null geodesics, and can thus be formulated over any Cauchy hypersurface; in particular, A is identified with the cotangent bundle of the hypersurface. The S-matrix is, almost by definition, a holographic object, defined in terms

f asymptotic states. It is thus suggestive to try to formulate the

ambitwistor string for an asymptotically flat spacetime with respect to null infinity I . In particular, this implies that ambitwistor space is identified with the cotangent bundle of (complexified) null infinity A = T∗I .

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Ambitwistor Strings at Null Infinity Review

BMS symmetries

This is of particular interest considering the recent work in this area:

[Strominger et.al.] identified the soft limits of scattering amplitudes as Ward

identities associated to the BMS symmetries of asymptotically flat spacetimes, and [Adamo-Casali-Skinner] proposed a 2d CFT on I realizing this correspondence. The BMS group is the group of asymptotic symmetries at null infinity I R × S2, and consists

f

supertranslations (super)rotations

Figure : Diagram of null infinity, I .

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Ambitwistor Strings at Null Infinity Review

From BMS to soft limits

[Strominger et.al.]

Soft theorems as Ward identities associated with the diagonal subgroup of BMS+ ⊗ BMS−:

ut| B+S − SB− |in = 0,

where B± are extended BMS generators acting at I ±. The soft gravitons emerge as Goldstone bosons; specializing on supertranslations T±, T− |in = F− |in +

k∈in Ekf (zk, ¯

zk) |in

ut| T+ = out| F+ +

j∈out Ejf (zj, ¯

zj) out| where F± are outgoing/incoming soft graviton operators. This yields directly the Weinberg soft graviton theorem:

ut| F+S − SF− |in =
k Ekf (zk, ¯

zk) −

j Ejf (zj, ¯

zj)

ut| S |in .

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Ambitwistor Strings at Null Infinity Review

Review: Soft Limits

[Weinberg, Cachazo-Strominger, Casali, ...]

In the soft limit, gravity scattering amplitudes behave as

Mn+1 =

S(0) + S(1) + S(2)

Mn ,

where S(0) =

n

a=1

[as]ξ a2 a sξ s2 ,

S(1) =

n

a=1

[a s]ξ a a sξ s ˜ λs · ∂ ∂˜ λa ,

S(2) = 1 2

n

a=1

[a s] a s ˜ λ ˙

α s˜

λ

˙ β s

∂2 ∂˜ λ ˙

α a∂˜

λ

˙ β a

.

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Ambitwistor Strings at Null Infinity Review

Review: Soft Limits

[Weinberg, Cachazo-Strominger, Casali, ...]

Similarly, for Yang-Mills, the soft limits are given by

An+1 =

S(0) + S(1)

An ,

(2) where S(0)

ym =

1 n s 1 s n

S(1)

ym =

1 s1 ˜ λs · ∂ ∂˜ λ1 +

1 ns ˜ λs · ∂ ∂˜ λn

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Ambitwistor Strings at Null Infinity Review

Four dimensional ambitwistor space at I

In four dimensions, no additional coordinates are required in the twistorial representation to implement A = T∗I . Thus, A is still represented as the quadric

A = {(Z, W) ∈ T × T∗|Z · W = 0}/{Z · ∂Z − W · ∂W} ,

and the symplectic potential is given by

Θ = i

2(Z · dW − W · dZ) . Introducing again coordinates (u, pα ˙

α) on I , the projection of A to I

is implemented by u = −iλ˜

µ , ˜

u = i[˜

λ, µ] ,

pα ˙

α = λα˜

λ ˙

α ,

In particular note that for N = 0, the ambitwistor constraint Z · W = 0 implies u = ˜ u.

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Ambitwistor Strings at Null Infinity Geometry and Symmetries

Geometry and Symmetries

All diffeomorphisms of a manifold have a hamiltonian lift to the cotangent bundle, so in particular all symmetries of I lift to A = T∗I . Supertranslations: Hf = f(λ, ˜

λ), for f of weight (1, 1).

Superrotations Hr = [µ,˜ r] + ˜

µ, r, where rα and ˜

r ˙

α are of weight (1, 0)

and (0, 1) respectively Consider the supertranslations: Hf generates the transformations

δ˜ µα = i ∂f ∂λα ,

so

δu = λα ∂f ∂λα = f ,

as claimed above.

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Ambitwistor Strings at Null Infinity Geometry and Symmetries

Geometry and Symmetries

As in the discussion above, we will introduce further fields (ρ, ˜

ρ) to

describe Einstein gravity, and gauge the currents

ρ · ˜ ρ = Z · ˜ ρ = W · ρ = Zρ = [W ˜ ρ] = ρ ρ = [˜ ρ ˜ ρ] = 0

As before, the symplectic potential the becomes

Θ = i

2 (Z · dW − W · dZ + ρ · d˜

ρ − ˜ ρ dρ) .

Extend the Hamiltonians to commute with these constraints by including a factor of 1 + ρ · ∂Z˜

ρ · ∂W, thus giving

supertranslations (1 + ρ · ∂Z˜

ρ · ∂W) Hf

superrotations (1 + ρ · ∂Z˜

ρ · ∂W) Hr

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Ambitwistor Strings at Null Infinity Worldsheet theory

Worldsheet theory

As before, the theory is constructed from the symplectic potential, so we get the following action for Yang-Mills: S = 1 2π

Z · ¯

∂W − W · ¯ ∂Z + aZ · W.

Introducing the additional (ρ, ˜

ρ) system, the action for gravity is

S = 1 2π

Z · ¯

∂W − W · ¯ ∂Z + ˜ ρ · ¯ ∂ρ − ρ · ¯ ∂˜ ρ + eaKa .

The amplitude calculations reduce trivially to those of the original four-dimensional ambitwistor string, and thus yield the expected tree-level scattering amplitudes.

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Ambitwistor Strings at Null Infinity Worldsheet theory

Symmetries and Diffeomorphisms

Note that basing the action on the symplectic potential implies in particular the singular parts of OPE of operators in the ambitwistor string theory precisely arise from the Poisson structure. Hamiltonians h generating diffeomorphisms on A preserve the symplectic potential, and thus define

perators via

Qh = 1 2πi

h.

Operators defined in this way generate symplectic diffeomorphisms in the ambitwistor string model. In particular, all BMS transformations have a Hamiltonian lift to A = T∗I , and thus define charges of the ambitwistor string model.

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Ambitwistor Strings at Null Infinity Soft limits

Soft limits

The general idea is now to expand the vertex operators in the soft limit. The leading and subleading terms in the expansion can then be identified as generators of supertranslations and superrotations. All further contributions generate diffeomorphisms of A, which will not correspond to diffeomorphisms of I . Gravity Using ¯

δ(λs λ(σs)) = ¯ ∂

1 2πiλs λ(σs), the soft vertex operator can be written

as

Vs =

(1 + ρ · ∂Z˜

ρ · ∂W) ξ λ(σs)2[˜ λ(σs) ˜ λs] ξ λs2λs λ(σs) e

i ξ λs[µ(σs)˜

λs] ξ λ(σs)

= V0

s + V1 s + V2 s + . . .

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 33 / 38

SLIDE 35

Ambitwistor Strings at Null Infinity Soft limits

In this expansion,

V0

s

=

(1 + ρ · ∂Z˜

ρ · ∂W) ξ λ(σs)2[˜ λ(σs) ˜ λs] ξ λs2λs λ(σs) V1

s

=

(1 + ρ · ∂Z˜

ρ · ∂W) iξ λ(σs)[˜ λ(σs) ˜ λs][µ(σs)˜ λs] ξ λsλs λ(σs) V2

s

=

(1 + ρ · ∂Z˜

ρ · ∂W) [˜ λ(σs) ˜ λs][µ(σs)˜ λs]2 λs λ(σs)

Indeed, we can identify V0

s as a supertranslation generator, and V1 s as a

superrotation generator. Inserting these contributions into the correlation function yields the soft theorems for gravity scattering amplitudes (i = 0, 1, 2),

V1...

VkVk+1...VnVi

s

= S(i)

V1... VkVk+1...Vn

.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 34 / 38

SLIDE 36

Ambitwistor Strings at Null Infinity Summary

Summary

Identifying A = T∗I with the cotangent bundle at null infinity, symplectic diffeomorphisms of I define charges of the ambitwistor string (since the action is based on the symplectic potential). When expanding vertex

perators in the ambitwistor string in the soft limit, the leading and

subleading term can then be identified as generators of supertranslations and superrotations. Ambitwistor strings at null infinity therefore confirm the relation between Ward identities of BMS symmetries and soft limits.

V1...

VkVk+1...VnVi

s

= S(i)

V1... VkVk+1...Vn

.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 35 / 38

SLIDE 37

Ambitwistor Strings at Null Infinity Further directions

Outlook and further directions

Ambitwistor string

Non-zero cosmological constant The formulation of the gravitational ambitwistor string model in four dimensions suggests a natural extension to non-zero cosmological constant Λ 0. Recall in this context that we obtained the theory for a flat spacetime as the degenerate limit of the infinity twistor I,

IIJIJK = ΛδI

K,

IIJ = ǫαβ Λǫ ˙

α˙ β

,

IIJ = Λǫαβ ǫ ˙

α˙ β

.

Comparison to the RNS ambitwistor string and the CHY formulae The correspondence to the CHY formulae is best established at the level of the correlators. On the support of the delta-functions,

the ambitwistor Hodges matrix det′(H) corresponds to one copy of the Pfaffian Pf′(Ψ). P(σ) :=

ki σ−σi = λ(σ)˜

λ(σ) . the ρ˜ ρ system can be understood as a spin representation of the ΨΨ current algebra.

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 36 / 38

SLIDE 38

Ambitwistor Strings at Null Infinity Further directions

Outlook and further directions

Ambitwistor string and Null infinity

Loop amplitudes: In the models presented here, anomalies pose an

bstruction to the extension to loop amplitudes. However, a critical

anomaly-free model exists in d = 10: the RNS ambitwistor string. It is thus likely that an anomlay free theory can be formulated by coupling to appropriate matter, and dimensional reduction should provide a guideline for its derivation. Conformal Gravity: It is possible to supplement the Yang-Mills ambitwistor string by (non-minimal) conformal gravity vertex

perators. It might be interesting to try investigate the possibility of a

formulation for the minimal model by introducing the (ρ, ˜

ρ) system

familiar from the Einstein gravity case, and only gauging the currents not involving the infinity twistor. Ambitwistor string at null infinity: Vertex operator algebra for the ambitwistor string at null infinity, both for general d and in d = 4

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 37 / 38

SLIDE 39

Thank you!

Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 38 / 38