From Twistor Strings to Quantum Gravity?

The twistor programme and twistor strings From twistor strings to quantum gravity?

L.J.Mason

The Mathematical Institute, Oxford [email protected]

Based on JHEP10(2005)009 (hep-th/0507269), hep-th/0606272 with Mohab Abou-Zeid & Chris Hull, and joint work with Rutger Boels and Dave Skinner. Outline

1 The twistor programme Twistor space Penrose-Ward transform The non-linear graviton

2 Twistor strings

3 A proof of the Berkovits-Witten conjecture String ﬁeld theory Twistor actions

4 Einstein (Super-) gravity Twistor Correspondences

Quantum gravity requires a pregeometry for space-time. Penrose’s Proposal: Twistor space is the fundamental arena for physics. Flat correspondence: 0 • Complex space-time M = C4, coords xAA , A=0,1,A0=00,10 2 AA0 BB0 ﬂat metric ds = dx dx εABεA0B0 , εAB = ε[AB] etc..

4 α A 0 • Twistor space T = C , coords Z = (ω , πA0 ), α=(A,A ). Projective twistor space ∗ 3 PT = {T − {0}}/{Z ∼ λZ, λ ∈ C } = CP . • Incidence relation A AA0 ω = ix πA0 . 1 {Point x ∈ M} ←→ {Lx = CP ⊂ PT}, hgs coords πA0 . Penrose transform

0 Massless ﬁelds ←→ cohomology on PT = PT−L∞ = {πA0 6= 0} I B0 1 0 φ(x) 0 0 0 = π 0 π 0 . . . π 0 f π 0 dπ , f ∈ H ( , O(−n−2)) A1A2...An A1 A2 An B PT Lx I n ∂ f B0 1 0 0 φ(x)A A ...An = πB dπ , f ∈ H ( , O(n−2)) 1 2 A1 A2 An PT Lx ∂ω ∂ω . . . ∂ω

∗ • homogeneous fns O(n) are f (Z), f (λZ) = λnf (Z ), λ ∈ C . • Cechˇ cohomology for open cover {Ui } 1 H (O) = {hol. fns fij on Ui ∩Uj , fij +fjk +fki = 0}/{fij = gi −gj , } • Dolbeault cohomology H1(O(n)) = {f ∈ Ω(0,1)(n)| ∂¯f = 0}/{f = ∂¯g} • n = 0, −4 for ASD, SD Maxwell • n = 2, −6 for linearized gravity (note parity asymmetry). Ward transform

H1(O): ASD Maxwell ﬁelds ↔ holomorphic line bundles.

For vector bundles: Theorem (Ward) ASD Yang-Mills ﬁelds, D = d + A with D2 = F with F + = 0, on E˜ → M are in 1 : 1 correspondence with holomorphic Bundles 0 E → PT trivial on each Lx . 0 Easiest proof is in Euclidean signature where p : PT → M; 0 Can pull back (E˜ , D) → M to give (E, ∂¯) → PT . ¯2 + For such bundles ∂E = F , so bundle is holomorphic ⇔ F + = 0.

[Reverse direction requires some complex analysis.] The non-linear graviton

Theorem (Penrose)   asd deformations of con- Deformations of complex 1−1   ←→ formal structure ( , η) ; structure, 0 ; M PT PT (M, [g]). 

For Ricci flat g ∈ [g], PT must have 1 • A holomorphic fibration p : PT → CP ∗ • A Poisson structure {, }I, on fibres of p, valued in p O(−2). Main ideas: We deform PT by plate tectonics or changing ∂¯. {,} I , Ricci-flat linearised deformations H1(O(2)) ,→ H1(T 1 0PT). 1 The CP s in PT survive deformation. Define

1 M = {moduli space of degree-1 CP s ⊂ PT }.

1 1 x, y ∈ M connected by a light ray ⇔ Incidence CPx ∩ CPy 6= ∅. ; ASD conformal structure, [g], Weyl+ = 0 on M. Supersymmetric extensions

3|N Super-twistor space [Ferber]: PT[N] = CP , homogeneous coords (Z α, ηi ), i = 1,..., N, ηi anticommuting. 1 AA0 iA0 + Correspondence: PT[N] ⊃ CP ←→ (x , θ ) ∈ M[N], + M[N] =chiral super-Minkowski space. Incidence relation

A AA0 i iA0 ω = x πA0 , η = θ πA0

1 0 Penrose-Ward transforms extends, e.g. as ∈ H (PT[4], O) expands:

i i j i j k ˜l 1 2 3 4 as = a + η ψi + η η φij + η η η ijkl ψ + η η η η b

˜ i + ↔ susy multiplet As := (A, Ψi , Φij , Ψ , B) helicity −1 to 1 on M[4]. Supersymmetric Ward transform

SUSY Ward transform: 0,1 For as ∈ Ω (ad ), take d-bar operator ∂¯a = ∂¯ + as GC s 0 + {Hol vector bundle E → PT[4]} ←→ {N = 4 SYM multiplet on M[4].}

But, the interactions are an ASD truncation: 1 2+ cf Chalmers-Siegel action for A ∈ Ω (adG) , B ∈ Ω (adG) Z + + Sasd[A, B] = B ∧ F , ; F = 0 , [d + A, B] = 0 . MR To extend to full YM: Z S[A, B] = B∧F +− B∧B , ; F + = B , [d+A, F +] = 0. 2 MR Supersymmetric Ward transform

SUSY Ward transform: 0,1 For as ∈ Ω (ad ), take d-bar operator ∂¯a = ∂¯ + as GC s 0 + {Hol vector bundle E → PT[4]} ←→ {N = 4 SYM multiplet on M[4].}

On-shell generating function for amplitudes with full Yang-Mills interactions is

∞ Z X ¯ A[As] = Det(∂as |C) dµ d d=1 M[4]

d M[4] = contour in moduli space of connected algebraic curves d C degree d in PT[4], C ∈ M[4], dµ some measure. Selection rules: • d M[4] contribution ↔ processes with d + 1 − l SD gluons. • l :=Number of loops, g := genus of C, then g ≤ l.

; concrete algebraic formulae for all tree amplitudes [RSV]. The Cachazo-Svrcek-Witten formulation The disconnected prescription

d + d If C = d lines then M[4] = (M[4]) /Symd, but must include propagators from holomorphic Chern-Simons action

Z 4 α β γ δ Y i Sasd[as] = CS(as)∧Ω[4] , Ω[4] = αβγδZ dZ dZ dZ dη PTs i=1 ; diagrammatic formalism with MHV diagrams glued together with Chern-Simons propagators.Gluing is done on-shell. Gukov, Motl & Nietzke argue ⇔ connected version. Path integral formulation: The above can be expressed as

„ « R 0 ∞ Z CS(as)∧Ω[4] Z 0 PT[4] X ¯ A[As] = Das e Det(∂a0 |C) dµ d s d=1 M[4]

0 Here as are off-shell, but → as, on-shell, at ∞. ; twistor action, see Boels talk. Underlying string theory

String models: PT[4] is Calabi-Yau • Witten: B-model in twistor space coupled to D1-instantons d (the holomorphic curves C of M[4]). • Berkovits: Open string action on Riem-surface C with bdy, I 1,0 ∗ 0,1 Z : C → T[4], YI : C → Ω (C) ⊗ T[4], A ∈ Ω (C) Z I ¯ I I S[Z , YI, A] = YI∂Z + AYIZ + Complex conjugate C

ﬁelds are real on boundary (↔ split signature for M). d (C ∈ M[4] are worldsheet instantons.) Conformal Supergravity

But: [Berkovits, Witten] Amplitudes contain N = 4 conformal supergravity; wont decouple from loops.

Theories depend on C-structure J of PT[4] and: 1,1 R Witten: B-ﬁeld b ∈ Ω (PT[4]) coupled in via C b I R I Berkovits: 1-form gI(Z)∂Z coupled in by ∂C gI∂Z . (J, gI) or (J, b) ←→ spectrum of N = 4 conf sugra.

This is a problem for gauge theory applications, but an opportunity for quantum gravity (although wrong theory....). Heterotic twistor-strings See talk by D.Skinner

Optimal formulation should be as a twisted (0, 2)-model.

• Model depends only on C-str, and b-ﬁeld, on PT[4] and bundle E → PT[4]. • Gives a Dolbeault formulation that allows off-shell ﬁelds.

• Allows C2(E) 6= 0 to incorporate instantons • Gives interpretation of b as twisting of Courant bracket in the context of Hitchin’s generalised complex structures. • Witten, hep-th/0504078 gives correspondence with Berkovits type models as sheaves of chiral algebras. • Directly gives generating functions of amplitudes as ¯ integrals of det ∂|C over instanton moduli spaces. String ﬁeld theory degree 0; Perturbative effects

We ‘prove’ Berkovits-Witten’s conjecture for conformal Sugra: Off-shell theory depends on

• PT [4], with Calabi-Yau almost complex structure J (1,1) • b ∈ ΩJ (PT [4]) Best guess (Berkovits-Witten) for string ﬁeld theory is action Z S[J, b] = (N(J) ∧ b) ∧ Ω[4] (1) PT [4]

¯2 (1,0) (0,2) where N(J) = ∂J ∈ T ⊗ Ω , is Nijenhuis tensor of J. Field equations: N(J) = 0 = ∂¯b ; so J is integrable. Gauge freedom: b → b + ∂χ¯ + ∂ξ. ; ∂ ∈ 1( , Ω1) PT [4] is a complex manifold, b H∂¯ PT [4] d . Gives: Spectrum of N = 4 conf sugra with ASD couplings. String ﬁeld theory Instanton effects

Instantons are pseudo-holomorphic maps Z : C → PT [4] and contribute Z X R b dµ e( C ) (2) d d M[4] d to string ﬁeld theory action, where C is an instanton and M[4] a contour in the moduli space of instantons of degree-d. d Disconnected prescription: C = ∪i=1Lxi , Lxi degree-1, xi ∈ M[4] = real space of degree-1 instantons so d d M[4] = (M[4]) /Symd . Thus instanton contribution is „ « d „ « Z R b Z R b X ∪i Lx X Y Lx 4|8 dµ e i = e i d xi d (M )d /Sym d M[4] d [4] d i=1 Z !d Z ! X 1 R b R b = e Lx = exp e Lx d! d M[4] M[4] Twistor action

Thus path integral becomes ! Z Z Z R b 4|8 DJ Db exp (N ∧ b) ∧ Ω[4] + e Lx d x PT [4] M[4] giving a string-ﬁeld theory action (incorporating instantons)

Z Z R b 4|8 S[J, b] = (N ∧ b) ∧ Ω[4] + e Lx d x (3) PT [4] M[4]

Theorem (M, hep-th/0507269)

Let the real slice of M[4] arise from Euclidean signature reality conditions, and assume that only spin-2 parts of N = 4 csugra spectrum are present Then: (3) is equivalent to conformal supergravity action R |C|2 on (Euclidean signature) space-time. M[4] [See Rutger Boels talk for gauge theory twistor actions.] Back to twistor programme Quantum gravity?

Returning to the twistor programme, this resolves old questions: • Quantize metrics on M ; well defined events but fuzzy lightcones—bad (fields dont know how to propagate), • Quantize on twistor space ; fuzzy events but well defined light rays—good. Note: we only have ASD (leg break) data. For physics, we need the SD (googly) part also. Lets quantize anyway! 1 But, do we quantize the CP s or the complex structures? 1 Answer: In string theory, quantization of the CP s leads to quantization of the complex structures. But: Above shows that this leads to conf sugra. How do we get Einstein (super-) gravity? Einstein Supergravity Gauging symmetries; work with Abou-Zeid & Hull

Basic idea: Conformal gravity contains Einstein gravity; eliminate superfluous fields by imposing symmetries. Berkovits string on T[4]: R ¯ I Consider strings on T[4], action S = YI∂Z ; I I reduce to PT[4] by gauging symmetry along Υ = Z ∂/∂Z by including field A ∈ Ω0,1(C) in the action Z ¯ I I S = YI∂Z + A ∧ JΥ, JΥ = YIZ ; C

¯ I −λ λ I gauge freedom A → A + ∂λ gives (YI, Z ) → (e YI, e Z ). Reduction to Einstein: 1 We gauge symmetries ↔ 1-forms pulled back from CP or 1|N 1|N 1 CP for projections PT[N] → CP → CP .

• For a general choice of such symmetries there is an anomaly. R|N 1|N • For CP → CP , we require R = 3, no restriction on N!. • In particular we obtain N = 4 and h, h˜ above give spectrum of N = 4 supergravity. 0 1 • For PT[4] → CP we obtain a theory with spectrum of N = 8 supergravity. • We obtain correct interactions for ASD couplings from string perturbation theory. • It is an open question as to whether the instanton contributions give couplings of the full theory as they do for conformal supergravity. Perhaps just ASD truncation, so back to square one??? Twistor action, self-dual case hep-th/0706.??; joint with Martin Wolf

1 0 linear gravitational field ↔ h ∈ H (PT , O(2)) 3|8 For N = 8, i.e., on CP , this gives full multiplet. ; inf. deformation {h, ·}I = Hamiltonian vector field using {, }I. In the full nonlinear ASD case, h can be regarded as a finite deformation of a ‘contact form’ and determines the C-structure. Full non-linear ASD field equations become ¯ ∂0h + {h, h} = 0 with Chern-Simons-like action Z 2 S[h] = (h ∧ ∂¯h + {h, h}h) ∧ Ω 3 PT Ω = natural wt -4 hol. volume form. It remains to relate this to (part of) a twistor-string theory.