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].} 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 Witten’s conjecture The connected prescription, after Roiban, Spradlin & Volovich 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.