Connecting the ambitwistor and the sectorized heterotic strings

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1 Connecting the ambitwistor and the sectorized heterotic strings Renann Lipinski Jusinskas February 11th Discussion Meeting on String Field Theory and String Phenomenology - HRI, Allahabad, India Renann Lipinski Jusinskas - AS CR - CEICO (1/27)

2 Outline ; ambitwistor strings; pure spinor formalism; the sectorized model; innite length limit; conclusion; Renann Lipinski Jusinskas - AS CR - CEICO (2/27)

3 CHY formulae Scattering equations Renann Lipinski Jusinskas - AS CR - CEICO (3/27)

4 CHY formulae CHY formulae Scattering equations In JHEP07(2014)033 (arxiv: [hep-th]), Cachazo, He and Yuan, based on previous results, proposed the following formula for the tree level scattering amplitudes of n massless particles: M s n = d n σ δ( s ab ) vol SL(2, C) σ a a σ b b a ( Tr(T a 1 T a 2 )... T an ) 2 s (σ 1 σ 2 ) (σ n σ 1 ) +... Pf (Ψ) s. (1) It is valid for gravitons (s = 2), gluons (s = 1) and scalars (s = 0). Renann Lipinski Jusinskas - AS CR - CEICO (4/27)

5 CHY formulae Scattering equations Scattering equations Motivation: "connection between the scattering data of n massless particles and maps from the n-punctured sphere into the null cone in D dimensional momentum space" ka m = p m (z) n b=1 (z σ b). (2) By requiring p m (z)p m (z) = 0, the so-called scattering equations are obtained: b a k a k b = 0. (3) (σ a σ b ) Is it possible to obtain such structures from a string theory? Renann Lipinski Jusinskas - AS CR - CEICO (5/27)

6 Bosonic case Supersymmetric case Renann Lipinski Jusinskas - AS CR - CEICO (6/27)

7 Bosonic case Supersymmetric case Mason and Skinner proposed them soon after CHY, JHEP07(2014)048 (arxiv: ); First order chiral string (only holomorphic elds), e.g. d 2 z { X m } X m d 2 z {P m X m }. (4) Massless condition P m P m = 0 is imposed as a constraint in the action; It can be seen as an α 0 limit of the (super)string (innite tension). Renann Lipinski Jusinskas - AS CR - CEICO (7/27)

8 Bosonic ambitwistor string Bosonic case Supersymmetric case The bosonic action, after gauge xing, is S = d 2 z {P m X m + b c + b c}, (5) with BRST charge Q = {ct bc c + c H {}}{ P m P m }. (6) The energy-momentum tensor T is written as T = P m X m b c (bc) b c ( b c), (7) and nilpotency of Q implies D = 26. Renann Lipinski Jusinskas - AS CR - CEICO (8/27)

9 Vertex operators Bosonic case Supersymmetric case The spectrum of the bosonic ambitwistor string consists of a symmetric traceless tensor, g mn, identied with the graviton. The unintegrated vertex operator is U = g mn c cp m P n e ik X. (8) The integrated vertex operator is more subtle and can be written as V = d 2 z δ(k P) g mn P m P n e ik X. (9) The operator δ(k P) is responsible for the localization of the scattering amplitudes as it appears in the CHY formulae! Renann Lipinski Jusinskas - AS CR - CEICO (9/27)

10 Heterotic ambitwistor strings Bosonic case Supersymmetric case If we add worldsheet supersymmetry, the heterotic model has the simple gauge-xed action S = d 2 z{p m X m ψ m ψ m + b c + β γ + b c + L C }. (10) with BRST charge Q = {ct + γg matter bc c 1 2 bγ2 + cp m P m } (11) : massless sector of the usual heterotic string plus an unexpected 3-form. The gluon amplitudes agree with the CHY formulae but the gravity sector resembles conformal gravity. Renann Lipinski Jusinskas - AS CR - CEICO (10/27)

11 Heterotic ambitwistor string in Berkovits' proposal Renann Lipinski Jusinskas - AS CR - CEICO (11/27)

12 of the pure spinor formalism Heterotic ambitwistor string in Berkovits' proposal "Super-Poincaré Covariant Quantization of the Superstring", N. Berkovits; JHEP04(2000) (arxiv:hep-th/ ). Based on a bosonic spinor variable satisfying (λγ m λ) = 0. Field content: X m, θ α, p α, λ α, w α. The postulated BRST charge Q λ α d α is nilpotent only for a pure spinor, Q 2 (λγ m λ)π m. (12) The cohomology is automatically described in terms of superelds; Unknown origin! Renann Lipinski Jusinskas - AS CR - CEICO (12/27)

13 Heterotic ambitwistor string in Berkovits' proposal Heterotic ambitwistor string in Berkovits' proposal The action of the model is given by S = d 2 z{p m X m + p α θ α + w α λ α + b c + L C }, (13) while the BRST charge is dened as Q d α {}}{ {λ α [ p α + (γ m θ) α P m ] + ct b c c}. (14) Fails to reproduce the expected gauge transformations for the gravity sector; Does not include the particle-like Hamiltonian H = P m P m ; Renann Lipinski Jusinskas - AS CR - CEICO (13/27)

14 I Heterotic ambitwistor string in Berkovits' proposal The cohomology includes the non-abelian super Yang-Mills elds and the would-be N = 1 supergravity. The former can be encoded by the vertex operator where A I α(θ) satises U SYM = λ α ca I α(θ)j I e ik X, (15) D α γmnpqra αβ I β = 0. (16) The gauge transformations of U SYM are described by the BRST-exact operator δu SYM {Q, cλ I (θ)j I e ik X }, = λ α c(d α Λ I )J I e ik X. (17) Renann Lipinski Jusinskas - AS CR - CEICO (14/27)

15 II Heterotic ambitwistor string in Berkovits' proposal As for the supergravity states, the vertex operator is U SG = λ α ca m α (θ)p m e ik X. (18) BRST-closedness implies the usual e.o.m. D α γmnpqra αβ β = 0, (19a) k m A m α = 0, (19b) However, the expected gauge transformation δa m α = D α Λ m + k m Λ α does not come from a BRST-exact state: δu SG λ α c(d α Λ m + k m Λ α )P m e ik X, {Q, something}. (20) Renann Lipinski Jusinskas - AS CR - CEICO (15/27)

16 Massive cohomology Innite length limit Renann Lipinski Jusinskas - AS CR - CEICO (16/27)

17 Massive cohomology Innite length limit of the sectorized model Background coupling modication of the ambitwistor model (Chandia and Vallilo); Reinterpretation of the variables in terms of two sectors, "+"and -", resembling the left and right movers of the string; Correct description of the massless heterotic spectrum; Background constraints obtained solely from nilpotency of the BRST charge; Quantum corrections are easily obtained; Problem: the construction of the integrated vertex remains unknown; Renann Lipinski Jusinskas - AS CR - CEICO (17/27)

18 Massive cohomology Innite length limit The two sectors are represented by O + = {P m, + b, c, J I }, (21) O = {Pm, d α, θ α, w α, λ α }, (22) with P m + = P m + X m, (23a) Pm = P m X m (θγ m θ), (23b) d α = p α 1 2 (γm θ) α Pm 1 4 (θγm θ)(γ m θ) α. (23c) The N = 1 supersymmetry of the heterotic string is explicit. Renann Lipinski Jusinskas - AS CR - CEICO (18/27)

19 Massive cohomology Innite length limit The two sectors have characteristic (pseudo) energy-momentum tensors dened by T 1 4 ηmn PmP n d α θ α w α λ α, (24a) T ηmn P mp + n + + T C b c ( b c), (24b) which combine to form the full energy-momentum tensor T = T + + T, (25) = P m X m p α θ α w α λ α 2 b c c b + T C. The BRST charge of the model is dened to be Q = {λ α d α + ct + b c c}, (26) and naturally incorporates the sector splitting. Renann Lipinski Jusinskas - AS CR - CEICO (19/27)

20 Massive cohomology Innite length limit In addition to super Yang-Mills, encoded in U SYM = λ α ca I αj I, (27a) γ αβ mnpqrd α A I β = 0, (27b) δ Σ A I α = D α Σ I, (27c) the N = 1 supergravity states are correctly described: U SG = λ α ca m α P + m, (28a) γmnpqrd αβ α A s β = 0, (28b) n n A m α m n A n α = 0, (28c) δ Σ A m α = D α Σ m + m Σ α. (28d) Renann Lipinski Jusinskas - AS CR - CEICO (20/27)

21 Massive cohomology Innite length limit Dimensionful parameters? Initially, the sectorized model was viewed as an innite tension string; Amplitude computations provide terms with dierent powers of momenta; "hidden"length parameter in the model: P + m = P m + X m P + m = P m + 1 l 2 X m Can there be extra (massive) states in the cohomology? Renann Lipinski Jusinskas - AS CR - CEICO (21/27)

22 Massive cohomology Innite length limit Massive cohomology The sectorized heterotic model is two-fold. Depending on the choice of the supersymmetric sector (+ or -), there is a massive state denoted by U = cu open, with U open λ α θ β B αβ + λ α d β C β α + Jλ α E α +λ α N mn F αmn + λ α G α + λ α P + mh m α. (29) This vertex was shown by Chandia and Berkovits to describe the rst massive level of the open string: S mn, A mnp and ψ mα. η mn G mn = 0, m ψ mα = 0, m G mn = 0, (γ m ψ m ) α = 0. m A mnp = 0, Renann Lipinski Jusinskas - AS CR - CEICO (22/27)

23 The innite length limit Massive cohomology Innite length limit It is straightforward to reintroduce the dimensionful parameter l in the sectorized model (global scale symmetry of the action): X m l 1 X m, θ α l 1 2 θ α, λ α l 1 2 λ α, c l 2 c, P m l +1 P m, p α l p α, w α l w α, b l +2 b. The BRST charge, for example, is rewritten as: Q = {λ α [p α 1 2 P m(γ m θ) α 1 2l 2 Πm (γ m θ) α ]} + { 1 4 cηmn PmP n + 1 l ct 2 C + 1 b c c}, l 2 = {λ α [p α 1 2 P m(γ m θ) α ] cpm P m } + O(l 2 ). Renann Lipinski Jusinskas - AS CR - CEICO (23/27)

24 The innite length limit Massive cohomology Innite length limit Surprisingly, the ambitwistor string can be seen as a tensionless (innite length) limit of the sectorized model. And what about the massive state? In this case, it is more convenient to identify explicitly the physical degrees of freedom. This can be achieved by using a DDF construction (M 2 = 2k + k ): { G mn, A mnp U j,i, U a,ȧ, ψ mα U j,ȧ, U a,i. Renann Lipinski Jusinskas - AS CR - CEICO (24/27)

25 The innite length limit Massive cohomology Innite length limit SO(8) decomposition of the physical d.o.f. illustrating the origin of the extra states of Mason and Skinner: U j,i U a,i U j,i U a,i qḃ qḃ q b q b l q b q b qḃ qḃ U j,ȧ U a,ȧ U j,ȧ U a,ȧ SO(8) indices: i, j (vector) and a, ȧ (spinor). Renann Lipinski Jusinskas - AS CR - CEICO (25/27)

26 Renann Lipinski Jusinskas - AS CR - CEICO (26/27)

27 Mason and Skinner: ambitwistor string is a string model for the CHY ampltudes; Pure spinor construction (explicitly supersymmetric); Generalization: sectorized model (can be extended to all superstring formalisms); Physical motivation? Open problems? Connection to twistor string theory; Renann Lipinski Jusinskas - AS CR - CEICO (27/27)