Higher-Order Corrections in String & M-theory and Generalised - PowerPoint PPT Presentation

Higher-Order Corrections in String & M-theory and Generalised Holonomy High Energy, Cosmology and Strings IHP, Paris, 11’th December 2006

Plan • Higher-Order Corrections in String Theory • Deformations of Special-Holonomy Backgrounds • Preservation of Supersymmetry • Higher-Order Corrections in M-Theory • Generalised Holonomy

Gravity from String Theory One of the miracles of string theory is that it embodies gen- eral covariance, and gravity, albeit in an a priori rather non- transparent way. It shows up even in perturbative string calulations around a flat Minkowski spacetime background. The 3-graviton scattering am- plitude in string theory is consistent with the 3-point interaction implied by tree-level scattering in Einstein gravity. The 4-graviton string scattering amplitude has a contribution that is also consistent with the Einstein-Hilbert term. However, there is an additional string term that is not explained by Einstein- Hilbert gravity. It is in fact the first indication of a higher-order correction to Einstein gravity: � � � d 10 x √− g R + c α ′ 3 (Riemann) 4 + · · · I = where (Riemann) 4 is quartic in curvature.

Corrections to Gravity Backgounds in String Theory Not only gravity, but the entire leading-order effective supergrav- ity action receives higher-order string corrections. The detailed forms of some of these corrections, even at the 4-point level, are not known. However, if we restrict attention to the gravity (and dilaton) sector, then all the corrections in the effective action up to order α ′ 3 are known. This allows, in particular, a detailed dis- cussion of the α ′ 3 corrections to purely gravitational backgrounds which, at leading order, satisfied the vacuum Einstein equations. One particularly interesting question concerns the fate of leading- order gravitational backgrounds with special holonomy, since these, at leading order, are supersymmetric. Examples are (Minkowski) 4 × K 6 , (Minkowski) 3 × K 7 , (Minkowski) 2 × K 8 where K 6 is a Ricci-flat Calabi-Yau 6-manifold, K 7 is a 7- manifold of G 2 holonomy, and K 8 is an 8-dimensional Ricci-flat Calabi-Yau manifold, a hyper-K¨ ahler manifold or a manifold of Spin(7) holonomy.

Tree-Level Corrections to Type IIA or IIB Strings In the gravity/dilaton sector, the corrected effective action up to order α ′ 3 is given by � � L = √− g e − 2 φ R + 4( ∂φ ) 2 − c α ′ 3 Y where c is a known pure-number constant (proportional to ζ (3)). Y is quartic in curvature. The equations of motion are c α ′ 3 X µν R µν + 2 ∇ µ ∇ ν φ = 2 c α ′ 3 ( Y − g µν X µν ) ∇ 2 φ − 2( ∂φ ) 2 1 = where � X µν = e 2 φ d 10 x √− g e − 2 φ Y δ √− g δg µν The quartic curvature correction Y is quite complicated, as a ten-dimensional Riemannian expression. With care, we can use a simpler eight-dimensionally covariant light-cone expression, for the special case of (Minkowski) × K backgrounds.

The Quartic-Curvature Correction The quartic curvature invariant is given, in light-cone gauge, by Y ∝ ( t i 1 ··· i 8 t j 1 ··· j 8 − 1 4 ǫ i 1 ··· i 8 ǫ j 1 ··· j 8 ) R i 1 i 2 j 1 j 2 · · · R i 7 i 8 j 7 j 8 and t i 1 ··· i 8 is defined by t i 1 ··· i 8 M i 1 i 2 · · · M i 7 i 8 = 24tr M 4 − 6(tr M 2 ) 2 , for all M ij = − M ji It was shown by Gross and Witten that Y could be written as a Berezin integral over SO (8) Majorana spinors ψ = ( ψ + , ψ − ): � � � d 16 ψ exp ψ + Γ ij ψ + )( ¯ ψ − Γ kℓ ψ − ) R ijkℓ ( ¯ Y ∝ Since the integrability condition for a covariantly-constant spinor η in the transverse 8-space is [ ∇ i , ∇ j ] η = 1 4 R ijkℓ Γ kℓ η = 0, it fol- lows that a leading-order supersymmetric background will have a spinor zero-mode for at least one of the right-handed or left- handed spinors in the Berezin integral, and hence Y = 0.

Corrections to ( Minkowski ) 4 × K 6 Corrections to Ricci-flat Calabi-Yau manifolds were analysed long ago. It was shown (Freeman & Pope, 1986) that the variation of Y , calculated from the Berezin integral, gives i ≡ J ij V j X ij = ∇ ˆ where for any V i , V ˆ i ∇ ˆ j S , J is the K¨ ahler form of the original CY background metric, and S = R ijkℓ R kℓmn R mnij − 2 R ijkℓ R kmℓn R minj is the 6-dimensional Euler density. (This agrees with sigma- model beta function calculations by Grisaru et al.) The corrected equations of motion then imply: R ij = c α ′ 3 ( ∇ i ∇ j + ∇ ˆ 2 c α ′ 3 S φ = − 1 i ∇ ˆ j ) S , (Quantities on RHS calculated using the leading-order back- corrections are valid to order α ′ 3 .) ground; In complex co- β = c α ′ 3 ∂ α ∂ ¯ ordinates this corrected Einstein equation is R α ¯ β S . The first Chern class still vanishes, but SU (3) → U (3) holonomy. What happens to supersymmetry?

Supersymmetry in Corrected ( Minkowski ) 4 × K 6 The leading-order supersymmetry transformation rules also re- ceive α ′ 3 corrections; their detailed form has recently been ob- tained (Peeters, Vanhove, Westerberg). There is a general ex- pectation that supersymmetry should survive the corrections. This was studied by Candelas, Freeman, Pope, Sohnius & Stelle (1986) for the 6-dimensional Calabi-Yau case: Can we at least conjecture an α ′ 3 correction that will make this happen? The modification of δψ µ = ∇ µ ǫ to δψ µ = D µ ǫ , where D i = ∇ i + i 2 c α ′ 3 ( ∇ ˆ i S ) has as integrability condition precisely the corrected Einstein equation in the CY background. In the corrected backgound, we shall have Killing spinors satisfying the corrected condition D i η = 0; hence supersymmetry. We can propose δψ µ = D µ ǫ as the corrected SUSY transformation in the CY background, but since it involves the explicit use of the K¨ ahler form (hidden in the hat), we must make sure that it is also expressible as a fully Riemannian expression, which specialises to D i in CY backgrounds.

Riemannian Form of Supersymmetry Correction A Killing spinor in the leading-order CY background satisfies Γ j η = − i Γ ˆ j η . This allows us to write a Riemannian expression 2 c α ′ 3 ( ∇ ˆ that reduces to D i = ∇ i + i i S ) in a six-dimensional CY background (CFPSS): 4 c α ′ 3 ∇ s R rikℓ R stmn R trpq Γ kℓmnpq D i = ∇ i + 3 An alternative form, obtained by dualising in the transverse 8- space, is D i = ∇ i − 6 c α ′ 3 ∇ s R ipkℓ R stℓn R tpnq Γ qk These, extended to the full index range, provide candidate ten- dimensional Riemannian expressions for the α ′ 3 correction to the gravitino transformation rule in string theory, that would satisfy the desideratum of implying that the supersymmetry of leading- order (Minkowski) 4 × K 6 backgrounds is preserved in the face of string corrections at order α ′ 3 . What about leading-order (Minkowski) 3 × K 7 or (Minkowski) 2 × K 8 backgrounds? Will these remain supersymmetric? What is D i for these?

Corrections to G 2 Holonomy ( Minkowski ) 3 × K 7 We can view these as (Minkowski) 2 × K 8 , where K 8 = R × K 7 . With K 7 having G 2 holonomy, we shall have one covariantly- constant SO (8) spinor zero-mode of each chirality. The Berezin integral for Y again vanishes in the background, and its variation can be nicely expressed in terms of special structures on the G 2 manifold (L¨ u, Pope, Stelle, Townsend): X ij = c ikm c jℓn ∇ k ∇ ℓ Z mn where c ijk = i ¯ η Γ ijk η is the associative 3-form and Z mn ≡ 1 32 ǫ mi 1 ··· i 6 ǫ nj 1 ··· j 6 R i 1 i 2 j 1 j 2 · · · R i 5 i 6 j 5 j 6 From the corrected string equations, we find that on K 7 we now have R ij = cα ′ 3 ( ∇ i ∇ j S + c ikm c jℓn ∇ k ∇ ℓ Z mn ) , 2 cα ′ 3 S φ = − 1 where S = g ij Z ij is the 6-dimensional Euler integrand again. Since G 2 manifolds are Ricci-flat, the correction here has de- stroyed the special holonomy completely. But, in a generalised sense, maybe it hasn’t...

Supersymmetry in Corrected ( Minkowski ) 3 × K 7 Can we again modify the supersymmetry transformation rule in such a way that the corrected G 2 background will again remain supersymmetric? We can again ask for a modification of the gravitino transformation rule, to δψ µ = D µ ǫ , where D i = ∇ i + c α ′ 3 Q i , and require that the integrability condition [ D i , D j ] ǫ = 0 give the corrected G 2 Einstein equation to order α ′ 3 . We find D i = ∇ i − i 2 c α ′ 3 c ijk ( ∇ j Z kℓ ) Γ ℓ This, and the corrected G 2 Einstein equation, both reduce to the previous CY results if we take K 7 = R × K 6 . Thus these G 2 holonomy results encompass the previous CY results. The corrected gravitino transformation was “cooked up” to re- tain supersymmetry in the corrected G 2 background. We must check that it at least admits a covariant Riemannian generalisa- tion, that does not make use of special tensors peculiar to G 2 backgrounds. This is more restrictive than the previous CY case. Remarkably, the previous 6-Gamma Riemmanian expression still works.