Four-graviton amplitudes in string theory Pierre Vanhove Amplitude 2011 Ann Arbor, Novembre 10-13, 2011 based on work 1111.2983 Michael B. Green, Stephen D. Miller Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 1 / 29 String theory S-matrix We are interested in the properties of BPS protected interactions in string the- ory String theory interactions are invariant under the action of the extended supersymmetries and duality symmetries These symmetries put strong constraints on I Non-perturbative contributions: black hole contributions I the ultraviolet behaviour of the low-energy limit Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 2 / 29 String theory interactionsI I At low-energy the four-graviton of type II string theory compactified on tori, the S-matrix can be separated into an analytic and non-analytic terms analytic non-analytic AD(s, t, u) = AD (s, t, u) + AD (s, t, u) I Due to the maximal supersymmetries the amplitudes factorize the dimension eight operators R4 analytic 4 I The analytic part AD (s, t, u) = TD(s, t, u) R is organized by 2n n n n increasing powers of σn := `D (s + t + u ) and σ1 = 0 E(D) (') (0,-1) (D) p q T (', σ ) = + E (') σ σ D 2,3 σ (p,q) 2 3 3 p q X, >0 Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 3 / 29 String theory interactionsII (D) I E (') = (0,-1) 1 : Einstein-Hilbert term for classical gravity I The coefficient are functions of the moduli ' in the coset Ed+1(R)=Kd+1 parametrizing the vacuum of the theory I The string theory is invariant under a discrete subgroup Ed+1(Z) E(D) (γ · '~ ) = E(D) ('~ ) γ 2 E (Z) (p,q) (p,q) ; d+1 Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 4 / 29 Constraints from supersymmetryI (D) (D) (D) I E R4 E @4R4 E @6R4 The first 3 corrections (0,0) , (1,0) and (0,1) are BPS protected I Supersymmetry implies they satisfy simple differential equation (k + 3)(8 + 2k - D)(D(k + 3)- k - 33) (D) (D) ∆ - E = S ; k = 2p+3q (k + 3)(D - 2) (p,q) (p,q) I Boundary conditions from string/M-theory allow to determine the solution [Green, Russo, Miller, Vanhove] Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 5 / 29 Constraints from supersymmetryII I The zero instanton sector (the constant term in a Fourier expansion) displays non-renormalisation theorems -2 8-D a E(D) = g D-2 tree + I (0,0) D 2 1-loop pert gD -4 7-D a 1 E(D) = g D-2 tree + I + I (1,0) D 4 2 1-loop 2-loop pert gD gD 6-D 1 (D) -6 D- atree 1 1 - E = g 2 + I + I + I + O(e gD ) (0,1) D 6 4 1-loop 2 2-loop 3-loop pert gD gD gD I Only the invariance under the continuous symmetry group Ed+1(R) was needed for these non-renormalisation theorems I This was enough to rule the existence of possible UV divergences for L 6 6 in D = 4 for N = 8 supergravity [Green et al.; Elvang et al.; Bossard et al.] Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 6 / 29 Constraints from supersymmetry III I Beside the perturbative sector these interactions receive non-perturbative contributions I They are instantons from various supersymmetric BPS configurations. They are necessary for the non-perturbative consistency of the theory I In this case we will need to make consider the discrete duality group Ed+1(Z) I The BPS protected operators are expected to have very particular Fourier coefficients. In this talk we will analyze how supersymmetry restricts the Fourier modes Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 7 / 29 Duality symmetries [Hull,Townsend] D E11-D(11-D)(R) KD E11-D(11-D)(Z) 10A R+ 1 1 10B Sl(2, R) SO(2) Sl(2, Z) 9 Sl(2, R) × R+ SO(2) Sl(2, Z) 8 Sl(3, R) × Sl(2, R) SO(3) × SO(2) Sl(3, Z) × Sl(2, Z) 7 Sl(5, R) SO(5) Sl(5, Z) 6 SO(5, 5, R) SO(5) × SO(5) SO(5, 5, Z) 5 E6(6)(R) USp(8) E6(6)(Z) 4 E7(7)(R) SU(8)=Z2 E7(7)(Z) 3 E8(8)(R) Spin(16)=Z2 E8(8)(Z) I E11-D(11-D) real split forms, KD maximal compact subgroup. I The vacuum of the theory is the scalar manifold '~ 2 MD = GD=KD Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 8 / 29 1 1 The 2 and 4 BPS interactions (D) (D) Ed+1(Z) E(0,0) E(1,0) E8 1 E8 E8(8)(Z) E 7 3 2 E 7 5 [1 0 ]; 2 [1 0 ]; 2 E7 1 E7 E7(7)(Z) E 6 3 2 E 6 5 [1 0 ]; 2 [1 0 ]; 2 E6 1 E6 E6(6)(Z) E 5 3 2 E 5 5 [1 0 ]; 2 [1 0 ]; 2 SO(5,5) 1 SO(5,5) 4 SO(5,5) SO(5, 5, Z) E 3 2 Eb 5 + 45 Eb[00001] 3 [10000]; 2 [10000]; 2 ; SL(5) 1 SL(5) 3 SL(5) SL(5, Z) E 3 2 Eb 5 + π3 Eb 5 [1000]; 2 [1000]; 2 [0010]; 2 SL(3) 1 SL(3) SL(3) SL(3, Z) × SL(2, Z) Eb 3 + 2Eb1(U) 2 E 5 - 4 E 1 E2(U) [10]; 2 [10]; 2 [10];- 2 3 4 5 9 - 7 7 1 - 7 2ζ(2) 7 SL(2, Z) E 3 (Ω) ν + 4ζ(2) ν ν E 5 (Ω) + ν 2 1 1 2 1 2 15 1 1 SL(2, Z) E 3 (Ω) E 5 (Ω) 2 2 2 Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 9 / 29 Part I Fourier expansion Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 10 / 29 Fourier expansionI I We are interested in type II string compactified on tori Td I The instantonic configurations are obtained by wrapping the world-volume of D-brane and NS-brane (or metric, M2-, M5- and Kaluza-Klein monopole from a M-theory point of view) m I If θi 2 U R is a set of continuous commuting charges of non-perturbative contributions we define the fourier modes (D) (D) 2iπQ·θ F(p q)[Q, '] := dθ E(p q) e , m , Z[0,1] I In low dimensions, the set of charges is not always commutative (when there are NS5-brane, and Kaluza-Klein monople contributions) but it is the integral over the abelianization of U, ie the commutator [U, U], which has a Fourier expansion Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 11 / 29 Fourier expansionII I We can examine instanton corrections to the perturbative regime, the strong coupling (M-theory) regime and decompactification regime I In this talk we will be concerned with the decompactification limit where the connection with the BPS states in higher dimension is visible I For a compactification of the theory in dimensions D + 1 to a theory in 2 dimension D = 10 - d on circle of radius r = rd=`D+1 the symmetry group becomes Pαd+1 = GL(1) × Ed × Uαd+1 ⊂ Ed+1 Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 12 / 29 Fourier expansion III I The Fourier coefficient is a sum over integer charges Q 2 Ed+1(Z) \ Uαd+1 that are conjugated to the angular variables θ 2 Uαd+1 (D) (D) 2iπQ·θ F(p q)[Q, '] := dθ E(p q) e , m , Z[0,1] I The fourier coefficients are function of the moduli of the symmetry group ' 2 Ed=Kd Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 13 / 29 Character variety orbitsI I The action of the symmetry group Ed on the angular variabes θ is irreducible if and only if Uα is abelian. This is the case of duality group until E8 where the charge system is non-abelian (due to the Kaluza-Klein Monopole contribution) Ed+1 Mαd+1 Vαd i E8 E7 q : 56, q : 1 i E7 E6 q : 27 E6 SO(5, 5) Sα : 16 SO(5, 5) SL(5) v[ij] : 10 SL(5) SL(3) × SL(2) via : 3 × 2 + SL(3) × SL(2) SL(2) × R vva : 2 Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 14 / 29 Character variety orbitsII I The (complexified) action of the symmetry group GL(1) × Ed acts on the angular variables θ according irreducible representations and therefore fills out character variety orbits o 2 Ed=Hd where Hd is the stability group I The characterity orbits are given by conditions on the representative of the action of the symmetry group Ed given in the previous table I The relevant continuous BPS orbits have been classified by [Ferrara, Maldacena; Pope, Lu, Stelle] I 1 One important orbits is the 2 -BPS orbits in dimension D Ed+1 O 1 -BPS = n 2 Ed n R d I Where nd is the number of BPS charges in dimension D - 1 I 1 The minimal orbit is the 2 -BPS orbits, and the next-to-minimal (NTM) 1 orbit is the 4 -BPS orbit Pierre Vanhove (IPhT & IHES) graviton scattering in string theory Ann Arbor Nov 2011 15 / 29 Character variety orbits III Mαd+1 = Ed BPS BPS condition Orbit Dim.