SUPERGRAVITY AMPLITUDES IN THE HORAVA-WITTEN BACKGROUND Michael B. Green University of Cambridge/Queen Mary, University of London CHRIS HULL AT 60 Imperial College, April 27 2017 Chris – thanks for the memorable times over many years INTRODUCTORY COMMENTS • Connections between superstring scattering amplitudes and perturbative supergravity. NO UV DIVERGENCES IN STRING THEORY. TECHNICAL FANTASY: H OW DO FIELD THEORY UV DIVERGENCES ARISE IN LOW ENERGY FIELD THEORY LIMIT? • SUPERGRAVITY DUALITY: Scalar fields parametrize coset space G(R)/K(R) where G ( R ) is the duality symmetry group of classical supergravity. Maximal compact subgroup Preserved in perturbation theory in D=4 dimensions for N>4 supergravity. • SUPERSTRING DUALITY GROUP IS G(Z): No continuous duality group – even at tree level. Moduli space G(Z) G(R)/K(R) \ • RICH DEPENDENCE OF STRING AMPLITUDES ON MODULI – strong coupling duality. Fascinating connections with mathematics of automorphic forms, Jacobi forms, …….DIFFERENT TALK AMPLITUDES WITH MAXIMAL SUPERSYMMETRY Low energy expansion involves low order higher-derivative BPS interactions 4 4 4 6 4 8 4 C1(µr) R C2(µr) D R C3(µr) D R C4(µr) D R ?? 1/2-BPS 1/4-BPS 1/8-BPS Where µ r are the moduli (coupling constants) for ten-dimensional type II theories compactified on tori. • The exact expressions for C 1 ,C 2 ,C 3 are known in all dimensions. • They are automorphic functions under the action of arithmetic duality groups E d +1 ( Z ). SL(2, Z) SL(2, Z) SL(3, Z) SL(2, Z) SL(5, Z) SO(5, 5, Z) E6(6)(Z) E7(7)(Z) E8(8)(Z) ⇥ 10B 9 8 7 . 6 5 4 3 D dimensions . • Rich content: -These only contribute up to 3 loops in string perturbation theory; - Correct spectrum of BPS instantons AMPLITUDES WITH ½-MAXIMAL SUPERSYMMETRY We would like to use the structure of string/M-theory to constrain properties of low energy expansion of HETEROTIC/TYPE I scattering amplitudes. This will make use of the Horava-Witten description of this background. MBG, Arnab RUDRA c,f, low order higher-derivative BPS interactions in maximal SUGRA : 4 4 4 6 4 8 4 C1(µr) R C2(µr) D R C3(µr) D R C4(µr) D R ?? 1/2-BPS 1/4-BPS 1/8-BPS Where µ r are the moduli for ten-dimensional type II theories compactified on tori. Fascinating connections with mathematics of automorphic forms, Jacobi forms, …….DIFFERENT TALK Issues to address: • The realization of S-duality between Heterotic and Type I theories. • Absence of a three-loop divergence in four-graviton scattering in N=4 SUGRA in 4 dimensions. Is this implied by supersymmetry? • Extensions of the strong coupling duality results of maximal (type II) superstring? There has been little discussion of this (but see Bossard, Cosnier-Horeau, Pioline) for discussion of ½- and ¼-BPS gauge interactions in D=3 heterotic string with duality group O (24 , 8 , Z ) ). M-THEORY AND N=1 SUPERSTRING DUALITIES Horava and Witten (1995): ELEVEN-DIMENSIONAL SUPERGRAVITY IN AN INTERVAL OF LENGTH L = π R11 ℓ11 Introduce E 8 gauge fields on each boundary – necessary for anomaly cancellation 11-dimensional Bulk E8 E8 quantum supergravity N=1 N=1 Super YM Super YM Compactify on a further circle of radius R 10 ℓ 11 to nine dimensions – Break symmetry to SO (16) with Wilson lines on circle. T-duality between SO(16) SO(16) E E × ⊂ 8 × 8 heterotic theories SO(16) SO(16) SO(32) × ⊂ DUALITIES BETWEEN M-THEORY AND N=1 SUPERSTRING THEORIES Heterotic R HE 11 11-dim. M-theory E E 8 × 8 M-THEORY ON ORBIFOLD R10 R10 OF A TORUS HO Heterotic SO(32) R11 Type I Type IA 9 1 1 M-theory on M S S /Z2 × × Based on a number of CLASSICAL SUPERGRAVITY/GAUGETHEORY arguments I will here consider quantum contributions in PERTURBATIVE SUPERGRAVITY/YANG-MILLS in order to address some long-standing questions (see e.g. Tseytlin, …..) MBG, Arnab RUDRA M-THEORY/STRING THEORY DICTIONARY ℓ11 ℓ11 String lengths: ℓhet = ℓI = √R11 √R10 Relations between couplings: 3/2 3/2 ghe = R11 gIA = R10 R11 1 HETEROTIC/TYPE I DUALITY gho = = R10 gI Relations between radii 1 1 T-DUALITY rhe = R10 R11 = rIA = R11 R10 = rho rI ! ! HE HO IA I ↔ ↔ A QUESTION ? Heterotic SO(32) and Type I theories should have the same amplitudes (or effective actions) in the Einstein frame when 1 g g− not naively BPS ho ↔ I e.g. In the STRING FRAME perturbative terms of order R 4 arise from: Type I world-sheet Sphere Disk Torus …. 1 2 g− 0 Type I string frame gI− I (gI ) 1 3 3 1 1 1 Einstein frame 2 2 2 2 2 (g )− 2 = g g− = g (gI ) = g− H0/type I duality I ho I ho ho 0 1 HO string frame gho (gho) gho− nonsense 1-loop nonsense 2 But HO perturbation theory is an expansion in powers of g ho . So naïve term-by-term HO/type I duality gives nonsense – apart from Type I disk/ HO torus. But the R 4 interaction must transform sensibly under HO/type I duality. HOW IS THIS REALISED ? It might be that HO/type I duality of these terms is extremely complicated. But note the very simple modular ( SL (2 , Z ) - invariant) function of the coupling that arises in type IIB theory (where this is a BPS interaction). FEYNMAN RULES IN HORAVA-WITTEN BACKGROUND INTERACTION VERTICES ORIGINATE FROM: • BULK ELEVEN-DIMENSIONAL SUPERGRAVITY. GMN ,CMNP M,N,P =1,...,11 a µ =1,...,10 • TEN-DIMENSIONAL GAUGE THEORY IN EACH BOUNDARY. Aµ • BOUNDARY INTERACTIONS OF BULK SUPERGRAVITY FIELDS Gauge and Lorentz Chern-Simons interactions induced by boundary – couple to bulk graviton and to C11µν Bµν ≡ FEYNMAN PROPAGATOR (MBG, Gaberdiel) m(x11 y11) 11 11 1 i − 1 m R11ℓ11 1 On circle: G(p, x y )= e m2 p = p1,...,p9, − 2L 2 R10ℓ11 m p + R2 ℓ2 ! " !∈Z 11 11 On orbifold: Identify x 11 with x 11 i.e. m with m − − 11 11 11 11 11 11 (p,x ,y )=G(p,x y )+G(p,x + y ) m G − p11 = 1 1 R11ℓ11 = cos(p x11) cos(p y11) L p2 + p2 11 11 m 11 !∈Z We want to determine the coupling constant dependence of low-lying terms in the low energy expansion of string scattering amplitudes. SUPERINVARIANTS Tseytlin, …. R4 SUPERINVARIANTS OF MAXIMAL SUPERGRAVITY/YM 4 1 4 Type IIA and IIB invariant J0 = t8t8R ϵ10ϵ10R − 8 1 = ϵ ϵ R4 +6ϵ BYvw(R) Type IIA invariant I2 −8 10 10 10 8 1 Y vw(R)=B tr (R R R R) tr (R R)tr(R R) ∧ ∧ ∧ ∧ − 4 ∧ ∧ 4 ½-MAXIMAL SUPERGRAVITY: - THREE COMBINATIONS OF R : t t R4 24 t trR4 +6t tr(R2)2 =0 8 8 − 8 8 SUPERINVARIANTS OF ½-MAXIMAL SUPERGRAVITY/YM: 4 4 2 2 1 2 2 X = t trR 1 B trR X2 = t8(trR ) 10B (trR ) 1 8 − 4 10 − 4 Parity-conserving + parity-violating. ANOMALY CANCELLING (PARITY-VIOLATING) INVARIANTS: gs 1 10BY (R, F ) Y gs(R, F )= 8trF 4 +trR4 + (trR2)2 trF 2 trR2 4 − 1 8-form In particular, note that Y gs(R, 0) = Y vw(R)+ (trR2)2 2 1 1 gs Parity violating part of X1 + X2 =(t8 ϵ10B) Y (R, 0) 4 − 4 We aim to determine the coupling constant dependence of low-lying terms in the low energy expansion of string scattering amplitudes. • PARITY VIOLATING HIGHER DERIVATIVE INTERACTIONS DETERMINED BY ANOMALY CANCELLATION One-loop five-point function 1 2 2 (vw) Boundary ϵ10 B (trR ) + ϵ11 CY8 (R) Bulk (Vaf a-Witten) 2 1 Y (R)(vw) = tr R4 (tr R2)2 8 − 4 ! " (gs) (gs) 4 1 2 2 Anomaly cancelling = ϵ BY (R) Y8(R) = tr R + (tr R ) 10 8 4 ! " (+ analogous gauge terms) • These higher derivative terms have PARITY CONSERVING PARTNERS that fill out ten-dimensional =1 supermultiplets. N Different kinds of TREE DIAGRAMS: (MBG, Gaberdiel) e.g. gauge bosons on Cµν11 GMN separate boundaries m R10 (1) (2) (3) (4) ( 1) 3 tr1(T1 T1 )tr2(T2 T2 ) − m2 R11 ℓ 11 m Z s + R2 ℓ2 !∈ − 11 11 2 2 LOW ENERGY EXPANSION: expansion in powers of ( s) R ℓ − 11 11 HE powers of ( s) g2 ℓ2 Type I powers of ( s) ℓ2 r2 − he H − I IA tree-level pole + higher loop corrections all terms arise from one loop (annulus) 0 m =0 m =0 in string perturbtion theory. (gIA) Reproduces HE theory up to terms of order s (tr F 2 )(tr F 2 ) - i.e. up to genus two. These are expected to be BPS “protected” terms. MULTIPLE PROPAGATOR CONTRIBUTIONS TO 4-GAUGE PARTICLE SCATTERING REPRODUCES HIGHER DERIVATIVETERMS OF FORM n 2 2 n = 1, 0, 1 s t8 (trF ) − in ten-dimensional HE - up to two loops – as expected from BPS conditions Cµν11 GMN Gµν Gµν MULTIPLE PROPAGATOR CONTRIBUTIONS TO 4-GRAVITON SCATTERING 4 2 2 This generates various expected combinations of t8 trR ,t8 (trR ) 4 GAUGE-PART ICLE AMPLITUDE GAUGE LOOP 9 1 in M S × 10-dimensional SYM 10 E8 gauge loop on x circle of radius r he =1 /r ho with appropriate Wilson lines • Massless SO (16) gauge particles + massive spinor states + KK recurrences • Zero winding number sector is UV divergent – killed by counterterm r /g2/3 ∼ he he HE Coefficient must vanish • Non-zero winding terms give finite expression 1/rhe ∼ Trace in the FUNDAMENTAL rho 4 rI 4 rep. of SO(16) ζ(2) tr1F = ζ(2) tr1F ℓhet gI ℓI 1-LOOP IN HETEROTIC SO(32) SO(16) SO(16) DISK AMP.