Supergravity Amplitudes in a Horava-Witten Background

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. I N TYPE I → × 4 gauge-particle amplitude

GAUGE LOOP

10 E8 gauge loop on x circle with appropriate Wilson lines

• Massless SO (16) gauge particles + massive spinor states + KK recurrences

• POISSON RESUMMATION expresses this as a sum over windings of loop

rhe 2 2 Renormalised • UV divergence in 0-winding sector C (tr1F )(tr1F ) 0 ∼ 2/3 → C =0 ghe ℓhet

rho 4 rI 4 • Sum over non-zero windings ζ(2) tr1F = ζ(2) tr1F ℓhet gI ℓI

1-LOOP IN HETEROTIC SO(32) SO(16) SO(16) DISK AMP. I N TYPE I → × 4-GRAVITON LOOP AMPLITUDES

GAUGE LOOP

BULK GRAVITON LOOP

4-GRAVITON amplitude

Sum over supergravity states in the eleven-dimensional loop. BULK GRAVITON LOOP

11 11 m Orbifold conditions on states: Ω : x x Ω m m p11 = →− | ⟩→| − ⟩ R11ℓ11

A A¯ Decompose SO (10 , 1) spinor into chiral SO (9 , 1) spinors =(S+ ,S ) 1 1 S − (1 + ) A ; m = ( A ; m + A ; m ) 2 | + 2 | + | + Projected states 1 1 (1 + ) A¯ ; m = ( A¯ ; m A¯ ; m ) 2 | 2 | |

(i) m =0 A¯ ;0 =0 =1 ten-dimensional superspace | − ⟩ N (ii) m =0 =2 eleven-dimensional superspace ̸ N (apart from m =0 sector) Evaluate loop using a world-line superspace vertex operator formalism (i) Total m =0 sector (adding gauge loop to m =0 bulk loop) gives HO effective action 1 1 1 ζ(2) d10x √ Gt trR4 + (trR2)2 = ζ(2) d10x √ Gt Y gs ℓ − 8 4 ℓ − 8 H ! " # H ! • Forms =1 supermultiplet with parity violating anomaly cancelling terms in the HO theory.N (not renormalised beyond 1 loop). • HO/type I duality with disk contribution in the type I theory. (ii) m =0 sector – after some work : ̸ 1 2 1 10 4 (g− E 3 (ig− ) 2ζ(2)) d x √ Gt8t8 R ho 2 ho − − ! Real analytic Eisenstein Series

(0) 1 (0) c.f. Type IIB coefficient E 3 C + ig − with C =0 2 IIB

! "3 2 1 gho where 3 E (igho )= 2 3 2 2 2 2 (m ,m )=(0,0) (m1 gho + m2) 1 2 3 1 2 n 2 2 g | | =2(3) g +2(2) g + 8 1( n ) e ho (1 + O(gho)) ho ho | | n + Z Tree-level One loop “D-instantons” (CANCELS with 2 ζ (2) ) −

1 1 • Invariant under HO/type I duality E 3 (ig− )=E 3 (igI )=E 3 (ig− ) 2 I 2 2 ho BUT NOT INVARIANT TERM BY TERM IN PERT. EXPANSION 4 • SUGGESTS NON-RENORMALISATION OF t8t8 R BEYONDTREE LEVEL IN HO THEORY. No further M-theory SUGRA diagrams that can contribute to order R 4 1 The one-loop action is (t + B) trR4 + (trR2)2 8 10 4 Agrees with explicit perturbative heterotic string calculations

• Explicit SUGRA calculations show absence of expected R4 three-loop counterterm. Bern, Davies, Dennen, Huang WHY? • Is there some symmetry protecting this (apparently non-BPS) interaction?

• The presence of TYPE I D-INSTANTON contributions is expected and results from KK modes in the x 10 direction with world-lines wound around the x 11 direction. These are Z2 instantons that break O(32) to SO(32).

• The HO D-INSTANTON correspond to KK modes in the x 11 direction 10 - these are UNSTABLE since p 11 is not conserved) with world-lines wound around x . COMMENTS

• More generally, moduli space is a symmetric space G ( R ) /K ( R ) with discrete identifications G(Z) G(R)/K(R) \ where G ( Z ) is a a discrete arithmetic subgroup of the supergravity duality group; Multi-graviton/gauge amplitudes transform as automorphic functions under G ( Z ).

• Non-perturbative information involving D-instantons. • Makes contact with BPS black hole counting formulas - D-instantons in D dimensions are wrapped euclidean world-lines of D=4 black holes.

• Well-studied in the low energy expansion of type II (maximal SUSY) but not in the heterotic/type I case.

• However suggested formulas for ½-BPS F 4 and ¼-BPS d 2 F 4 interactions in the D=3 dimensional SUGRA/YM theory Bossard, Cosnier-Horeau, Pioline G(Z)=O(24, 8, Z) Instantons correspond to the Dikgraaf, Verlinde, Verlinde genus-two modular form that counts ¼-BPS black hole states in D=4 dimensions.

MULTIPLE PROPAGATOR CONTRIBUTIONS TO 4-GAUGE PARTICLE SCATTERING

2 2 2 2 Power series in ℓ11 s /R11 = ℓhet s = gI ℓI s

Tree l evel HE Multiloop type I

n 2 2 REPRODUCES HIGHER DERIVATIVETERMS OF FORM s t (trF ) n = 1, 0, 1 8 − in ten-dimensional HE - up to two loops – as expected from BPS conditions

n 2 2 n 2 2 n 2 2 s t8 (tr1F ) s t8 (tr1F )(tr2F ) s t8 (tr2F )