Strings on Celestial Sphere. Stephan Stieberger, MPP München

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1 Strings on Celestial Sphere Stephan Stieberger, MPP München String Theory from a Worldsheet Perspective Galileo Galilei Institute, Firenze April 15-19, 2019

2 based on: St.St., T.R. Taylor: Strings on Celestial Sphere arxiv: Nucl. Phys. B95 (2018) Symmetries of Celestial Amplitudes arxiv: to appear in Phys. Lett. B + work to appear

3 Recap: studying scattering amplitudes: deep connections between gravity and gauge interactions e.g.: KLT, BCJ, EYM (double-copy-construction) (in momentum or twistor space) traditional momentum space description: p μ k, k = 1,, N p 2 k = m2 k amplitudes specified by asymptotic wave functions, which transform simply under space-time translations with manifest translation symmetry traditional amplitudes describe transitions between momentum eigenstates D=4 Minkowski probably not the right space to see all symmetries of scattering amplitudes

4 R 1,D+1 Lorentz group in is identical to Euklidian D-dimensional conformal group SO(1,D+1) Scattering amplitudes in R 1,D+1 interpretation as Euklidian D-dimensional conformal correlators p in 1 I I + p out 2 p in D=2: celestial sphere p out 1 p in 2 Can 2D CFT on celestial sphere offer some new insight into gauge-gravity connections?

5 N particles on celestial sphere p k (E k, z k, z k ) z k represent points on CS 2 with: z k = p1 k + ip 2 k p 0 k + p k, E k = p 0 k, ( p k ) 2 = (p 0 k )2 p μ k = Ek ( 1, z k + z k 1 + z k, i(z k z k ), 1 z k z k z k 2 ) := ω k q μ k ω k = 2 E k (1 + z k 2 ) E.g.: ij = 2 (ω i ω j ) 1/2 (z i z j ) [ij] = 2 (ω i ω j ) 1/2 (z i z j )

6 Lorentz symmetry: z i az i + b cz i + d global conformal symmetry on CS 2 Amplitudes = conformal correlators of primary fields on CS 2 p g p 2 z k = p1 k + ip2 k p 0 k + p k = z x z x 1 x z 2 p 1 D = 2 D = 4 g z 1 z 2 h 1+h 2 h z 2 z h 2+h h 1 z 1 z h 1+h h 2 D=4 space-time QFT correlators D=2 Euklidian CFT correlators D=2 CFT correlators involve conformal wave packets

7 <latexit sha1_base64="sfbqfarh6pbit9mjbmq7+a/usm4=">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</latexit> in momentum basis: plane waves with momentum p = ωq(z) in conformal basis: conformal primary wave functions Δ = h + h C construct complete set of on-shell wave functions in D=4: solves D=4 wave (Maxwell) equations and transforms as SL(2,Z) conformal primaries Pasterski, Shao arxiv: bases plane waves conformal primary wavefunctions vector fields A µ`(x; p) = µ`(p) exp { ip µ x µ } V ± µj (x; z, z) =(@ Jq µ )( q µ x µ i ) continuous p µ =1+i ( 2 R) parameters p 2 =0, p > 0 z 2 CS 2 2 discrete 4d helicity ` = ±1 2d spin J = ±1 parameters incoming vs. outgoing incoming vs. outgoing in the massless case the change of basis is furnished by Mellin transform of plane wave (or plus a shadow transform): V Δ ± μj (xμ ; z, z) = J q μ 2 0 = ( i) Δ Γ(Δ) 2 dω ω Δ 1 e ± iωq x ϵω J q μ ( q μ x μ iϵ) Δ specified by x and conformal dimension Δ = 1 + iλ, λ R no dependence on D=4 momentum p μ J q μ = zq μ = 2ϵ μ +(q) = (z,1, i, z) z q μ = 2ϵ μ (q) = (z,1, + i, z) Pasterski, Shao, Strominger, 2017

8 N-point amplitude on celestial sphere N A({p i, ξ j }) = i(2π) 4 δ (4) ( p 1 + p 2 p k ) M({p i, ξ j }) k= In the massless case, with or without spin, transition from momentum space to conformal primary wavefunctions with conformal dimension Δ j is implemented by Mellin transform: ϕ(δ) = dω ω Δ 1 ϕ(ω) Mellin transform, with: Δ j = 1 + iλ j 0 A {λn }(z n, z n ) = ( N 0 ω iλ n n dω n ) δ(4) (ω 1 q 1 + ω 2 q 2 N k= ω k q k ) M(ω n, z n, z n )

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Mostly-plus three-gluon amplitude M(,, + ) = = ω 1 ω 2 z12 ω z 1 z 2 A(,, + ) = 4 z 1 i(λ 1+λ 2 ) 21 Three-point Amplitudes z iλ 1 1 z iλ logarithmically divergent in the infra-red and in ultra-violet any cutoff would violate SL(2,C) symmetry conformal transformation properties, read off: δ( z 1 )δ( z 2 ) 0 ω i(λ 1+λ 2 +λ ) 1 dω {z } =2 ( ) h 1 = i 2 1, h1 =1+ i 2 1, h 2 = i 2 2, h2 =1+ i 2 2, h = 1+ i 2, h = i 2, 9 >= >; Δ n = 1 + iλ n J 1 = J 2 = 1, J = + 1 Pasterski, Shao, Strominger, 2017

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sha1_base64="e/xm2wb8jymntw2vinw027otwk=">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</latexit> (ii) Mostly-plus three-graviton amplitude M(,, + + ) = = ω2 1 ω2 2 ω 2 z 6 12 z 2 1 z2 2 A(,, + + ) = 4 z 2 i(λ 1+λ 2 ) 21 z iλ 1 1 z iλ δ( z 1 ) δ( z 2 ) 0 ω i(λ 1+λ 2 +λ ) dω {z } = UV divergent The degree of this divergence will grow with the number of external gravitons, reflecting the violation of unitarity bounds at each order of perturbative Einstein's gravity conformal transformation properties, read off: h 1 = h 2 = i 2 1, h1 = 2 + i 2 1, i 2 2, h2 = 2 + i 2 2, h = 2 + i 2, h = i 2 9 >= >; Δ n = 1 + iλ n J 1 = J 2 = 2, J = + 2

11 <latexit sha1_base64="e/xm2wb8jymntw2vinw027otwk=">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</latexit> <latexit sha1_base64="e/xm2wb8jymntw2vinw027otwk=">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</latexit> <latexit sha1_base64="e/xm2wb8jymntw2vinw027otwk=">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</latexit> <latexit sha1_base64="e/xm2wb8jymntw2vinw027otwk=">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</latexit> (iii) Mostly-plus EYM (one graviton, two gluon) amplitude M(,, + ) = = ω2 1 ω 2 ω z 4 12 z 2 2 A(,, + ) = 4 z 1 i(λ 1+λ 2 ) 21 z iλ z iλ 2 1 δ( z 1 ) δ( z 2 ) 0 ω i(λ 1+λ 2 +λ ) dω {z } = UV divergent three-point amplitudes in string-theory: same!

12 <latexit sha1_base64="a1ibgroquz7mvlvfik4dtpp7ilo=">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</latexit> <latexit sha1_base64="a1ibgroquz7mvlvfik4dtpp7ilo=">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</latexit> <latexit sha1_base64="a1ibgroquz7mvlvfik4dtpp7ilo=">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</latexit> <latexit sha1_base64="a1ibgroquz7mvlvfik4dtpp7ilo=">aaagyhicdvrbbxm5fj4uccusl8ibvfgbbbcrazuprfqbjkbaqzxkpfrcltqnpb7pjfvprbytkbxmh/frefstt/wszlw8bqpryclvvmd2+f4jl1mckuhg9bczdumrfxrztufvhvfsphi49ollprfj2sforymopkcz4wg4114idz5kr2bpsie+vfi/tjlupe0o9cxjo5iecq84jrqo8vlrh+yxkcegselmurfu5bsc5ujmln6jhjwxpjmyeoeepuvrpzx+8/6zpo+fxyng4tzxxhhh0mnfmrq9gjeske7lcm9ngyrma1iccmeempxtywdf9mk5kvw/jk404hnwwycoyqfnf8bsyopoppzb81rmeowtaaiyjcaopouf5ajtmmm8gg2lzrlbal6rkbgknzvjyetgqkqfmhdxhpmth9k7n+iwdsi5ymz7b62cw+hxdxg+7g7vbodcvwk1b16nhp/hszqxspqss0rtqzq6cqezhsfymlpbiiojyhmh5yrkjwatajuzmflcc/qxmd4kugkz0ahkr0yyeis1iz1qxkrh6rqvih/no5noyhnkejjnnetotveweuinqoge5hpjqbyzairkdmdfncksua09nreliizbcc0r48v5b7qlbvgebgsw6lwyjtcww8jacmzfw5judkandmcb2nxljwimvgvmwuymk6npry84vooks0hvszisxt7udmhdemzxvealbenfsubsbypkdiwfwzfqipymiuuwxfj0wt5vqsqfxvx7kpovdkcqudhvdletuedtzrtw/9eyh/8/e5ibq0qwkb/zxwvovahptxyq0juzv6bbmmygs/xksjjdkzfh9wgiz5xtl7lb1n99nzqogtk/hq2r8agtijlv8rhsfm2rgtk62l6zvgmpqjfth781lma2vcrpb/msesbb6ysneloeoto0iq2lbntmv1umlvgtsrlhboslc+vwmxvff+2euwaytjziyxlulwa7vu4ldwtl7mdtbcvyumf1+7hophx+dfyc1npvhg2nu/oounbx1vfwv+1vt/60v5sp2gvvdkfvhz2jkb7sc/as7smje=</latexit> Four-point Gauge Amplitudes r = z 12 z 4 z 2 z 41 conformal invariant cross-ratio on CS 2 actually: s 2 s 12 = 1 r = u s = sin2 ( θ 2 ) s = s 12 = (p 1 + p 2 ) 2 u = s 2 = (p 2 p ) 2 θ = scattering angle in center of mass frame Ã(,, +, +) = 8 (r r) 4Y i<j z h h i h j ij 4X z h ij n h i hj! r 5 (r 1) 2 (r 1) Pasterski, Shao, Strominger, 2017

13 <latexit sha1_base64="flwgtmksyweaiibnm2xpq2yscc4=">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</latexit> <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit sha1_base64="fqkfp2ujf54pmupkh47tqkjiwjk=">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</latexit> type I superstring theory: Ã I (,, +, +) = 4 ( 0 ) (r r) (r 1) r 5 (r 1) 2 I(r, ) 4Y i<j z h h i h j ij z h ij h i hj! 4 β := i 2 λ n I(r, β) = Γ(1 β) r dx x [r ln x ln(1 x)] β 1 4X I(r, ) = 2 n + i 2 ( r) 1 sinh 1 2 4X n 1 1 X k=0 ( r) k i 2 4X n k, {1} k

14 Remarks: <latexit sha1_base64="5rpqglkbhvyjjjj6ovthuyefdw=">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</latexit> <latexit sha1_base64="5rpqglkbhvyjjjj6ovthuyefdw=">aaafuhicdzrlb9qwemfdsowypnrckyvfcsfplbslxw4tvbrs2/telk5jrnr1u4i27vs1so4dnwhrmvgkcb5oh2ojozn5ztjj+dtykhkmtof9mzl90ph7+mz+efpfy1evfxaxpyrzcwjpsmjt+rlgbxllkznmmlol1njsqg4vqiutvl4xyrkxzl4ve9t2hd4gloieawtgiyuiu4juybhblyycnxndoc8kyjiei9n4h+e9e4rejisufio2yw2planpnvbmmrff5dwwty4of/fw4v+2yuefqiew8fsyxafcrklgmvcsvi90t1/5wm8kpltdwnmxkcg9pz5oxflt1tdfgbj9yesiokxbeghb09gydhvjtedhmgfviy/l8kfyb6yjtt+wob1rgpoyudtmuccwvwugtfki+dqamehm1wrjcetmtnx0thu1gkfrvuzmiljme4u0qjwyhdjz6upmsikrx1lfsahalr8rgyjbpniitgiya4s4qeyqcqqztjqdltom6ok4mxg8r8ofrisztxcqilmtakdoahfsxybqoisqsmtucwxxcnbk26jpqt2xxfjlmdat+xsdr9jm7zczkay7jiaew5snzwfv01/w86/iphdl4vbocwoj5osifs4fek9gjlfvkltx6zoy46yomvl77yjyjzydrv1nemfulpx+uekfuto7wyxycgn7pb9fj7if7lim4zljufztojcuziudaqjszk5zmyrpyhtro/nsr26a9u5mlmsrum45xze07ci2i7vm1lva6pmemnvahsxm5xa+dfnxhv6ejliivhashzhfgu5qgi8b58tt2v7r6utja/v/tepoh4bpwwtryanvgejwbar6dh+an+nx4fjb+dcu6bovl/wftx55pr/auim1f4=</latexit> <latexit sha1_base64="5rpqglkbhvyjjjj6ovthuyefdw=">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</latexit> <latexit sha1_base64="5rpqglkbhvyjjjj6ovthuyefdw=">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</latexit> no α - expansion (trivial dependence on α )! instead expansion in small scattering angle r 1 = sin 2 θ ( 2 ) all heavy string modes participate on same footing field-theory is recovered in the limit of forward scattering θ = 0 Question: celestial CFT_2 string world-sheet CFT_2 9 >= >; any relation?

15 String world-sheet as celestial sphere M I (,, +, + ) = M(,, +, + ) F I (s, u) with string formfactor: F I (s, u) = α s 12 B( α s 12,1 + α s 2 ) = s B( s,1 u) = consider high-energy limit: B( s,1 u) = 1 saddle-point approximation: x 0 = 1 world-sheet vertex position = point on celestial sphere = solutions to scattering equations 0 Γ(1 s)γ(1 u) Γ(1 s u) dx x 1 s (1 x) as 1 a CS2 a = r 1 < 0 celestial sphere = world-sheet CFT on celestial sphere related to free world-sheet CFT

16 <latexit 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Four-point Closed String Amplitudes heterotic gauge amplitude: Ã H (,, +, +) = 4( 0 ) (r r) (r 1) r 5 (r 1) 2 H(r, ) 4Y i<j z h h i h j ij z h ij h i hj! H(r, β) = Γ(1 β) r 2π C d 2 z z 2 (1 z) [r ln z 2 ln 1 z 2 ] β 1 H(r, 4X )=2 n + i 2 ( r) 1 sinh 1 2 4X n 1 1 X k=0 ( r) k S c i 2 4X n k 1,k+1

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Four-point Gravity Amplitudes heterotic graviton amplitude: Ã H (,, ++, ++) = 4 ( 0 ) 1 (r r) (r 1) 4Y i<j z h h i h j ij z h ij h i hj! r 11 (r 1) 1 G(r, ) G(r, β) = H(r, β 1) H(r, β) = Γ(1 β) r 2π C d 2 z z 2 (1 z) [r ln z 2 ln 1 z 2 ] β 1 H(r, 4X )=2 n + i 2 ( r) 1 sinh 1 2 4X n 1 1 X k=0 ( r) k S c i 2 4X n k 1,k+1

18 Properties: Finite result for any r! ultra-soft high energy behaviour of string formfactors ensures the convergence of energy integrals UV completion provided by string theory finite result for integration from IR to UV first calculation of graviton amplitudes in the conformal basis Alert: Divergent for r (field-theory limit) every order in the perturbative expansion of gravity violates the unitarity bounds by growing powers of energy. This uncontrollable growth at large energies poses an obstacle for transforming gravitational amplitudes to celestial sphere

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sha1_base64="dvhgje+lt6j2e+lesnkbeozox0s=">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</latexit> Single-valued Nielsen polylogarithms Nielsen's polylogarithm functions (real): S n,p (t) = ( 1)n+p 1 (n 1)!p! Z 1 0 dx x lnn 1 x ln p (1 xt), t 2 C in particular: S n,p (1) = (n +1, {1} p 1 ) Single-valued descendants: S c (n, p) = 1 ( 1)n+p 1 (n 1)!p! ζ(n + 1,{1} p 1 ) = ζ(n + 1, 1,,1) = Z C p 1 1 n n 1 >n 2 > >n 1 n+1 n 2 n p p d 2 z z 2 (1 z) 1 ln n 1 z 2 ln p 1 z 2 S c (n, p) = sv S n,p (1) = sv ζ (n + 1,{1} p 1 )

20 Concluding remarks explicit and compact expressions for string amplitudes on celestial sphere: intriguing examples for the study of flat space holography string amplitudes on celestial sphere: no α - expansion (trivial dependence on α ) all heavy string modes participate on same footing high-energy limit: string world-sheet = celestial sphere first calculation of graviton amplitudes in the conformal basis (with gravity UV completed): - ultra-soft high energy behaviour of string formfactors ensures the convergence of energy integrals - important for the soft graviton theorem in this basis

21 Can 2D CFT on celestial sphere offer some new insight into gauge-gravity connections? on celestial sphere celestial gravitational amplitudes appear from space-time translations of pure gauge amplitudes, Ã EYM (1,2,, N, G ±± ) = κ g N 1 l=1 (ϵ ± G X l ) Ã YM (1,2,, l, G ±, l + 1,, N) indicating that celestial CFT will be helpful in studying connections between gauge theories and gravity! understanding the nature of 2D CFT on celestial sphere would enable a holographic description of flat spacetime

22 ds 2 = dt 2 + dx 2 with : x 2 = r 2 ds 2 = du 2 2 du dr + 2 r 2 γ zz dzdz, γ zz = 2(1 + z 2 ) 2 dr 2 dt 2 u = t r, x 1 + ix 2 = S 2 2rz 1 + z 2, x = r (1 z 2 ) 1 + z 2