Superstring Amplitudes as a Mellin Transform of Supergravity. Tomasz Taylor Northeastern University, Boston

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1 Esse est percipi Research supported by the National Science Foundation grant PHY Opinions expressed are those of the authors and do not necessarily reflect the views of NSF.

2 Superstring Amplitudes as a Mellin Transform of Supergravity Tomasz Taylor Northeastern University, Boston Based on work with Stephan Stieberger (MPI, Munich) 2

3 Superstring Amplitudes as a Mellin Transform of Supergravity N + D N gauge bosons MHV ( + + : : : +) \partial amplitude" disk level all orders in 0 = M 2 S = 4 N ++ N gravitons MHV \mostly plus" tree level (Einstein) \ A S R N ( 0 s) = s 0 x 0 A G (x) dx " N Do not confuse with KLT! 3

4 2 3+ Superstring = \ R Q N i=4 dz i " V 2 (0) V ( ) D N + = : : : V ( )V 2 (0)V 3 () R z z 3 dz 4 V 4 (z 4 ) R E z z N dz N V N (z N ) Example: N=4 (Veneziano-Virasoro) four-gluon amplitude # u V 3 () V 4 (z 4 ) V 5 (z 5 ) V N (z N ) s! 4 + A S 4 = h2ih23ih3i s ij MS! 2 s 23 = u = 2p 2 p 3 ; s 34 = s = 2p 3 p 4 In general, s ij = 2p i p j ; dz 4 jz 24 j s 24 jz 34 j s 34 z {z 34 } z ij = z i z j h2j3j4] h24i B(s 23 ; s 34 ) = s 23 + s 34 + : : : h2ih23ih34ih4i 4

5 Superstring A S 4 = dz 4 jz 24 j s 24 jz 34 j s 34 h2ih23ih3i z 34 h2j3j4] h24i All N MHV formula (Stieberger, Taylor, 202, 2 3+ A S N = N h2ih23ih3i and a general formula of Mafra, Schlotterer, Stieberger, 20) d¹(z; s) = d¹(z; s) X dz 4 z 4 dz 5 : : : NY z Perm(4;:::;N) k=4 (k )k z N dz N Y 2<i<j<N jz ij j s ij {z } Koba-Nielsen factor h2j3 + : : : + (k )jk] h2ki (N 3)! terms Each term contains integral over N 3 vertex positions ) generalized hypergeometric functions of kinematic variables Transcendental can be represented by certain tree graphs ) Special properties of low-energy ( 0! 0 limit) expansions 5

6 Superstring A S N = h2ih23ih3i d¹(z; s) X NY Perm(4;:::;N) k=4 z (k )k h2j3 + : : : + (k )jk] h2ki Supergravity (Mason, Skinner, 200; Berends, Giele, Kuijf, 988, Bern, Dixon, Perelstein, Rozowsky, 999, Nguyen, Spradlin, Volovich, Wen, 200, ) N ++ A G N = h2i 2 h23i 2 h3i X Perm(4;:::;N) hni NY k=4 h(k )ki h2j3 + : : : + (k )jk] h2ki Very Similar 6

7 Graphs for superstrings (and supergravity) A S N = h2ih23ih3i d¹(z; s) X NY z Perm(4;:::;N) k=4 (k )k h2j3 + : : : + (k )jk] h2ki De ne z 0 i h2xi hixi hi2i =) z0 ij = z0 i z0 j = hiji h2iih2ji (Schouten) A S N = d¹(z; s) X trees Y edges s ij z ij zij N (N 2) (N 4) trees (Cayley), after partial fractioning =) (N 3)! chains (Hamilton paths) rooted at ² N 5 7 7

8 Unified description: Hodges determinant 8 s ij >< z ij z 0 if i 6= j ij Laplacian N N Matrix à ij = (Feng, He, 202; X s ij >: z ij z 0 if i = j j6=i ij based on Hodges, 202) De ne M N (z; z 0 ; s) rst ijk z ij z jk z ki z 0 rs z0 st z0 tr jªj rst ijk A S;G N = R d¹ S;G N R d¹ G N (z 0 ; ) M N (z; z 0 ; s) rst ijk d¹ S N(z; s) ijk = z ij z jk z ki D d¹ G N (z; ) = N Y d¹ G N(z 0 ; ) = µ Y 0 Y dzl l N m<n N i= Y N i= 0 jzmn j s mn µ dz i ± hxii 2 z i hxiihyii hxyi µ dzi 0 ± hyii 2 zi 0 hxiihyii hyxi 8

9 From Koba-Nielsen d¹ S N(z; s) = dz 4 dz 5 : : : dz N z 4 z N Y jz ij j s ij 2<i<j<N to Mellin Transform M f (s) = R 0 us f(u)du f(u) = 2¼i R +i+c i+c u s M f (s)ds N(N 3) kinematic invariants 2 more precisely, 3N 0 in D=4 but we'll take them all e.g. s 2;3 and s 3;4 for N = 4 N(N 3) MÄobius invariants 2 but only N vertex positions ) algebraic constraints e.g. u 2;3 u 3;4 = 0 for N = 4 s i;j = (p i + p i+ + : : : + p j ) 2 Ã! u i;j = (z i z j )(z i z j+ ) (z i z j+ )(z i z j ) 0 u i;j d¹ S N(z; s) = ½ i = 2; j = 3; : : : ; N i = 3; : : : ; N < j = 4; : : : ; N Y i;j du i;j u s i;j i;j Jacobian ±(fu i;j g) constraints 9

10 Pascal s Triangles of Constraints i = 2; j = 3; : : : ; N ) ¾ kl (u) = l Y u 2;n n=k i = 3; : : : ; N < j = 4; : : : ; N ) ½ kl (u) = u k; l Y ancestors(uk; l ) (3; N) (3;4;:::;N) (3; N ) (3;4;:::;N ) (4; N) (4;5:::;N) (3; N 2) (3;4;:::;N 2) (4; N ) (4;5:::;N ) (5; N) (5;6;:::;N) (3; 5) (3;4;5) (4; 6) (4;5;6) (5; 7) (5;6;7) (N 2; N) (N 2;N ;N) (3; 4) (4; 5) (5; 6) (N ; N) ½ kl + ¾ kl = 0 polynomial constraints 0

11 Summary: Superstring /Supergravity Correspondence Superstring Amplitudes are Mellin Transforms of Supergravity, directly from the worlsheet into the dual space of kinematic invariants, thus bypassing space-time M S (s) = Y i;j du i;j u s i;j i;j Jacobian ±(fu i;j g) M G (u(z); z 0 )

12 In retrospective, this is not totally unexpected: (2¼i) 2 +i+c i+c ds 2;3 +i+c i+c ds 3;4 u s 2;3 2;3 u s 3;4 3;4 B(s 2;3 ; s 3;4 ) = ±( u 2;3 u 3;4 ) µ( u 2;3 ) µ( u 3;4 ) Inverse (multiple) Mellin transforms of hypergeometric string formfactors are simple delta functions localizing on the worldsheet. Very similar to SYM amplitudes in twistor string or Grassmanian formulations. Mellin trivialize string amplitudes 2

13 To be done Proof of Superstring/Supergravity correspondence for all treelevel amplitudes, beyond MHV (in progress) Mellin transforms are not completely trivial because the integrations are over a constrained surface (Pascal s triangle). Understanding the nature of this embedding will allow a deeper understanding of the correspondence and to establish a supergravity description of stringy features: Regge resonance poles etc. For the moment, we can say that Superstring Theory is Supergravity in a Brilliant Disguise (Bruce Springsteen, 987) 3

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