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Research supported by the National Science Foundation grant PHY-0757959.Opinions expressed are those of the authors and do not necessarily reflect the views of NSF.

Esse est percipi

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Superstring Amplitudes as a Mellin Transform of Supergravity

Tomasz Taylor Northeastern University, Boston

Based on work with Stephan Stieberger (MPI, Munich)

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Superstring Amplitudes as a Mellin Transform of Supergravity

1¡

2¡ 3+

4+

5+

N + D = 4

2¡ ¡

1¡ ¡ N ++

5++

4++

3++

N gauge bosonsMHV (¡ ¡ + +:::+)\ partial amplitude"disk levelall orders in ®0= 1

M 2S

N gravitonsMHV \ mostly plus"tree level (Einstein)

\ ASN (®0s) = R1

0 x®0s¡ 1AGN (x) dx "

Do not confuse with KLT !

4

1¡

2¡3+

4+

5+

N +

V1(¡ 1 )

V2(0)

V3(1)V4(z4)

V5(z5)

VN (zN )

= :::DV1(¡ 1 )V2(0)V3(1) Rz1

z3dz4V4(z4) ¢¢¢Rz1

zN ¡ 1dzN VN (zN )

E

= \ RQ Ni=4 dzi "

Superstring

Example: N=4 (Veneziano-Virasoro) four-gluon amplitude

1¡

2¡ 3+

4+

s !

#us23 = u = 2p2p3; s34 = s = 2p3p4

In general, si j = 2pi pj ; zi j = zi ¡ zj

AS4 = 1

h12ih23ih31iZ 1

1dz4 jz24js24 jz34js34 1

z34| {z }h2j3j4]h24i

B(s23;s34) = 1s23

+ 1s34

+ :::si j ¿ M 2

S¡ ! 1h12ih23ih34ih41i

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1¡

2¡3+

4+

5+

N+

Superstring

All N MHV formula (Stieberger, Taylor, 2012, and a general formula of Mafra,

Schlotterer, Stieberger, 2011)

AS4 = 1

h12ih23ih31iZ 1

1dz4 jz24js24 jz34js34 1

z34

h2j3j4]h24i

Zd¹ (z;s) =

Z 1

1dz4

Z 1

z4dz5 : : :

Z 1

zN ¡ 1dzN

Y

2<i<j <Njzi j jsi j

| {z }Koba-Nielsen factor

ASN = 1

h12ih23ih31iZ

d¹ (z;s)X

P erm(4;:::;N )

NY

k=4

1z(k¡ 1)k

h2j3+ ::: + (k¡ 1)jk]h2ki

(N ¡ 3)! termsEach term contains integral over N ¡ 3 vertex positions) generalized hypergeometric functions of kinematic variablesTranscendental ´ can be represented by certain tree graphs) Special properties of low-energy (®0! 0 limit) expansions

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SuperstringAS

N = 1h12ih23ih31i

Zd¹ (z;s)

X

P erm(4;:::;N )

NY

k=4

1z(k¡ 1)k

h2j3+ ::: + (k¡ 1)jk]h2ki

Supergravity (Mason, Skinner, 2010; Berends, Giele, Kuijf, 1988,Bern, Dixon, Perelstein, Rozowsky, 1999,Nguyen, Spradlin, Volovich, Wen, 2010,…)

2¡ ¡

1¡ ¡ N ++

5++

4++

3++

AGN = 1

h12i2h23i2h31iX

P erm(4;:::;N )

1h1N i

NY

k=4

1h(k ¡ 1)ki

h2j3+ :::+ (k¡ 1)jk]h2ki

Very “Similar”

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Graphs for superstrings (and supergravity)AS

N = 1h12ih23ih31i

Zd¹ (z;s)

X

P erm(4;:::;N )

NY

k=4

1z(k¡ 1)k

h2j3+ ::: + (k¡ 1)jk]h2ki

De ne z0i ´ 1

h2xihixihi2i =) z0

i j = z0i ¡ z0

j = hij ih2iih2j i (Schouten)

ASN = ¢¢¢

Zd¹ (z;s)

Xtrees

Y

edges

si jzi j z0

i j 7

8

N9

5

4 3

3 6 N 5 7

(N ¡ 2)(N ¡ 4) trees (Cayley), after partial fractioning=) (N ¡ 3)! chains (Hamilton paths) rooted at ² 3

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Unified description: Hodges determinant

Laplacian Matrix (Feng, He, 2012; based on Hodges, 2012)

Ãi j =

8>>><>>>:

si jzi j z0

i jif i 6= j

¡X

j 6=i

si jzi j z0

i jif i = j

De ne M N (z;z0;s)r sti j k ´ 1

zi j zj k zk i1

z0r s z0st z0

t rjª jr st

i j k

N £ N

AS;GN = Rd¹ S;G

NRd¹ G

N (z0;¸) M N (z;z0;s)r sti j k

Zd¹ G

N (z;¸) =Z NY

i=1dzi ±

µhxii2zi ¡ hxiihyii

hxyi¶

Zd¹ G

N (z0;¸) =Z NY

i=1dz0

i ±µ

hyii2z0i ¡ hxiihyii

hyxi¶

Zd¹ S

N (z;s)i j k = zi j zj kzki

Z

D

µ Y

l· N

0dzl

¶ Y

m<n· N

0 jzmn jsm n

Mellin Transform

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From Koba-Nielsen to

Zd¹ S

N (z;s) =Z 1

1dz4

Z 1

z4dz5 : : :

Z 1

zN ¡ 1dzN

Y

2<i<j <Njzi j jsi j

M f (s) = R10 us¡ 1f (u)du f (u) = 1

2¼iR+i1 +c

¡ i1 +c u¡ sM f (s)dsN (N ¡ 3)

2 kinematic invariantsmore precisely, 3N ¡ 10 in D=4but we'll take them alle.g. s2;3 and s3;4 for N = 4

si ;j = (pi + pi+1 + ::: + pj )2 Ã! ui ;j = (zi ¡ zj )(zi ¡ 1 ¡ zj +1)(zi ¡ zj +1)(zi ¡ 1 ¡ zj )

N (N ¡ 3)2 MÄobius invariants

but only N vertex positions) algebraic constraintse.g. 1¡ u2;3 ¡ u3;4 = 0 for N = 4

0 · ui ;j · 1½ i = 2; j = 3;:: : ;N ¡ 1

i = 3;: :: ;N ¡ 1 < j = 4;:: : ;NZ

d¹ SN (z;s) =

Z Y

i ;jdui ;j usi ; j ¡ 1

i ;j £ J acobian£ ±(fui ;j g)constraints

Pascal’s Triangles of Constraints

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i = 2; j = 3;: : : ;N ¡ 1 ) ¾kl(u) =l¡ 1Y

n=ku2;n

i = 3;: : : ;N ¡ 1 < j = 4;:: : ;N ) ½kl(u) = uk; lY

ancestors(uk; l)

(3;N¡ 1)(3;4;:::;N ¡ 1)

(4;N)(4;5:::;N )

(3;N)(3;4;:::;N )

(5;N)(5;6;:::;N )

(4;N¡ 1)(4;5:::;N ¡ 1)

(3;N¡ 2)(3;4;:::;N ¡ 2)

(N¡ 2;N)(N ¡ 2;N ¡ 1;N )

(3;5)(3;4;5)

(4;6)(4;5;6)

(5;7)(5;6;7)

(5;6)(4;5)(3;4) (N ¡ 1;N)

½kl + ¾kl ¡ 1 = 0 polynomial constraints

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…

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M S (s) =Z Y

i ;jdui ;j usi ; j ¡ 1

i ;j £ J acobian£ ±(fui ;j g) £ M G (u(z);z0)

In retrospective, this is not totally unexpected:

Inverse (multiple) Mellin transforms of hypergeometric string “formfactors” are simple delta functions localizing on the world-sheet. Very similar to SYM amplitudes in twistor string or Grassmanian formulations. Mellin “trivialize” string amplitudes

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1(2¼i)2

+i1 +cZ

¡ i1 +c

ds2;3

+i1 +cZ

¡ i1 +c

ds3;4 u¡ s2;32;3 u¡ s3;4

3;4 B(s2;3;s3;4)

= ±(1¡ u2;3 ¡ u3;4) µ(1¡ u2;3) µ(1¡ u3;4)

To be done

Proof of Superstring/Supergravity correspondence for all tree-level 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, 1987)

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