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Transcript of 1 Research supported by the National Science Foundation grant PHY-0757959. Opinions expressed are...
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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 !
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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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