the quantum structure of the type ii effective action and uv behavior of maximal supergravity jorge...
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The quantum structure of the type II effective action
and UV behavior of maximal supergravity
Jorge Russo
U. Barcelona – ICREA
[Based on work in coll. with M.B. Green and P. Vanhove,
hep-th/0610299, hep-th/0611273, and in progress]
Strings 2007, Madrid – June 2007
•Explicit string theory calculations like graviton scattering amplitudes have never carried out beyond genus 2.
•Such calculations involve extremely complicated integrals over vertex operator positions and integration over the supermoduli space of Riemann surfaces.
•How to get information about string perturbation theory beyond genus 2 ?
•Strategy: USE THE CONNECTION WITH D=11 SUPERGRAVITY ON S1 AND T2.
•L-loop scattering amplitudes depend on the radius R11 of S1.One has A4
(L) (S,T,R11). BUT R11 = gs
Expanding this at small S,T, one gets a Laurent expansion in powers of (R112 S),
(R112 T) or (gs
2 s), (gs2 t) .
•In this way the expansion generates higher genus string-theory coefficients to
arbitrary order !! One gets explicit results at genus 3, 4, 5, 6, 7,…etc. … no chance to compute them using string perturbation theory with the present knowledge.
WHAT IS STRING THEORY IN THE DEEP QUANTUM REGIME?
The tree-level four-graviton effective action:
4233341243222412
4123272412
419
41032848
217
46232644543210
)(~
,)(
)]~
))9()3(2()9(['
)5()3(')7('
)3(')5(')3(2'(
RutsRDRutsRD
RDRD
RDRD
RDRDRRegxdS
In type IIB, each D2kR4 term in the exact effective action will contain, in addition, perturbative and non-perturbative corrections. The zeta functions will be replaced by a modular function of the SL(2,Z) duality group.
THE MAGIC OF L LOOP 11D SUPERGRAVITY
L-LOOP = +…
EXAMPLE: ONE LOOP IN 11D SUPERGRAVITY ON S1
=...
6
)6(
2
)4())3(
1(
)1(
)2())3((
464442432
42
2
)1(2432
RDgRDgRg
RDkk
keRe
AAA
k
k
k
This L=1 amplitude determines the genus k coefficient of all D2kR4 terms in type IIA superstring theory.
•[Green,Gutperle,Vanhove]
•[Russo,Tseytlin]
•[D’Hoker,Gutperle,Phong]
•[Green,Vanhove]
ONE LOOP IN 11D SUPERGRAVITY ON T2
=
42
2
2
42/222
2
2
4/224329
422/1
2
2/142/3
21439
)1(..
))()22()12((
))()2(2)3(((
(
RDkgenusgenusrei
RDeOgkkrc
ReOgRrgrdx
RDZcRZRgdx
k
k
kB
kgkB
k
kBk
gBBIIBB
kk
k
kk
/-
B
B
-
V VV-
[Green, Russo, Vanhove, to appear]
Note that it is finite in the ten dimensional limit
12
212
2
211,
)(122
)0,0(),(2
2
)exp(
)exp(
,)/1(exp()(0),(
)2()2cos()2(2
),(
s
srr
kwn
knw
rk
r
nmr
r
r
g
i
gOrgenusgenusZ
wnKwncr
nmZ
The modular functions Zs are the so-called non-holomorphic Eisenstein series
)(,0))1(( 2222 21 rZrr
The Zr are unique modular function satisfying this differential equation (having at most polynomial growth at infinity)
TWO LOOPS IN 11D SUPERGRAVITY ON S1
=
...)~
1411(log45
)3(2
)~
()6()3(
1527
)4()3(log)2()3(
15
16
3
)3(2)5(
4124124
4122
4121
4
410248
462
244
2
RDRDgg
RDnRDng
RDgRDg
RDg
RDg
ss
s
ss
ss
•[Green,Kwon,Vanhove]
•[Green,Vanhove]
•[Green,Russo,Vanhove, to appear]
TWO LOOPS IN 11D SUPERGRAVITY ON T2
=
...)
)~
),(),((
),(),(
),(),((
4122
4121
6
4104482
46)2/3,2/3(
442/5
RDGRDGr
RDFrRDEr
RDERDZr
B
BB
B
•[Green,Kwon,Vanhove,2000]
•[Green,Vanhove, 2005]
•[Green, Russo,Vanhove, to appear]
))/1(exp(
))/1(exp(
42
23),(
2/12/32
),(
46),(
...1
)3(1
)2(4)3(2,6)12(
:
23
23
23
23
23
23
23
23
sgO
sgO
sss
s
ss
ggg
Eg
ggZZE
RDE
The modular function is uniquely determined by the differential equation with the simple boundary condition that alpha = 0.
Thus one finds [Green,Vanhove, 2005]
D6R4 : genus 0 + genus 1 + genus 2 + genus 3 + non-pert.
THE COUPLING D6R4
E, F, G(i), G(ii) (multiplying D8R4, D10R4, D12R4, D’12R4) are new modular functions exhibiting a novel structure: they are sums of modular functions satisfying Poisson equations with the same source term but different eigenvalues [GRV, to appear]
21
23
21
23
21
23
3
2
1
3210
)42(
)20(
)6(
ZZE
ZZE
ZZE
EEEEE
90,56,30,12,2
)(21
21
23
23
43210
j
jjj ZZwZZF
FFFFFF
156,110,72,42,20,6,0)12(2
)(21
21
25
23
6543210,
jj
ZZcZZfG
GGGGGGGG
j
xj
xj
xjj
yx
Conjecture: all modular functions in the type IIB effective action are determined by Poisson equations.
)similar(
)16()14(
)12()10()8()6(3
280)log)4(1760(
)4()3(5005
357856)2()3(
77
28096)5()2(
77
8828)5()3(96
)12()10(1875
304)8(
405
896
)6(9
98)4(
3
3248)log
3
1456
135
91721()2()3()2(
15
7168)3(180
)8(496125
512)6(
945
4)4(
9
4)2(
2205
1114log)3(
5
16)3(
75
118
4
1614
1210861
4
4224
12108
642223
86422
nonpert
nonpert
nonpert
xs
ss
ssssss
sssx
s
sss
ssssss
ssssss
Gg
gcgc
gcgcgcggcg
gggGg
ggg
gggggFg
gggggEg
Strikingly, the term proportional to 1760 zeta(4) exactly matcheswith a genus 3 string theory result determined by the genus one amplitude and unitarity:
the term Z5/2 s6 log s
1) The L = 1 11d supergravity amplitude on a 2-torus contains genus one terms
They exactly match with the corresponding terms in the 9d genus one amplitude computed in string theory.
In a large r expansion for each given k, they represent the leading term.
2) The L = 2 11d supergravity amplitude on a 2-torus gives the subleading term
where bk is a product of zeta functions. We have checked that they exactly match with the corresponding terms in the 9d genus one amplitude up to k=6.
Log rA is obtained from Log gB term after using gB=gA/rA
4212
2
)12()1( RDrkcsugraL kkIIA
kk
4272
2
)2( RDrbsugraL kkIIA
kk
Genus 1 from D=11 supergravity on T2
String theory 42127212
2
))(...log...( RDeOrcrrdrbra krkIIAkIIAIIAk
kIIAk
kIIA
kk
IIA
)),(),(),(( 46),(
2/12
442/5
2/12
42/3
2/12
10
23
23 RDERDZRZgxdS
So far the terms that have been determined are:
D=10 TYPE IIB EFFECTIVE ACTION
R4 : genus 0 + genus 1 + non-pert.
D4R4 : genus 0 + genus 2 + non-pert.
D6R4 : genus 0 + genus 1 + genus 2 + genus 3 + non-pert.
It suggests
D2kR4 : genus 0 + genus 1 + ...+ genus k + non-pert.
4)(4 )'(1 RsOsA hh
In other words, that the genus h amplitude has the form
What would be the expected contributions from M-theory?
String theory analogy: integrating out string excitations gives rise to an effective action of the general form:
'2
1,4)3(210
3
TRDgxdTcS nn
nn
11328
)(][, 32
4211
10112
0,
nmk
lTRDgRxdRTcS Pkmn
knnk
For M5-branes, [T5] = L-6, leading to corrections h = k – 2n.
Similarly, in d=11 on S1 one would expect that membrane excitations should give contributions of the form
32311
211211
2111
2 ,)( PAIIA lgRdxCdxRdsRds
!!
)( 4210)1(2
0 0
knkh
RDgxdgcS IIAknk
sk
nk
k
n
Convert to type IIA variables:
L
rmnL
L
Ln
rRDgxdRS
RsOsconstA
L
L
2690
...,2,1,0,
))(1(
4221111
)1(211
44
0)1(3
0,3
21)1(3
3
)( 4210)2
33(2
nifkLkhgenus
nifkLn
khgenus
RDgxdgS
L
Ak
nLk
An
Using
khgeneralinso
nforkhL
03
12
TYPE IIA EFFECTIVE ACTION
An infinite sequence of non-renormalization conditions
Convert to type IIA variables:
Thus we find a maximum genus for every value of k. This is irrespective of the value of L. The highest genus contribution is h = k and comes from L = 1.
[Berkovits, 2006]:
Direct string theory proof based on zero mode counting that D2kR4 are not renormalized beyond genus k, for k=2,3,4,5.
4268)1(2)10(
443)1(2)10(
4)1(214
))'(1(
))'(1('
))'(1('
RsOs
RsOs
RsOseA
hh
hh
hh
hh
hh
hh
Genus h four-graviton amplitude in string theory:
•Explicit calculations show beta1 = 0 [Green et al], beta2 = 2 [Bern et al] and, more recently, beta3 = 3 [0702112]
•Berkovits (2006): also beta4 = 4 and beta5=5
•Our arguments based on the use of M-theory/IIA duality indicate betah = h for all h.
The presence of Sbeta R4 means that the leading divergences Lambda8h+2 of individual h-loop Feynman diagrams cancel and the UV divergence is reduced by a factor Lambda – 8 – 2beta
Is N=8 Supergravity a finite theory?
h-loop supergravity amplitudes are obtained by taking the limit alpha’ to zero.The inverse string length is interpreted as a UV cutoff.
Since the discovery of the theory, it was clear from various superspace arguments that the high degree of symmetry of the theory would significantly improve the UV behavior
•Kallosh, 1981•Howe, Lindstrom, 1981•Howe, Stelle and Townsend, 1984•Howe, Stelle, 1989
In the last few years, a number of results have appeared which suggest that N=8 might even be finite.
•Bern, Dixon, Dunbar, Perelstein, Rozowsky, 1998.
•Howe, Stelle, 2002
•Bern, Bjerrum-Bohr and Dunbar [0501137]
•Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager, 0610043.
•Bern, Dixon and Roiban [0611086]
•Bern, Carrasco, Dixon,, Johansson, Kosower and Roiban, 0702112.
Maximal supergravity in lower dimensions:
Compactify string loop amplitude on a 10-d torus. Consider low energy limit alpha’ -> 0 with the radii of the torus proportional to sqrt(alpha’), so that all massive KK states, winding numbers and excited string states decouple.
(Subtle limit for non-perturbative states [see Green, Ooguri, Schwarz])
In terms of the d-dimensional gravitational constant kappad2 ,
','
,))'(1(
1022/)2(2
426)2()1(2)(,4
rradiire
RsOsA
ddd
hdhd
hd
hh
Therefore UV divergences are absent in dimensions satisfying the bound
hd h 62
2
Now using IIA/M theory duality[GRV]: betah = h for all h > 1
Hence there would be no UVdivergences if1,
64 h
hd
d = 4: UV FINITE FOR ALL h !!
Remarkably, this is the same condition as maximally supersymmetric Yang-Mils in d dimensions [Bern, Dixon, Dunbar, Perelstein, Rozowsky, 1998], [Howe,Stelle, 2002]
One can be more conservative and strictly use the non-renormalization theorems that are obvious from superstring perturbation theory in the pure spinor formalism [Berkovits, 2006].
One finds
betah = h for h =2, 3, 4, 5
betah > or = 6 for h > 5.
Hence there would be no UVdivergences if
5,...,2,/64
5,/182
hhd
hhd
d = 4: UV finite at least up to h = 8 !!
M-graviton amplitudes
Our non-renormalization theorems equally apply to terms D2k RM
They state that these terms do not get any genus corrections beyond h = k + M – 4
In other words, the M-graviton amplitude at genus h should have the form
))'(1(4 sORsA MMhhM
A simple generalization of the previous analysis shows that there is no UV divergence in the M-graviton supergravity amplitude for
1,/64 hhd
According to this, the d=4 N=8 supergravity theory would be completely finite in the UV.
2/4,2/)14(2
2/41,/64
MhMd
Mhhd
In d = 4, this implies finiteness if h < 7 + M/2 (hence M > 4 amplitudes are even smoother in the UV)
The Berkovits non-renormalization theorems can also be generalized to the M-graviton amplitude. This leads to the conditions
KNOWN CASES:
h = 1: UV finite for d < 8 (beta1=0)
h = 2: UV finite for d < 7 (beta2=2)
h = 3: UV finite for d < 6 (beta3=3)
h > 3: expected betah > or = 3. This gives d < 2 + 12/h. Thus d = 4 finite up to h < 6.
•d = 4: UV finite for all h !!
•d = 5 : UV finite for h = 1,2,3,4,5
•d = 6 : UV finite for h = 1,2
•d = 7: UV finite for h = 1
•d = 8,9,10: UV divergent at h = 1 already
•d = 4: UV finite for h < 9 (i.e. up to h = 8)
•d = 5 : UV finite for h = 1,2,3,4,5
•d = 6 : UV finite for h = 1,2
•d = 7: UV finite for h = 1
•d = 8,9,10: UV divergent at h = 1 already
SUMMARY
betah = h, for all h>1
betah = h, up to h=5 and betah > 5 for h>5
Some final remarks:
-There is an intriguing structure underlying the superstring effective action and thus underlying the quantum structure of string theory.
Much of this structure can be understood by combining L loops in eleven dimensions, supersymmetry and duality.
-We have seen that non-renormalization theorems for D2kR4 couplings indicate finiteness of N=8 supergravity.
Conversely, note that finiteness of N=8 d=4 supergravity would imply that all terms D2kR4 in the 10d type II effective action are not renormalized.
362
2)(4
h
hdSA h
hh h
3 h
beta
3
allowed
GRV
UV div