Type II string effective action and UV behavior of maximal supergravities

Jorge Russo

U. Barcelona – ICREA

Based on work in coll. with M.B. Green and P. Vanhove, [PRL (2007), JHEP 0702 (2007), JHEP 0802:020 (2008); GRV june 2008; and work in progress]

Wonders of Gauge Theory and Supergravity Paris – June 23 - 27

Plan

1. Review: Non-renormalization theorems for higher derivative couplings in 10 dimensions and implications for the UV behavior of N=8 supergravity.

2.The type II string effective action according to L loop eleven dimensional supergravity.

3. Conclusions

UV behavior of N=8 Supergravity from string theory

Consider the four-graviton amplitude in string theory at genus h.By explicit calculation one finds (*)

By using the pure spinor formalism, one further finds (+)

These results alone imply that there should not be any ultraviolet divergence in N=8 4d supergravity until at least 9 loops.

They suggest the pattern:

42)2(4

4)1(4

))'(1(:2

))'(1(:1

RsOsAgenus

RsOAgenus

(*) [Green, Schwarz, Brink], [Green, Vanhove, 2000],[D'Hoker, Gutperle, Phong](+) [Berkovits, 2006]

46)6(4

45)5(4

44)4(4

43)3(4

))'(1(:6

))'(1(:5

))'(1(:4

))'(1(:3

RsOsAgenus

RsOsAgenus

RsOsAgenus

RsOsAgenus

4)(4 ))'(1(: RsOsAhgenus hh

242

)10(443)1(2

)10(

4)1(214

',))'(1('

))'(1('

eRsOs

RsOseAhh

hh

hh

hh

Genus h four-graviton amplitude in string theory:

The presence of S R4 means that the leading divergences 8h+2 of individual h-loop Feynman diagrams cancel and the UV divergence is reduced by a factor – 8 – 2

Supergravity amplitudes from string theory

h-loop supergravity amplitude is obtained by taking the limit alpha’ to zero.The inverse string length is interpreted as a UV cutoff.

4268)1(2)10(4 ))'(1( RsOsA hhhhh

Maximal supergravity in lower dimensions:

Start with string loop amplitude on a 10-d torus with the radii = sqrt(alpha’).

In the limit alpha’ -> 0 all massive KK states, winding numbers and excited string states decouple.

In terms of the d-dimensional gravitational constant kappad2 ,

','

,))'(1(

22/)2(2

426)2()1(2)(,4

radiie

RsOsA

dd

hdhd

hd

hh

Therefore UV divergences are absent in dimensions satisfying the bound

hdhd h 62

2026)2(

KNOWN CASES:

•h = 1: beta1=0 , therefore UV finite for d < 8

•h = 2: beta2=2 therefore UV finite for d < 7

•h = 3: beta3=3 [Bern, Carrasco, Dixon, Johansson, Kosower and Roiban 2007]. Therefore UV finite for d < 6

•h > 3: expected betah > or = 3. This gives d < 2 + 12/h. Thus d = 4 finite up to h < 6.

For h > 3 we use the non-renormalization theorems obtained in the pure spinor formalism [Berkovits, 2006].

One finds

•betah = h for h =2, 3, 4, 5 ,6

•betah > or = 6 for h > 5.

Hence there would be no UVdivergences if

5,...,2,/64

6,/182

hhd

hhd

d = 4:

9/1824 hh

d = 4: UV finite at least up to h = 8

Now using IIA/M theory duality [Green, J.R., Vanhove, 2007] : betah = h for all h > 1

Hence there would be no UV divergences if

1,6

4 hh

d

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]

h

h

hd h 62

262

2

Checked explicitly up to h = 3 [Bern, Carrasco, Dixon, Johansson, Kosower and Roiban 2007].

M-graviton amplitudes

))'(1(4 sORsA MMhhM += −+

Duality arguments now show

Following previous analysis, this 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.

The Berkovits non-renormalization theorems can easily be generalized to the M-graviton amplitude. This leads to the conditions for finiteness of an R^M term,

D = h, up to h=3(explicit calculation)

= h, up to h=5(Berkovits theorems)

= h, for all h(string dualities)

4 6 9 finite

5 4 6 6

6 3 3 3

7 2 2 2

8 1 1 1

9 1 1 1

10 1 1 1

Summary: loop order at which we expect the first divergence to occur in d dimensional Maximal supergravities

Superspace arguments:- Howe, Stelle: D = 4 is finite up to 6 loops- Kallosh: D = 4 is finite up to 7 loops (constructs 8 loop counterterm).

NON-RENORMALIZATION THEOREMS IN STRING THEORY

is equivalent to saying that

The term sh R4 = D2h R4 does not receive any quantum correction beyond genus h

(since A(h+1) starts contributing to sh+1 R4)

...

)'(1

)'(141)1(

4

4)(4

RsOsA

RsOsAhh

hh

Type II effective action: Consider the terms

Terms involving other supergravity fields, or terms with the same number of derivatives (like e.g. Rm or |G|4nR4 terms), are expected to be connected to these terms by N=2 supersymmetry.

The statement that

424644424 ,,,,, RDRDRDRDR k

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 π=−= −−

∞

=∫∑ 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 M2 brane excitations should give contributions of the form

32311

211211

2111

2 ,)( PAIIA lgRdxCdxRdsRds =−+= − μμ

!!

)( 4210)1(2

0 0

knkh

RDgxdgcS IIAknk

sk

nk

k

n

≤−=⇒

−=⇒ ∫∑∑ −−∞

= =

Convert to type IIA variables:

Genus:

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:

)(

22

4210)2

33(2

∫

nifkLkh

nifkLn

khghgenus

RDgxdgS

LhA

Ak

nLk

An

Using that S2 R4 = D4R4 is not renormalized beyond L=2

khgeneralinso

nforkhL

03

12

TYPE IIA EFFECTIVE ACTION FROM ELEVEN DIMENSIONS

Convert to type IIA variables:

Thus we find a maximum genus for every D2kR4. This is irrespective of the value of L. The highest genus contribution is h = k and comes from L = 1.

The argument only uses power-counting and the relation between type IIA and M-theory parameters. So the argument equally applies to all interactions of the same dimension as D2kR4, in particular, R4+k, etc.

Thus the highest-loop contribution for a term of dimension 2k+8, with k > 1, is genus k.

As an aside remark, note that this implies that the M-theory effective action can only contain terms with 6n+2 derivatives (e.g. R3n+1 or D6n-6R4) [J.R. and Tseytlin, 1998]

...103

72

41

11 ∫ RcRcRcRgxdS

Reason: a term R4+k produces a dependence on the coupling gA2k/3. For terms where k is not a multiple

of 3, one gets 1/3 powers of the coupling. Such dependence cannot possible arise if the genus expansion stops.

The type II string effective action according to

L loop eleven dimensional supergravity.

L LOOP = +…

Regularize supergravity amplitudes by a cutoff = 1/lP

- Eleven-dimensional supergravity on S1 = Type IIA

A(S,T,R11) = A(s,t,gs)

- Eleven-dimensional supergravity on T2 = Type IIB

Low momentum expansion gives analytic terms Sk R4 and massless threshold terms sk Log(s) R4

12

44

)exp(,)exp(

),,,(),,,(

s

B

gi

rtsAVTSA

TREE-LEVEL FOUR GRAVITON AMPLITUDE

4',4

',0,""][

...)(,)()(

...))3(21(1

)](12

)12(2exp[

1

)1()1()1(

)1()1()1(1

4

41

'''''

4'

3'

2'

1

121212

14

μμμμμμ

ssutsRK

stkKkKkKK

stustu

K

utsk

k

stuK

uts

uts

stuKA

iii

kkk

k

)()...(

3,

),(

1),(

),(

3333

2222

0,32),(32

kqp

kqp

qp

qpqp

rrcT

stuutsuts

TA

Remarkably, it is a polynomial in two kinematical structures and :

The tree-level four-graviton effective action is

)]~

))9()3(2()9(['

)5()3(')7('

)3(')5(')3(2'(

4123272412

419

41032848

217

46232644543210

∫

RDRD

RDRD

RDRDRRegxdS

• In type IIB, each D2kR4 term in the exact effective action will contain, in addition, perturbative (higher genus) and non-perturbative corrections. For the exact coupling, the zeta functions must be replaced by a modular function of the SL(2,Z) duality group.

•In type IIA, there are perturbative corrections only.

422212

042 ))(...()( nonpert RDgccg

cRDgf k

ss

ksk

To the extent we checked, all coefficients in the higher genus amplitudes are always products of Riemann zeta functions.(this includes e.g. up to genus 9 coefficients)

L = 1 eleven-dimensional supergravity amplitude on T2

42

2

2

422/1

2

242/3

43291

)1(..(

)()((

nonpert RDkgenusgenusrei

RDZrcRZRrgrdxS

k

k

kB

kk

k

kBkBIIBB

L

∫

-

)(,0))1(( 2222 21 rZrr

12

211,

)(122

)0,0(),(2

2

)exp(,)exp(

)2()2cos()2(2

),(

s

kwn

knw

rk

r

nmr

r

r

gi

wnKwncr

nmZ

ππ

[J.R., Tseylin, 1997], [Green, Kwon, Vanhove, 2000]

where

L = 2 eleven-dimensional supergravity amplitude on T2

...~

)()()(

)()()(),(

412)(6

4412)(

6

4410

4

3

482

246

)2/3,2/3(44

2/52

421

RDGr

gRDG

r

gRDF

r

g

RDEr

gRDEgRDZgrtsA

ii

IIB

si

IIB

s

IIB

s

IIB

sssIIB

L

21

23

21

23

21

23

3

2

1

321048

)42(

)20(

)6(

:

ZZE

ZZE

ZZE

EEEEERDE

2

),(46

),( 23

23

23

23

23 6)12(: ZERDE

90,56,30,12,2

)(

:

21

21

23

23

43210410

j

jjj ZZwZZF

FFFFFFRD

156,110,72,42,20,6,0)12(2

)(

:

21

21

25

23

6543210,412

jj

ZZcZZfG

GGGGGGGGRD

j

xj

xj

xjj

yx

E, F, G(i), G(ii) are new [Green, J.R.,Vanhove, june 2008]. The source terms that appeared could have been predicted using supersymmetry.

[Green, Vanhove, 2005]

L LOOPS IN ELEVEN DIMENSIONS

42)()(

95261

42)()(

93

42)(),(

912236933

]3/2[

42)(),(

936933

]3/2[

),(

),(

),(

),(

23

29

2

23

RDfgxdrg

RDfGVxdVS

RDfgxdrg

RDfGVxdVS

SSS

kLk

kLB

kB

kLk

kfiniteL

kLkm

kmB

kB

mLL

km

kLkm

kmLL

km

divL

finiteL

divLL

L

L

m

m

For L = 1 and L = 2, these expressions reproduce the several terms previously obtained by explicit calculation.

The terms that decompactify in ten dimensions are linear with rB .

For the finite part, this is the term k = 3L – 3. For the divergent part, it is m = k + 1

...)()5(),( 482/1532/7

2/346534 RDEgrEgrEgrRDrtsA YBBXBBBBBL

This was not computed explicitly. Combining the general formula with some genus 1 dataone gets

L = 3 eleven-dimensional supergravity amplitude on T2

Similarly, one can write down the expected L loop modular functions for general L.

Consider the IIB effective action in nine dimensions.

The modular group is SL(2,Z) x R+ , where SL(2,Z) acts on the complexified coupling constant and R+ acts on the size of the circle rB .

Its action is naturally defined on the dimensionless volume nu of the compactification manifold measured in the ten-dimensional Planck length unit

22/1 / BB rg

μμμ

μ

μμμ

μμ

∫

∫

4922

47)9(

22

212

72)9(9

9

9)10(

22

21)10(10

10

)(

0)(

)(

21

21

281

RgdxS

eee

RgdxS

Ed

rr

Ed

-

-

TAKE GENERAL L LOOP FORMULA APPLIED TO R4, D4R4 AND D6R4

22/1

4922

47)9(

076)9(

40

9422/3

94

/,

0

),())2(4)((:

BB

BIIBBBIIB

rg

M

RrMgdxRrZrgdxR

∫∫

--

0

),(

))()3()2()((:

4730)9(

444

9

442/315

441582

2/5944

∫

∫

M

RDrMgdx

RDZrrZrgdxRD

BIIB

BBBIIB

-

-

(L = 1)

(L = 2) 3 (L = 1)(L = 2)

(L = 1) 3

2067

90)9(

466

9

46442/5

62/3

2),(

946

6

),(

))2(5

48)5(

63

24)(

63

12)()((:

23

23

MM

RDrMgdx

RDrrZrZrErgdxRD

BIIB

BBBBBIIB

∫

∫

-

-

(L = 1)(L = 2) (L = 2)(L = 3)3(L = 2)

These results can be compared to the 8d modular function of R4 [Kiritsis, Pioline, 1997] and the 8d modular functions of D4R4 and D6R4 [Basu, 2007]

Taking the 9d limit on these modular functions, we reproduce the above results.

The differential equations that determine the different modular functions also dictate that they contain a finite number of perturbative contributions.Namely that the corresponding higher derivative coupling is not renormalized beyond a given genus

?

),(...),(),((?)

),(

)525

)7()((:

)(8

)9(

)(8

)(88

488

9

486422/72/1

422/7

8315

2948

1

∫

∫

i

n

M

rMrMrM

RDrMgdx

RDrErErZZrErZrrgdxRD

i

BBB

BIIB

BYBXBBBBBIIB

-

-

Problem: presence of terms O(exp(-1/rB)) in genus one amplitude [GRV 2008].

This implies that the complete M8 contains terms O(exp(-1/rB)).

These are difficult to calculate. But they could be generated automatically once the correct differential equation is known.

(L = 1) (L = 2) (L = 2) (L = 3) (L = 3) (L = 3) (L = 4)

CONCLUSIONS

1. UV BEHAVIOR OF N = 8 SUPERGRAVITYWe have argued that for all D2kR4 couplings the perturbative expansion stops at genus k (Agreement up to 18 derivatives with the non-renormalization theorems derived by Berkovits).

This seems to imply finiteness of N= 8 supergravity.

Conversely, finiteness of N = 8 supergravity would imply that all D2kR4 are not renormalized in string theory beyond some given genus.

2. TYPE II EFFECTIVE ACTIONWith a few string theory inputs one can determine the full string-theory quantum S matrix just from loops in 11D supergravity.

New modular functions for higher derivative couplings up to D12 R4. They obey Poisson equations with different eigenvalues.It would be important to see if also modular functions arising from L = 3 supergravity are determined by differential equations having as source terms modular functions of lower derivative terms (this would give independent evidence on non-renormalization theorems).

General structure of modular functions arising from L loop supergravity (combining power counting and string dualities).The complete modular function multiplying a given coupling R4, D4R4 or D6R4, in nine dimensions, (which is a sum of several pieces originating from different loop orders in 11d supergravity) is determined by a simple differential equation.

)6()0()3()3()0()6(

)5(5)3(3)0(

)5(5)3(3)0(

0

...''

...)''(

SSS

SSSS

Structure of differential equations expected from supersymmetry

This leads to the Poisson equation defining E(3/2,3/2) with source term Z3/2Z3/2

Similarly, one can predict the source terms for E,F,Gi,Gii