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Toward the Determination of Effective Action
in Superstring Theory and M-Theory
Yoshifumi Hyakutake (Osaka Univ.)
1. Introduction
• Divergences in 4 dim. quantum gravity
It is important to formulate gravitational interaction in a
way consistent with quantum mechanics.
Perturbative approach to quantum gravity, however, is
problematic. Loop corrections give divergences in UV.
t’Hooft, Veltman
Non-susy gravity: divergence at 1 or 2 loop
Supergravity
: divergence at 3 loop (except N=8)
• Superstring theory as a quantum gravity
In order to cure the divergences, we consider not a
point particle object but a string object.
Then interactions are smeared around the string scale,
and no UV divergence appears in superstring theory.
• Superstring theory in low energy region
Superstring theory is perturbatively defined around
10 dim. flat space-time.
And the low energy limit of superstring theory is
described by supergravity.
Superstring
Supergravity
Yoneya
• Quantum correction to supergravity
Loop calculations in superstring theory give corrections
to the low energy supergravity.
Gross, Witten
Superstring
Supergravity + higher
derivative corrections
Since corrections become important to analyze black
hole physics or singularities in classical gravity.
And higher derivative corrections in string theories are
considerably investigated in various ways
• String scattering amplitude
• Non linear sigma model
• Superfield method
• Duality
• Noether’s method
… and so on
I will review the current knowledge on the higher
derivative terms in string theory and M-theory.
I also discuss the recent progress on the finiteness of
N=8 supergravity.
Plan :
1. Introduction
2. Effective Action from Superstring Amplitude
3. Effective Action from Local Supersymmetry
4. Perturbative and Non-perturbative Terms via Duality
5. Non-renormalization Conditions in Type IIA
6. SYM vs. SUGRA
7. Summary
2. Effective Action from Superstring Amplitude
• Corrections in Type II superstring
Closed oriented superstring amplitudes can be evaluated by
inserting vertex operators of external states on Riemann
surfaces.
0
0
sugra
sugra
stringy
0
0
quantum
Nontrivial corrections start from 4pt amplitude
• Evaluation of 4pt Graviton Amplitude
Green, Schwarz
Gross, Witten
Gross, Sloan
Momentum and two (left and right) polarization vectors are
assigned at each external leg.
The same kinematical factor
Here
arises for tree and 1-loop.
( ) is a kinematical factor for left (right) mover.
: SO(8) generator
4pt graviton amplitude is given by
where
and
are functions of Mandelstam variables
Constant terms of
and
become coefficients of
Remark : Others will give terms with more derivatives
• Effective Action in type II
By the analysis so far, bosonic part in type II is written as
• Effective Action in M
M-theory is described by 11 dim. N=1 Supergravity.
Cremmer, Julia, Scherk
Higher derivative corrections in M-theory can be obtained by
lifting the IIA result.
We will check these forms by local supersymmetry
3. Effective Action from Local Supersymmetry
• Strategy
Hyakutake, Ogushi
Since perturbative methods in M-theory are not developed,
it is impossible to obtain higher derivative terms by
evaluating scattering amplitudes of membranes. The best
way to determine this structure is to use the invariance
under local supersymmetry
Noether method + computer programming
Here we mainly concentrate on bosonic terms and consider
cancellation of O(1) and O(F) terms step by step.
Variation
O( )
O(
)
O(1)
O(F)
O(F^2)
…
Q. How many R^4 terms ?
By using computer program, we obtain 7 independent terms.
Q. How many AR^4 terms ?
There are two terms
• Cancellation
In order to cancel variations of these bosonic terms, it is
necessary to add fermionic terms to the ansatz
Variations of the ansatz are expanded by the following terms
The cancellation mechanism up to O(F) is sketched as
244 Equations among 126 Variables
• Two superinvariants
After miraculous cancellation, the bosonic part is
determined as
tree
1-loop
The first term corresponds to tree level and the second does
to one-loop part in type IIA superstring
• The next step is to examine the invariance
under local supersymmetry up to O(F^2)
In order to execute the cancellation to this order, we have
to add
The variations of this ansatz are expanded by
The cancellation mechanism up to O(F^2) is sketched as
4169 Equations among 1544 Variables
From this cancellation we obtain
1-loop
The structure of R^4 terms is completely determined by
local supersymmetry
• Vanishing theorem
Tree and one-loop amplitudes only contribute to
terms
Proof :
In 11 dim. there is only one superinvariant which contain
terms. These become tree level or 1-loop terms in type
IIA by Kaluza-Klein reduction. No terms more than oneloop.
11d
10d
Green, Gutperle, Vanhove
Sum of KK non-zero modes
KK zero mode
4. Perturbative and Non-perturbative Terms via Duality
• 11 dim. 1-loop amplitude on a circle
Green, Gutperle, Vanhove
Russo, Tseytlin
Let us reexamine 1-loop amplitude for 4 gravitons in the low
energy limit
This expression contains the integral of loop momentum, and it
is implicitly included in
In order to lift this to 11 dimensions, the sum of KK momentum
should appear.
• 11 dim. 1-loop amplitude on a torus
Momentum along 9th dimension should be discretized.
10 dimensional type IIB is realized if
is a non-holomorphic Eisenstein series
D-instanton
Summary
Higher derivative corrections in Type II and M-theory
are considered via
• String scattering amplitude
• Local Supersymmetry
• Duality
Recent Arguments on the finiteness of
N=8 Supergravity
Yoshifumi Hyakutake (Osaka Univ.)
5. Non-renormalization Conditions in Type IIA
• 11 dim. L-loop amplitude on a torus
Green, Russo, Vanhove
Calculation of 11 dim. L-loop amplitude is difficult. By using
power counting, however, we can restrict its form. The L-loop
amplitude on a torus which includes
term will be
subdivergences
where
And following constraints are imposed
derivatives
• Higher derivative action in type IIA
By considering the duality between M and IIA,
Effective action in IIA from genus
Terms which are linear to
can be derived
survive after decompactification
or
gives
Then we obtain strong conditions.
• Conditions for higher derivative terms in type IIA
h-loop amplitude in type IIA contributes as follows.
1. No contributions to
2.
3.
can be determined by 1-loop in 11 dim.
are permitted and may arise from
L-loop in 11 dim. (L>1)
Leading term in low energy can be given by
• Dimensional reduction
If the same property holds for lower dimensions, the
leading contributions can be given by
If this is true, UV divergences are absent in dimensions
which satisfy
4 dim. N=8 supergravity seems to be UV finite
6. SYM vs. SUGRA
• Tree level (KLT relation)
Kawai, Lewellen, Tye
Tree level 4 graviton amplitude is expressed as a ‘‘product’’ of
two tree level 4 point gluon amplitudes.
By using the relation of gamma functions,
we obtain KLT relation (closed)~(open)^2
Low energy
• 1-loop 4pt in the low energy limit
Green, Schwarz, Brink
Tree level 4pt and 1-loop 4pt amplitudes share the same
kinematical factor. Then the Low energy limit of 4pt 1-loop
amplitudes become
SUGRA~(SYM)^2
Here
is given by the scalar box integral
1-loop
UV divergence arises in 8 dimensions
Bern, Rozowsky, Yan
• 2-loop 4pt in N=4 SYM
Bern, Dixon, Dunbar,
Perelstein, Rozowsky
2-loop 4pt amplitudes for N=4 SYM can be computed in
terms of scalar integral functions via cutting method.
Calculations are done by employing spinor helicity formalism
Rung rule
UV divergence arises in 7 dimensions
Bern, Rozowsky, Yan
• 2-loop 4pt in N=8 SUGRA
Bern, Dixon, Dunbar,
Perelstein, Rozowsky
2-loop 4pt amplitudes for N=8 SUGRA can be obtained by
squaring the SYM factor
Rung rule
UV divergence arises in 7 dimensions
• Estimation of UV divergences by rung rule
Let us evaluate L-loop diagram
• N=4 SYM
Divergence free
• N=8 SUGRA
Divergence free?
• 3-loop calculation tells …
4pt graviton amplitude at 3-loop is divergent in 6 dimension.
This is the same as N=4 SYM. Thus the conjecture is
for N=8 SUGRA.
It seems that D=4 N=8 SUGRA is UV finite.
7. Summary
Higher derivative corrections in Type II and M-theory
are considered via
• String scattering amplitude
• Local Supersymmetry
• Duality
Finiteness of N=8 supergravity is reviewed