1. Toward the Determination of Effective Action in Superstring Theory and M-Theory Yoshifumi Hyakutake (Osaka Univ.)

2. 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. 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. t’Hooft, Veltman

3. 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. SuperstringSupergravity Loop calculations in superstring theory give corrections to the low energy supergravity. Quantum correction to supergravity SuperstringSupergravity + higher derivative corrections Yoneya Gross, Witten

4. 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 Since corrections become important to analyze black hole physics or singularities in classical gravity. 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.

5. 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 Plan :

6. 2. Effective Action from Superstring Amplitude Closed oriented superstring amplitudes can be evaluated by inserting vertex operators of external states on Riemann surfaces. sugra 0 000 stringy quantum Nontrivial corrections start from 4pt amplitude Corrections in Type II superstring

7. Evaluation of 4pt Graviton Amplitude Momentum and two (left and right) polarization vectors are assigned at each external leg. Green, Schwarz Gross, Witten Gross, Sloan The same kinematical factor arises for tree and 1-loop. Here ( ) is a kinematical factor for left (right) mover. : SO(8) generator

8. 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

9. By the analysis so far, bosonic part in type II is written as M-theory is described by 11 dim. N=1 Supergravity. We will check these forms by local supersymmetry Effective Action in type II Effective Action in M Higher derivative corrections in M-theory can be obtained by lifting the IIA result. Cremmer, Julia, Scherk

10. 3. 3. Effective Action from Local Supersymmetry 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. O(1) O(F) O(F^2) … O( ) Variation Strategy Hyakutake, Ogushi

11. By using computer program, we obtain 7 independent terms. Q. How many AR^4 terms ? There are two terms Q. How many R^4 terms ?

12. The cancellation mechanism up to O(F) is sketched as 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 244 Equations among 126 Variables Cancellation

13. After miraculous cancellation, the bosonic part is determined as The first term corresponds to tree level and the second does to one-loop part in type IIA superstring Two superinvariants tree 1-loop

14. 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

15. The cancellation mechanism up to O(F^2) is sketched as 4169 Equations among 1544 Variables

16. From this cancellation we obtain The structure of R^4 terms is completely determined by local supersymmetry 1-loop

17. 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 one- loop. 11d10d Sum of KK non-zero modes KK zero mode Green, Gutperle, Vanhove

18. 4. Perturbative and Non-perturbative Terms via Duality 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 circle Green, Gutperle, Vanhove Russo, Tseytlin

19. Momentum along 9 th dimension should be discretized. 11 dim. 1-loop amplitude on a torus 10 dimensional type IIB is realized if is a non-holomorphic Eisenstein series D-instanton

20. Summary Higher derivative corrections in Type II and M-theory are considered via String scattering amplitude Local Supersymmetry Duality

21. Recent Arguments on the finiteness of N=8 Supergravity Yoshifumi Hyakutake (Osaka Univ.)

22. 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 11 dim. L-loop amplitude on a torus where derivativessubdivergences Green, Russo, Vanhove And following constraints are imposed 5. Non-renormalization Conditions in Type IIA

23. By considering the duality between M and IIA, Effective action in IIA from genus can be derived Higher derivative action in type IIA Terms which are linear to survive after decompactification or gives Then we obtain strong conditions.

24. Conditions for higher derivative terms in type IIA h-loop amplitude in type IIA contributes as follows. 1.No contributions to 2. can be determined by 1-loop in 11 dim. 3. are permitted and may arise from L-loop in 11 dim. (L>1) Leading term in low energy can be given by

25. 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

26. 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

27. 1-loop 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 Here is given by the scalar box integral UV divergence arises in 8 dimensions SUGRA~(SYM)^2

28. 2-loop 4pt in N=4 SYM Bern, Rozowsky, Yan 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 UV divergence arises in 7 dimensions Rung rule

29. 2-loop 4pt in N=8 SUGRA Bern, Rozowsky, Yan Bern, Dixon, Dunbar, Perelstein, Rozowsky 2-loop 4pt amplitudes for N=8 SUGRA can be obtained by squaring the SYM factor UV divergence arises in 7 dimensions Rung rule

30. Estimation of UV divergences by rung rule Let us evaluate L-loop diagram N=4 SYM N=8 SUGRA Divergence free Divergence free?

31. 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.

32. 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