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Toward the Determination of Effective Action in Superstring Theory and M-Theory

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.

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Toward the Determination of Effective Action in Superstring Theory and M-Theory

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  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. 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.

  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. 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

  4. 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.

  5. Plan : • Introduction • Effective Action from Superstring Amplitude • Effective Action from Local Supersymmetry • Perturbative and Non-perturbative Terms via Duality • Non-renormalization Conditions in Type IIA • SYM vs. SUGRA • Summary

  6. sugra sugra stringy 0 0 0 0 quantum 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. Nontrivial corrections start from 4pt amplitude

  7. 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 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. 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

  10. 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(1) O(F) O(F^2) … O( ) O( )

  11. 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

  12. 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

  13. 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

  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 1-loop The structure of R^4 terms is completely determined by local supersymmetry

  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. 11d 10d Green, Gutperle, Vanhove Sum of KK non-zero modes KK zero mode

  18. 4. Perturbative and Non-perturbative Terms via Duality Green, Gutperle,Vanhove Russo, Tseytlin • 11 dim. 1-loop amplitude on a circle 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.

  19. 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

  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. 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 derivatives where And following constraints are imposed

  23. Higher derivative action in type IIA By considering the duality between M and IIA, Effective action in IIA from genus can be derived 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. • No contributions to • 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

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

  28. Bern, Rozowsky, Yan Bern, Dixon, Dunbar, Perelstein, Rozowsky • 2-loop 4pt in N=4 SYM 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

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

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

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