1. On Recent Developments in N=8 Supergravity Renata Kallosh Renata Kallosh EU RTN Varna, September 15, 2008 Stanford university

2. Outline Talks of Dixon, Cachazo, Green, RK at Strings 2008 UV properties of gravity and supergravity: Light-cone counterterms versus the known covariant ones, equivalence theorem New results on E 7(7) symmetry of N=8 SG, Noether current Status of the triangle anomaly Comment to the next talk by K. Stelle

3. A possibility of UV finite N=8 SG Recent work RK The Effective Action of N=8 SG, arXiv:0711.2108 RK, Soroush, Explicit Action of E 7,7 on N=8 SG Fields, arXiv:0802.4106 RK, T. Kugo, A footprint of E 7,7 symmetry in tree diagrams of N=8 SG, work in progress RK +… Counterterms in helicity formalism, work in progress Most relevant work Bern, Carrasco, Dixon, Johansson, Kosower, Roiban, Three-Loop Superfiniteness of N=8 SG, hep-th/0702112, 0808.4112 Bianchi, Elvang, Freedman, Generating Tree Amplitudes in N=4 SYM and N = 8 SG Drummond, Henn, Korchemsky and Sokatchev, Dual superconformal symmetry of scattering amplitudes in N=4 SYM, arXiv:0807.1095 RK, arXiv:0808.2310

6. In N=8 SG

7. (h), (i) graphs have 3- and 4-particle cuts!

10. Since has mass dimension 2, and the loop-counting parameter G N = 1/M Pl 2 has mass dimension -2, every additional requires another loop, by dimensional analysis Counterterms in gravity On-shell counterterms in gravity should be generally covariant, composed from contractions of Riemann tensor. Terms containing Ricci tensor and scalar removable by nonlinear field redefinition in Einstein action One-loop  However, is Gauss-Bonnet term, total derivative in four dimensions. So pure gravity is UV finite at one loop (but not with matter) ‘t Hooft, Veltman (1974)

11. Pure gravity diverges at two loops Relevant counterterm, is nontrivial. By explicit Feynman diagram calculation it appears with a nonzero coefficient at two loops RK 1974, van Nieuwenhuizen, Wu, 1977 Goroff, Sagnotti (1986); van de Ven (1992)

12. Pure supergravity (N ≥ 1): Divergences deferred to at least three loops However, at three loops, there is a perfectly acceptable counterterm, for N=1 supergravity: The square of the Bel-Robinson tensor, abbreviated, plus (many) other terms containing other fields in the N=8 multiplet. Deser, Kay, Stelle (1977) RK (1981); Howe, Stelle, Townsend (1981) : N=8 only linearized form cannot be supersymmetrized – it produces a helicity amplitude (-+++) forbidden by supersymmetry Grisaru (1977); Tomboulis (1977) 2007 N=8 SG is finite at the 3-loop order A 3 =0 Bern, Carrasco, Dixon, Johansson, Kosower, Roiban

13. Maximal supergravity Most supersymmetry allowed, for maximum particle spin of 2 Theory has 2 8 = 256 massless states. Multiplicity of states, vs. helicity, from coefficients in binomial expansion of (x+y) 8 – 8 th row of Pascal’s triangle SUSY charges Q a, a=1,2,…,8 shift helicity by 1/2 DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978,1979) Ungauged theory, in flat spacetime

14. Basic Strategy N = 4 Super-Yang-Mills Tree Amplitudes KLT N = 8 Supergravity Tree Amplitudes Unitarity N = 8 Supergravity Loop Amplitudes Bern, Dixon, Dunbar, Perelstein and Rozowsky (1998) Divergences Kawai-Lewellen-Tye relations: sum of products of gauge theory tree amplitudes gives gravity tree amplitudes. Unitarity method: efficient formalism for perturbatively quantizing gauge and gravity theories. Loop amplitudes from tree amplitudes. Key features of this approach: Gravity calculations mapped into much simpler gauge theory calculations. Only on-shell states appear.

15. Kawai-Lewellen-Tye relations Derive from relation between open & closed string amplitudes. Low-energy limit gives N=8 supergravity amplitudes as quadratic combinations of N=4 SYM amplitudes, consistent with product structure of Fock space, KLT, 1986

17. Unitarity Method Two-particle cut: Generalized unitarity: Three- particle cut: Apply decomposition of cut amplitudes in terms of product of tree amplitudes. Bern, Dixon, Dunbar and Kosower

18. UV divergent diagrams in unitarity cut method Integral in D dimensions scales as  Critical dimension D c for log divergence (if no cancellations) obeys N=8 N=4 SYM BDDPR (1998)

19. Current Summary of Dixon et al Old power-counting formula from iterated 2-particle cuts predicted At 3 loops, new terms found from 3- and 4-particle cuts reduce the overall degree of divergence, so that, not only is N=8 finite at 3 loops, but D c = 6 at L=3, same as for N=4 SYM! Will the same happen at higher loops, so that the formula continues to be obeyed by N=8 supergravity as well? If so, N=8 supergravity would represent a perturbatively finite, pointlike theory of quantum gravity Not of direct phenomenological relevance, but could it point the way to other, more relevant, finite theories? N=8 N=4 SYM

20. 3-loop manifestly supersymmetric linearized counterterm ( known since 1981) 2007 N=8 SG is finite at the 3-loop order A 3 =0 Bern, Carrasco, Dixon, Johansson, Kosower, Roiban RK; Howe, Stelle, Townsend SUPERACTION ( Integral over a submafifold of the full superspace)

21. L-loop counterterms Howe, Lindstrom, RK, 1981 : Starting from 8-loop order infinite # of non-linear counterterms is available: 8-loop example with manifest E 7,7 symmetry Analogous candidates for the L-loop divergences, integrals over the full superspace What could be the reason for A L =0 to provide all-loop order UV finite N=8 SG?

22. A possible explanation of the 3-loop computation with UV finite answer A miracle! Howe, Stelle, 2002: if a harmonic superspace of the type of Galperin, Ivanov, Kalitsyn, Ogievetsky,Sokatchev exists for N=8SG with manifest >16 supersymmetries, the onset of divergences will start at L=5 Our new explanation: if valid for the 3–loop order, is also valid for all-loop orders. Main idea: to compare the candidate counterterms in 4+32 covariant on- shell superspace with those in 4+16 light-cone superspace. Transform both results for the S-matrix into helicity formalism, easy to compare We have found a mismatch between the covariant and light-cone UV divergent answers for the S-matrix. If gauge anomalies are absent, we may apply the S-matrix equivalence theorem for different gauges. All-loop finiteness of N=8 SG follows.

23. Light-cone superspace Light-cone chiral superfield for the CPT invariant N=8 supermultiplet Brink, Lindgren, Nilsson, 1983 Brink, Kim, Ramond, 2007 Only helicity states! Most suitable for helicity formalism amplitudes! Nobody ever constructed the light-cone superfield counterterms.

24. N=8 SG divergences in helicity formalism 4-graviton Covariant superfields / helicity formalism

25. If the equivalence theorem for the S-matrix in light- cone versus covariant gauges is valid (if there are no anomalies in the BRST symmetry of N=8 SG) we consider this as a possible explanation of the recent 3-loop computation as well as a prediction for the all-loop finitness of the perturbative theory. We were not able to match a complete answer for the 4-scalar UV divergent amplitude in the light-cone gauge together with with the superfield partners, including a complete SU(8) structure and a 4-graviton amplitude deduced from the covariant superfield counterterms.

26. Important note: the number of scalars before gauge-fixing a local SU(8) is 133, the number of independent parameters in the group element of E 7(7) After gauge-fixing SU(8) the number of physical scalars is 70!

27. A hope that a better understanding of symmetries may help to understand the situation with higher loops. N=8 local supergravity has a gauge SU(8) chiral symmetry and a global E 7,7 (R) symmetry This is a continuos symmetry in perturbative supergravity (as different from string theory where it is discrete) conserved Noether current! It was poorly understood until recently (not acidentally called “hidden”) Based on R.K. and M. Soroush, We have constructed the conserved Noether current

31. We have found an explicit, closed form exact in all orders in  E 7,7 transformations of all fields in the theory.

33. of the work with M. Soroush

34. What is known about N=8 anomalies? 1984 SU(8) is chiral, anomalies possible Gauge theory anomalies make QFT inconsistent! Computed the contribution to SU(8) anomaly from fermions: 8 gravitini and 56 gaugini, found a non-vanishing answer Later in 1985 N. Marcus computed the contribution from vectors to triangle anomaly: Found an exact cancellation! During Strings 2008 many discussions. The main point is: can we trust this computation? Preliminary conclusion, most likely the claim about the cancellation of SU(8) anomaly is correct.

35. In 4d the consistent anomaly is associated with the 6-form (10)

36. SU(8) subgroupT X Orthogonal to SU(8 )

37. Implications of the one-loop SU(8) anomaly cancellation E 7(7) is not anomalous due to consistency condition for anomalies Supersymmetry is not anomalous since the algebra of two local supersymmetries has a local SU(8) symmetry Maybe all this will help to understand the status of UV divergences in N=8 supergravity.

38. Stelle et al, work in progress Proposal: take the known 3-loop or L-loop covariant counterterm and convert it into the light-cone one This should give a candidate counterterm responsible for the UV divergences in N=8 SG in the light-cone gauges We agree that as of now the light-cone counterterms have not been constructed so far as the explicit functions of the light-cone superfields The light-cone counterterm should agree with the known form of divergent amplitudes in covariant gauges

39. 4-graviton divergent amplitude 4-vector divergent amplitude

40. 2-vectors, 2-scalars divergent amplitude

41. 4-scalar divergent amplitude

42. If the light-cone superfield counterterms (to be constructed) will reproduce the known covariant answer for the divergent amplitudes, we will have to use other tools to study UV divergences in N=8 SG If they will not agree with expected structures, this will prove A 3 =0 and eventually A L =0 and N=8 SG finiteness All p+ and 1/p+ should disappear and all helicity brackets should appear!

43. Back up slides

44. Compare spectra 2 8 = 256 massless states, ~ expansion of (x+y) 8 SUSY 2 4 = 16 states ~ expansion of (x+y) 4

45. Light-cone superspace Light-cone chiral superfield for the CPT invariant N=8 supermultiplet New results: Simplest example of the Lorentz covariant 4-scalar UV divergent amplitude Incomplete SU(8) structure, the second SU(8) structure is not covariant Brink, Lindgren, Nilsson, 1983 Brink, Kim, Ramond, 2007

46. KLT only valid at tree level. To answer questions of divergences in quantum gravity we need loops. Unitarity method provides a machinery for turning tree amplitudes into loop amplitudes. Unitarity cuts in gravity theories can be reexpressed as sums of products of unitarity cuts in gauge theory. Allows advances in gauge theory to be carried over to gravity. Apply KLT to unitarity cuts: N = 8 Supergravity from N = 4 Super-Yang-Mills

47. N=8 supergravity in four dimensions during the last 25 years was believed to be UV divergent. During the last few years studies of multi-particle amplitudes in QCD were simplified using N=4 super Yang-Mills theory. This, in turn, led to significant progress in computation of amplitudes in N=8 supergravity. Some spectacular cancellations of UV divergences were discovered. Is N=8 supergravity UV finite? If the answer is "yes" what would this mean for Quantum Gravity?

48. UV finite or not? Quantum gravity is nonrenormalizable by power counting, because the coupling, Newton’s constant, G N = 1/M Pl 2 is dimensionful String theory cures the divergences of quantum gravity by introducing a new length scale, the string tension, at which particles are no longer pointlike. Is this necessary? Or could enough symmetry, e.g. supersymmetry, allow a point particle theory of quantum gravity to be perturbatively ultraviolet finite? If the latter is true, even if in a “toy model”, it would have a big impact on how we think about quantum gravity. If N=8 massless QFT will be found UV finite one would compare it with early days of massless Yang-Mills theory before one knew how to add a Higgs mechanism and to describe the realistic particle physics.