Supergravity Infinity Cancellations and Ultraviolet Puzzles

K.S. Stelle Imperial College London

Quantum Gravity in the Southern Cone Maresias, Brazil Wednesday, September 11, 2013

G. Bossard, P.S. Howe & K.S.S. 0901.4661, 0908.3883, 1009.0743 G. Bossard, P.S. Howe, K.S.S. & P. Vanhove 1105.6087 G. Bossard, P.S. Howe & K.S.S., 1212.0841 G. Bossard, P.S. Howe & K.S.S., 1304.7753 1 Gell-Mann’s Totalitarian Principle: “Everything not forbidden is compulsory” So why are expected UV divergences not occurring on schedule in maximal supergravity? Are miracles happening?

(Quotation actually stolen from T.H. White, The Sword in the Stone)

Saint Augustine: “Miracles are not contrary to nature, but only contrary to what we know about nature”

2 A very useful approach to the analysis of UV divergence problems in supersymmetric theories has been the combination of superspace techniques with the background field method for calculating Feynman diagrams. When one has an “off-shell” formalism for some degree of extended supersymmetry, the Feynman rules can be organised in such a way that background fields on external lines appear only through certain “geometrical” entities, e.g. superspace vielbeins and gauge connections. The introduction of prepotentials for the quantum fields which correspond to the internal lines of superspace Feynman diagrams allows all terms used in the calculation of 1PI diagrams at loop orders L>1 to be written as full- superspace integrals. Counterterms for L>1 must be writeable as full-superspace integrals of expressions involving the background fields. 3 The degree of off-shell supersymmetry is the maximal supersymmetry for which the algebra can close without use of the equations of motion. Knowing the extent of this off-shell supersymmetry is tricky, and may involve formulations (e.g. harmonic superspace) with infinite numbers of auxiliary fields. Galperin, Ivanov, Kalitsin, Ogievetsky & Sokatchev For maximal N=4 Super Yang-Mills and maximal N=8 supergravity, the fraction of off-shell realizable supersymmetry is known to be at least half the full supersymmetry of the theory, but the maximum realizable fraction in harmonic superspace is not currently known. Assuming that the maximal fraction is 1/2 led originally to the expectation that the first allowable counterterms would have “1/2 BPS” structure. 4 Dimension D 11 10 8 7 6 5 4 Loop order L 2 2 1 2 3 4 5 Gen. form 12R4 10R4 R4 4R4 6R4 6R4 4R4 The 3-loop R⁴ maximal supergravity candidateDimension countertermD 10 8 7 6 5 4 Loop order L 1 1 2 3 Dimension6? D 11 10 8 7 6 5 4 ⇥ 1 1 1 1 Loop1 order1 L 2 2 1 2 3 6? 5? BPS degree Dimension D 11 10 8 7 6 5 4 4 2 4 4 BPS4 degree4 0 0 1 1 1 0 1 Gen. form 2F 4 F 4 2F 4 2F 4 Loop2F order4 finiteL 2 2 1 2 43 6?8 5? 4 12 4 10 4 4 6 4 6 4 12 4 4 4 has a structure under linearized supersymmetry that is very Gen. form R R1 R1 R1 R 1R R BPS degree 0 0 2 4 8 0 4 12 4 10 4 4 6 4 6 4 12 4 4 4 Gen.Dimension form D R 11 R 10R 8R 7R 6R 5R 4 DimensionLoop orderD L11 2 10 28 17 26 35 46 7 similar to that of an F⁴ N=4 super Yang-Mills invariant. BothLoop BPS order degreeL 2 0 2 01 12 13 16 70 0 4 4 4 4 2 4 8 ¯ BPS degree 0 12 40 10 1 4 14 6 1 4 60 4 120 4 8 4 I = d x(d d ) tr(⇥ ) Gen. form R R1052 R4 R8 R R R SYM 105 105 12 4 10 4 4 6 4 6 4 12 4 8 4 Dimension D 11 10 8 7 Gen.6 form 5 R 4 R R R R R R of these are 1/2 BPS invariants, involving integrationLoop order L 2 over2 1 just2 3 6? 5? 1 1 1 1 4 BPS degree8 8 ¯0 0 2 44 8 0 4 ISG = d x(d d 12)2328484 10 4 (4W6 4)2328486 4 12 4 4 4 232848 Gen.Howe, form K.S.S. R & RTownsendR R 1981 R R R half the corresponding full superspaces: DimensionKalloshD 198111 10 8 7 6 5 4 Loop order L 2 2 1 2 3 6 7 BPS degree 0 0 1 1 1 0 0 4 4 4 2 4 8 ¯ 12 4 10 4 4 6 4 6 4 12 4 8 4 DISY M = (d qd q)105 tr(f )105 105 Gen.f formij R 6R ofR SU(4) R R R R Z 8 8 ¯ 4 2 DISG = (d qd q)232848(W )232848 232848 Wijkl 70 of SU(8)

3 Z 3 Versions of these supergravity and SYM counterterms indeed do occur at one loop in D=8. This implies that, at least in that spacetime dimension, the full nonlinear structure of such 3 counterterms exists and is consistent with all symmetries.

5 Bern, Carrasco, Dixon, Unitarity-based calculations Johansson, Roiban et al. 2007 ... 2011 The calculational front has now made substantial progress since the late 1990s.

This has led to unanticipated and surprising cancellations at the 3- and 4-loop orders, yielding new lowest possible orders for super Yang-Mills and supergravity divergence onsets.

Dimension D 11 10 8 7 6 5 4 Loop order L plus2 462 more1 2 topologies3 4 5 DimensionGen. form D @1112R4 @1010R4 8R4 @74R4 @66R4 @56R4 @44R4 Loop order L 2 2 1 2 3 4 5 Gen.Dimension form D12R4 1010R48 R4 7 4R4 6 6R4 56R4 44R4 Loop order L 1 1 2 3 6 Max. SYM first divergences, Dimension D 10 8 7 6 5 41 BPS degree 1 1 1 1 1 1 current lowest possible orders Loop order L 14 12 24 34 6?4 4 2 4 4 2 4 2 4 2 4 Gen. form @1F F1 @1 F @1 F @1 F finite1 (for integral spacetime BPS degree 4 2 4 4 4 4 dimensions). Gen. form 2F 4 F 4 2F 4 2F 4 2F 4 finite Dimension D 11Blue: known10 8 divergences7 6 5 4 Loop order L 2 2 1 2 3 6? 5? Max. supergravity first Dimension D 11 10 81 71 6 1 5 4 1 LoopBPS degree order L 20 20 12 24 3 8 6?0 5?4 divergences, current lowest 12 4 10 4 4 4 4 6 4 12 4 4 4 BPSGen. degree form @ 0R @ 0 R 1R @1 R @1 R @0 R @1 R possible orders (for integral 2 4 8 4 12 4 10 4 4 4 4 6 4 12 4 4 4 spacetime dimensions). Gen. form R R R R R R R 6

2 2 Algebraic Renormalization and Ectoplasm

Dixon; Howe, Lindstrom & White; Piguet & Sorella; Hennaux; Stora; Baulieu & Bossard; Voronov 1992; Gates, Grisaru, Knut-Whelau, & Siegel 1998 Berkovits and Howe 2008; Bossard, Howe & K.S.S. 0901.4661, 0908.3883, 1009.0743 The construction of supersymmetric invariants is isomorphic to the construction of cohomologically nontrivial closed forms in

superspace: I = D is invariant (where is a pull-back to M0 L a section of the projection map down to the purely bosonic “body” subspace M0) if is a closed form in superspace, and it is LD nonvanishing only if is nontrivial. LD Using the BRST formalism, one can handle all gauge symmetries and space-time diffeomorphisms by the nilpotent BRST operator

s. The invariance condition for D is s D + d 0 D 1 =0 , L L L 2 where d 0 is the usual bosonic exterior derivative. Since s =0

and s anticommutes with d 0 , one obtains from Poincaré’s lemma s D 1 + d 0 D 2 =0 , etc. L L 7 For example, the cocycle structure of the SYM Lagrangian turns out to match that of a full-superspace integral of a gauge- invariant integrand, showing that the latter are fully acceptable as counterterms. Examples of operators that are ruled out by the ectoplasm/ algebraic renormalization analysis include half-BPS counterterms such as the tr( F 4 ) or (tr( F 2 )) 2 SYM counterterms. In D dimensions, the generator component of such a 1/2 BPS cocycle is an (0, D) form of dimension 8-D/2. Since the structure of this cocycle is different (i.e. it is longer) from than that of the SYM Lagrangian, the corresponding 1/2 BPS counterterm is illegal. Similar considerations allow one to analyse the R 4 counterterm in N=8 supergravity, although the density character of supergravity invariants complicates analysis of their non-leading Bossard, Howe & K.S.S. 0901.4661, 0908.3883 structure. 8 Duality invariance constraints Maximal supergravity has a series of duality symmetries which extend the automatic GL(11-D) symmetry obtained upon dimensional reduction down from D=11. The classic

example is E7 in the N=8, D=4 theory, with the 70=133-63

scalars taking their values in an E7/SU(8) coset target space.

The N=8, D=4 theory can be formulated in a manifestly E7 Bossard, Hillman & Nicolai 2010 covariant (but non-manifestly Lorentz covariant) formalism. Marcus 1985 Anomalies for SU(8), and hence E7, cancel. Combining the requirement of continuous duality invariance with the superspace cohomology requirements gives further powerful restrictions on counterterms. Other approach to duality analysis from string amplitudes: Broedel & Dixon 2010 Elvang & Kiermeier 2010; Beisert, Elvang, Freedman, Kiermaier, Morales & Stieberger 2010 9 rigid E7(7), the measure will be E7(7) invariant whereas the integrand will necessarily transform 6 4 non-trivially with respect to E7(7). It would then follow that the ⌅ R invariant is not E7(7) invariant, in agreement with the conclusion of the preceding section. Note that this is not in contradiction with the existence of BPS duality invariants in higher dimensions (such as R4 in D = 8, ⌅4R4 in D = 7 and ⌅6R4 in D = 6), since the BPS invariants are not unique in dimensions D>5. The non-existence of harmonic measures for the 1/2 and the 1/4 BPS invariants is not in contradiction with the existence of these non-linear invariants in the full non-linear theory. Indeed as we will discuss in the next section, not all supersymmetry invariants can be written as harmonic superspace integrals, and some are only described in terms of closed super-D-form.

Non-linear consequences of linear invariants

A more general approach to the construction of superinvariants is aorded by the ectoplasm formalism [28, 29, 30]. In D-dimensional spacetime, consider a closed super-D-form, LD,inthe corresponding superspace. The integral of the purely bosonic part of this form over spacetime is then guaranteed to be supersymmetric by virtue of the closure property. Moreover, if LD is exact it will clearly give a total derivative so that we are really interested in the Dth superspace cohomology group. As we have seen in the preceding section, one cannot define a harmonic measure for every invariant, and in particular, not for the 1/2 and 1/4 BPS invariants in N =8 supergravity. However, according to the algebraic Poincar´eLemma, any supersymmetry invari- ant necessarily defines a closed super-D-form. In order to analyse superspace cohomology, it is convenient to split forms into their even and odd parts. Thus a (p, q)-form is a form with p even and q odd indices, totally antisymmetric on the former and totally symmetric on the latter. The exterior derivative can likewise be decomposed into parts with dierent bi-degrees,

d = d0 + d1 + t0 + t1 , (13) where the bi-degrees are (1, 0), (0, 1), ( 1, 2) and (2, 1) respectively. So d and d are basically 0 1 even and odd derivatives, while t0 and t1 are algebraic. The former acts by contracting an even index with the vector index on the dimension-zero torsion and then by symmetrising over all of the odd indices. The equation d2 = 0 also splits into various parts of which the most relevant components are

2 2 t0 = 0; d1t0 + t0d1 = 0; d1 + t0d0 + d0t0 =0. (14)

p,q The first of these equations allows us to define t0-cohomology groups, Ht [31], and the other two p,q p,q+1 allow us to introduce the spinorial derivative ds which maps Ht to Ht by ds[⇥p,q]=[d1⇥p,q], where the brackets denote Ht cohomology classes, and which also squares to zero [32, 33]. The pointSupergravity of this is that one Densities can often generate closed super-D-forms from elements of these cohomology groups. In a curved superspace, an invariant is constructed from the top In the context of curved superspace it is important to note that the invariant is constructed from the(pure top component “body”) in component a coordinate in basis, a coordinate basis: 1 I = dDx ⇤mD...m1 E AD E A1 L (x, = 0) . (15) mD m1 A1...AD A One transformsD to! a preferred basis by means··· of the supervielbein EM .At⇥ = 0 we can a a identify Em with the spacetime vielbein em and Em with the gravitino field ⇤mA (where OneincludesReferring transforms both space-timethis to a to preferred a ,preferred˙ and basis internal by“flat”i indices means basis for ofN theand= 8). supervielbein identifying In four dimensions, E M we .At therefore⇥ = 0 we can a 6 a identifyhave Em with the spacetime vielbein em and Em with the gravitino field ⇤m (where includescomponents both space-time with vielbeins, ˙ and internal and gravitinos,i indices for oneN = has, 8). In e.g. four in dimensions, D=4 we therefore 1 a b c d a b c a b ⇥ have I = e e e e Labcd +4e e e ⇤ Labc +6e e ⇤ ⇤ Lab ⇥ 24 ⇤ a ⇥ ⇤ ⇥ ⇤ ⌅ 1 +4e ⇤ ⇤ ⇤ La ⇥ ⇤ + ⇤ ⇤ ⇤ ⇤ L ⇥ ⇤⌅ . (16) a b c d a b c a b ⇥ I = e e e e Labcd +4e e e ⇤ Labc +6e e ⇤ ⇤ Lab ⇥ 24Thus the “soul” components of the cocycle also contribute⇥ to By definition,⇤ each component Labcd,Labc,Lab ⇥,La ⇥ ⇤,L ⇥ ⇤⌅ is supercovariant at ⇥ = 0. a ⇥ ⇤ ⇥ ⇤ ⌅ This is a useful formula because one can directly+4 reade ⇤ o ⇤the⇤ invariantLa ⇥ ⇤ in+ components⇤ ⇤ ⇤ ⇤ in thisL ⇥ ⇤⌅ . (16) basis.the local supersymmetric covariantization. By definition,In N = 8 supergravity, each component all the non-trivialL ,L t -cohomology,L ,L classes,L lie in H is0,4. supercovariant Invariants are at⇥ ⇥ = 0. Since the gravitinos do notabcd transformabc0 ab under⇥ a the⇥ ⇤ D=4 ⇥ ⇤⌅ Et 7 duality, Thistherefore is a useful completely formula determined because by one their can (0, 4) directly components readL o⇥⇤⌅the, and invariant all non-trivial in componentsL0,4 sat- in this isfying [d L ]=0int -cohomology define non-trivial invariants. H 0,4 is the set of functions basis.the LABCD1 0,4 form components0 have to be separatelyt duality of fields in the symmetric tensor product of four 2 8 2 8 of SL(2, C) SU(8) without ⇥ 0,4 In NSUinvariant.=(8) 8 contractions supergravity, (since all such the functions non-trivial would thent0-cohomology be t0-exact). Because classes of lie the in reducibilityHt . Invariants of are the representation, it will be convenient to decompose L into components of degree (0,p,q10 ) therefore completely determined by their (0, 4) components ⇥ ⇤⌅ L ⇥ ⇤⌅, and all non-trivial L0,4 sat- (p + q = 4) with p 2 8 and q 2 8 symmetrised indices. 0,4 isfying [d1L0,4]=0int0-cohomology define non-trivial invariants. Ht is the set of functions 0,4 2 of fieldsWe will in classify the symmetric the elements tensor of Ht productinto three of generations. four 2 8The2 first8 generationof SL(2 corresponds, C) SU(8) without to elements that lie in the antisymmetric product of four2 ⇥8 2 8 of SL(2, C) SU (8), SU(8) contractions (since such functions would then bet0-exact).⇥ Because of the reducibility of and can therefore be directly related to the top component L through the action of the the representation, it will be convenient to decompose L 4,0 into components of degree (0,p,q) superderivatives. We will write M0,p,q for the corresponding components ⇥ ⇤⌅ of a given L0,4.They (p +lieq = in 4)the with followingp 2 irreducible8 and q representations2 8 symmetrised of SL(2, C indices.) SU(8): 0,4 2 We will classify the elementsM :[0 of, 0 H0200000]t into threeM¯ generations.:[0, 0 0000020]The first generation corresponds 0,4,0 | 0,0,4 | to elements that lie inM the antisymmetric:[1, 1 1100001] productM¯ of:[1 four, 1 1000011]2 8 2 8 of SL(2(17), C) SU(8), 0,3,1 | 0,1,3 | ⇥ and can therefore beM directly:[2, 0 related2000010] to the topM¯ component:[0, 2 0100002]L4.,0 through the action of the 0,2,2 | 0,2,2 | superderivatives. We will write M0,p,q for the corresponding components of a given L0,4.They In order to understand the constraints that these functions must satisfy in order for L to lie in the following irreducible representations of SL(2, C) SU(8): 0,4 satisfy the descent equation [d1L0,4]=0, (18) M0,4,0 :[0, 0 0200000] M¯ 0,0,4 :[0, 0 0000020] 0,5 it is useful to look at the possible representations| of d1L0,4 which define H| t cohomology classes M0,3,1 :[1, 1 1100001] M¯ 0,1,3 :[1, 1 1000011] (17) in general, without assuming any `a| p r i o r i constraint. We will split d1 =| d1,0 + d0,1 according to the irreducible representationsM :[2 of, 0SL2000010](2, C) SU(8). OneM¯ computes:[0 that, 2 0100002] . 0,2,2 | 0,2,2 | [d M ]:[1, 0 1200000] [d M¯ ]:[0, 1 0000021] In order1,0 to0,4 understand,0 | the constraints that these0,1 functions0,0,4 | must satisfy in order for L0,4 to [d M ]:[0, 1 0200001] [d M¯ ]:[1, 0 1000020] satisfy0 the,1 0 descent,4,0 | equation 1,0 0,0,4 | [d M ]:[0, 1 0200001] [2, 0 2100001] [d M¯ ]:[1, 0 1000020] [0, 2 1000012] 1,0 0,3,1 | ⇥ | [d1L0,4]=00,1 0,,1,3 | ⇥ | (18) [d M ]:[1, 0 1100010] [1, 2 1100002] [d M¯ ]:[0, 1 0100011] [2, 1 2000011] 0,1 0,3,1 | ⇥ | 1,0 0,3,1 | ⇥ 0,5| it is useful[d M to look]:[1, at0 1100010] the possible[3, 0 representations3000010] [d of dM¯1L0,]:[04 which, 1 0100011] define H[0t, 3 0100003]cohomology classes 1,0 0,2,2 | ⇥ | 0,1 0,2,2 | ⇥ | in general, without assuming any `a p r i o r i constraint. We will split d1 = d1,0 + d0,1 according to [d0,1M0,2,2]:[2, 1 2000011] [d1,0M¯ 0,2,2]:[1, 2 1100002] . the irreducible representations| of SL(2, C) SU(8). One computes| that (19) 2 0,4 We will avoid discussing the elements of Ht of degree (0, 2, 2) in the [0¯, 0 0200020] representation, which do [d1,0M0,4,0]:[1, 0 1200000] [d0,1M0|,0,4]:[0, 1 0000021] not play any role. | | [d M ]:[0, 1 0200001] [d M¯ ]:[1, 0 1000020] 0,1 0,4,0 | 1,0 0,0,4 | [d1,0M0,3,1]:[0, 1 0200001] [2, 0 2100001] [d0,1M¯ 0,1,3]:[1, 0 1000020] [0, 2 1000012] | ⇥ | 7 | ⇥ | [d M ]:[1, 0 1100010] [1, 2 1100002] [d M¯ ]:[0, 1 0100011] [2, 1 2000011] 0,1 0,3,1 | ⇥ | 1,0 0,3,1 | ⇥ | [d M ]:[1, 0 1100010] [3, 0 3000010] [d M¯ ]:[0, 1 0100011] [0, 3 0100003] 1,0 0,2,2 | ⇥ | 0,1 0,2,2 | ⇥ | [d M ]:[2, 1 2000011] [d M¯ ]:[1, 2 1100002] . 0,1 0,2,2 | 1,0 0,2,2 | (19)

2 0,4 We will avoid discussing the elements of Ht of degree (0, 2, 2) in the [0, 0 0200020] representation, which do | not play any role.

7 At leading order, the E7/SU(8) coset generators of E7 simply produce constant shifts in the 70 scalar fields. This leads to a much easier check of invariance than analyzing the full superspace cohomology problem. Howe, K.S.S. & Townsend 1981

Although the pure-body (4,0) component L abcd of the R⁴ counterterm has long been known to be shift-invariant at lowest order (since all 70 scalar fields are covered by derivatives), it is harder for the fermionic “soul” components to be so, since they are of lower dimension.

Thus, one finds that the maxi-soul (0,4) L ↵ component is not invariant under constant shifts of the 70 scalars. Hence the D=4,

N=8, 3-loop R⁴ 1/2 BPS counterterm is not E7 duality invariant, so it is ruled out as an allowed counterterm. G. Bossard, P.S. Howe & K.S.S. 1009.0743 11 L=7 and Vanishing Volume G. Bossard, P.S. Howe, K.S.S. & P. Vanhove 1105.6087 The above type of analysis knocks out all the candidates in D=4, N=8 supergravity through L=6 loops. This leaves 7 loops (Δ=16) as the first order where a fully acceptable candidate might occur, with the volume of superspace as a prime candidate: d 4 xd 32 E ( x, ) . Z Explicitly integrating out the volume into component fields using the superspace constraints implying the classical field equations would be an ugly task. However, using an on-shell implementation of harmonic superspace together with a superspace implementation of the normal-coordinate expansion, one can in fact evaluate it, but one then finds that the volume vanishes: d 4 xd 32 ✓ E ( x, ✓ )=0 on-shell Z 12 Normal coordinates for a 28+4 split Kuzenko & Tartaglino-Mazzucchelli 2008 G. Bossard, P.S. Howe, K.S.S. & P. Vanhove 2011 One can define normal coordinates ˆ ⇥A = ⇥ = ⇤µui , ⇥¯˙ = ˙ u8 ⇤¯µi˙ ,zr ,z8 ,z8 { µ i 1 µ˙ i 1 r 1} ˆ associated to an involutive set of vector fields E A ˆ which allow for an on-shell harmonic superspace formalism based on the flag manifold S ( U (1) U (6) U (1)) SU (8) . ⇥ ⇥ \ Expanding the superspace Berezinian determinant in these, one finds the flow equation ↵ˆ 1 ↵ ˙ 1 ↵ ↵˙ ˙ 1ij ⇥ ln E = B ˙ ¯ + B ˙ B ¯ ¯ B ˙ =¯ 8ij ↵ˆ 3 ↵ 18 ↵ ↵↵˙ ⇥ ⇥˙ Integrating this, one finds the expansion of the superspace determinant in the four fermionic coordinates ↵ ˆ =( ↵ , ↵ ˙ ) : 1 ↵ ˙ E(ˆx, , ¯)= (ˆx) 1 B ˙ E 6 ↵ However, since this has only✓ ⇣ 2 terms,◆ integration over the four ⇣ ↵ˆ vanishes, so d 4 xd 32 ✓ E ( x, ✓ )=0 . Z 13 1/8 BPS E7 invariant candidate notwithstanding Despite the vanishing of the full N=8 superspace volume, one can nonetheless use an on-shell harmonic superspace formalism to construct a different manifestly E7 -invariant but

1/8 BPS candidate: Bossard, Howe, K.S.S. & Vanhove 1105.6087 8 ↵˙ B =¯1ij I := dµ(8,1,1) B↵˙ B ⇥˙ ⇥˙ 8ij Z At the leading 4-point level, this invariant, of generic ∂⁸R⁴ structure, can be written as a full superspace integral with respect to the linearized N=8 supersymmetry. It cannot, however, be rewritten as a non-BPS full-superspace integral with a duality-invariant integrand at the nonlinear level.

Non-BPS full-superspace and manifestly E7-invariant candidates do exist in any case from 8 loops onwards.

Howe & Lindstrom 1981 Kallosh 1981 14 Current outlook for maximal supergravity So far, things seem under control for maximal supergravity from a purely field-theoretic analysis: what is prohibited does not occur, and what is not prohibited has occurred, as far as one can see.

As far as one knows, the first acceptable D=4 counterterm for maximal supergravity occurs at L=7 loops (Δ = 16); if not that, then they clearly exist at L=8 loops (Δ = 18) and beyond. Howe & Lindstrom 1981 Kallosh 1981 The current divergence expectations for maximal supergravity are consequently:

Dimension D 11 10 8 7 6 5 4 Loop order L 2 2 1 2 3 6 7 1 1 1 1 BPS degree 0 0 2 4 8 0 8 Gen. form 12R4 10R4 R4 4R4 6R4 12R4 8R4 Blue: known divergences Green: anticipated divergences D E11 D(11 D)(R) KD E11 D(11 D)(Z) 15 10A R+ 1 1 10B Sl(2, R) SO(2) Sl(2, Z) 9 Sl(2, R) R+ SO(2) Sl(2, Z) ⇥ 8 Sl(3, R) Sl(2, R) SO(3) SO(2) Sl(3, Z) Sl(2, Z) ⇥ ⇥ ⇥ 7 Sl(5, R) SO(5) Sl(5, Z) 6 SO(5, 5, R) SO(5) SO(5) SO(5, 5, Z) ⇥ 5 E6(6)(R) USp(8) E6(6)(Z) 4 E7(7)(R) SU(8)/Z2 E7(7)(Z) 3 E8(8)(R) SO(16) E8(8)(Z)

D 4 + n(32 D n) fn()=0 (1) D 2 ⇥

3 The N=4 Supergravity L=3 surprise Not everything is perfect in the understanding of supergravity divergences, however. A surprize has occurred in an unexpected sector: D=4, N=4 supergravity at L=3. The expected 3-loop R⁴ divergence (Δ=8) does not occur

in that theory. Bern, Davies, Dennen & Huang 2012 Yet, the L=7 candidate counterterm of N=8 supergravity has a natural analogue here as a 1/4 BPS (4,1,1) G- 4 ↵˙ 1 analytic invariant: I = dµ(4,1,1) B↵˙ B B↵˙ = ↵¯˙4 Expanding the content ofZ this N=4 invariant at linearized level, one finds a leading R⁴ structure undressed by the SL (2 , R ) /U (1) complex scalar field: it is perfectly duality invariant, just like the 1/8 BPS candidate 7-loop N=8 counterterm. Bossard, Howe, K.S.S. & Vanhove 1105.6087 22 Some aspects of this N=4 case:

Marcus 1985 There are anomalies at one loop in the U(1) R-symmetry. These eventually destroy the SL(2, ℝ) duality symmetry. Happily, these anomalies do not yet affect the structure of the 3-loop divergences, for which the requirement of duality invariance still holds. Bossard, Howe & K.S.S.1304.7753

Tourkine & Vanhove 2012 Genus-1 and genus-2 asymmetric-orbifold string calculations likewise show that R⁴ divergences do not appear in N=4 supergravity models coupled to 4≤nᵥ≤22 vector multiplets. Note that such matter-coupled models are already divergent at L=1, so there are subdivergence Fischler 1979 subtractions to worry about, but the absence of R⁴ divergences at D=4, L=3 is nonetheless confirmed.

17 Vanishing volumes and their consequences Another aspect of this story needs to be clarified. The vanishing of a superspace volume can open the door to another representation of candidate counterterms. Consider the cases where superspace volumes vanish on- shell: The full superspace volumes of all D=4 pure supergravities vanish, for any extension N of supersymmetry. In D=5, the volume of maximal (32 supercharge) supergravity does not vanish, but the volume of half- maximal (16 supercharge, i.e. N=2, D=5) supergravity does.

18 Half-maximal D=5, L=2 Bern, Davies, Dennen & Huang 2012 Unitarity-based calculations in D=5 half-maximal supergravity show cancellation of R⁴ divergences at the 2-loop level similar to those found in half-maximal D=4, L=3.

This cancellation is equally surprising as in the N=4, D=4 case, because there is an available 1/4 BPS D=5 (4,1) G-analytic Sp(2)/(U(1)×Sp(1)) counterterm:

↵ 1 1 1 1 dµ(4,1)⌦ ⌦ ↵ Z ⇣ ⌘ where ⌦ ↵ is the D=5 Lorentz Sp(1,1) symplectic matrix.

Moreover, in D=5 there are no complications from anomalies to the “duality” shift symmetry for the single scalar φ of half- maximal D=5 supergravity, unlike in the D=4, N=4 case.

19 The vanishing volume of half-maximal D=5 supergravity invites another way to write a candidate Δ=8 counterterm in D=5. One can write simply

4 16 I 0 = d ✓E where Φ is the D=5 field-strengthZ superfield containing the scalar φ as its lowest component field. Also, this candidate is clearly invariant under the rather minimalistic D=5 duality symmetry Φ → Φ + constant, since d 16 ✓ E =0 . Z Moreover, this candidate turns out to be just a rewriting of the above (4,1) G-analytic manifestly duality invariant 1/4 BPS candidate counterterm. In this sense, the D=5 Δ=8 (4,1) R⁴ counterterm is of marginal F/D type. G. Bossard, P.S. Howe & K.S.S., 1212.0841, 1304.7753 20 The D=4 (4,1,1) G-analytic counterterm has the same marginal F/D character.

The D=4, N=4 theory has as its lowest-dimension physical component a complex scalar field τ taking its values in the Kähler space SL(2,ℝ)/U(1). In terms of τ, the Kähler potential is K[⌧]= ln(Im[⌧]) and the N=4, Δ=8 (4,1,1) counterterm can equally well be written d16✓EK[⌧] Z As in the D=5 case, although this full-superspace integral is duality invariant, its integrand is not duality invariant. The integrand varies as follows: (E ln(Im[⌧])) = 2hE + fE(⌧ +¯⌧)

21 Superspace nonrenormalization theorems: refinement of the duality-invariance requirement

G. Bossard, P.S. Howe & K.S.S., 1212.0841, 1304.7753

The marginal F/D structure of the Δ=8 counterm candidates in half-maximal D=4 and D=5 supergravities requires a more careful treatment of the Ward identies for duality. If one makes the assumption that there exist off-shell full 16- supercharge superfield formulations for the half-maximal theories, then one can derive a stronger invariance requirement: not only must the integrated counterterm be duality invariant, but also the counter-Lagrangian superfield integrand must itself be duality invariant.

22 Off-shell half-maximal supergravity From the point of view of field-theoretic nonrenormalization theorems, a key question is whether there exists an off-shell linearly realised formulation of half-maximal supergravity. If so, then the nonrenormalization theorem would require a full- superspace ∫ d16θ integral with a duality-invariant integrand, thus ruling out the F/D marginal D=4 and D=5 R⁴ counterterms. Unfortunately, the answer to this question is not currently known. But there is a closely related off-shell formulation for linearized D=10, N=1 supergravity, with a finite number of component fields: Howe, Nicolai & Van Proeyen 1982 1 = V V V D¯ DS : D16, @D14, etc. L10 2 abc abc,def def abc abc Upon dimensional reduction to D=4, the N=1, D=10 theory yields D=4, N=4 supergravity plus 6 N=4 super-Maxwell multiplets. So one has something close to the required formalism, at least in the

linearised theory. 23 The most recent developments The divergences in half-maximal supergravity in D=4 and D=5 have now been calculated up through 3 loops including also nᵥ “matter” vector multiplets. D=4 result: a mixture of 1/ε and 1/ε² divergences. Some of these will be consequences of known divergences from matter inclusion at L=1, so the implications of this result are still under debate. More clear is the D=5 situation: in addition to the cancellations for nᵥ = 0 (pure half-maximal supergravity), there is just one other case with 2-loop cancellations: nᵥ = 5, giving precisely the dimensionally reduced content of off-shell D=10, N=1 supergravity. Out just today: pure half-maximal supergravity at L=4 loops has divergences ~ ∂² R⁴. In D=4, there is a clearly expected duality 16 i j invariant full superspace counterterm d ✓ E ¯ i ¯ j . End of the miracles story? Maybe not: the one-loopZ U(1) anomaly can now be causing 4-loop divergences. So is this result purely caused by the anomaly? Bern, Davies & Dennen, 1305.4876; Bern, Davies, Dennen, Smirnov & Smirnov, out today 24