N=4 and N=8 Supergravity Counterterms

K.S. Stelle

Workshop on the Breaking of Supersymmetry and Ultraviolet Divergences in Extended Supergravity

INFN FrascatiThursday, March 28 2013

1

G. Bossard, P.S. Howe & K.S.S. 0901.4661, 0908.3883, 1009.0743G. Bossard, P.S. Howe, K.S.S. & P. Vanhove 1105.6087G. Bossard, P.S. Howe & K.S.S., 1212.0841G. Bossard, P.S. Howe & K.S.S., in preparation

March 28, 2013

2

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)

March 28, 2013

Determining whether one has ever encountered a real “miracle”:the Piazza dei Miracoli, Pisa

March 28, 2013

Outline

Power counting and the originally anticipated R⁴ type divergences versus the unitarity-based calculations.Algebraic renormalization, ectoplasm and constraints from duality invarianceVanishing superspace volumesThe N=4 supergravity L=3 surpriseMarginal F/D invariants and refinement of the duality invariance requirements.The current situation and suggestions for further tests.

4March 28, 2013

Ultraviolet Power Counting in Gravity

Simple power counting in gravity and supergravity theories leads to a naïve degree of divergence

in D spacetime dimensions. So, for D=4, L=3, one expects Δ=8. In dimensional regularization, only logarithmic divergences are seen (1/ε poles, ε=D-4), so 8 powers of momentum would have to come out onto the external lines of such a diagram.

8

Figure 11. A sample diagram whose divergencepart would need to be evaluated in order to deter-mine the ultra-violet divergence of a supergravitytheory. The lines represent graviton propagatorsand the vertices three-graviton interactions.

ready been used to show that at least for the caseof maximally supersymmetric gravity the onset ofdivergences is delayed until at least five quantumloops [49,50].

4. STATUS OF LOOP CALCULATIONS

Before surveying the main advance since thelast ICHEP conference, it is useful to survey thestatus of quantum loop calculations. Here we donot discuss tree-level calculations which have alsoseen considerable progress over the years.

4.1. Status of one-loop calculations

In 1948 Schwinger dealt with one-loop three-point calculations [18] such as that of the anoma-lous magnetic moment of leptons described inSection 2. It did not take very long be-fore Karplus and Neuman calculated light-by-light scattering in QED in their seminal 1951paper [51]. In 1979 Passarino and Veltman pre-sented the first of many systematic algorithms fordealing with one-loop calculations with up to fourexternal particles, leading to an entire subfield de-voted to such calculations. Due to the complexityof non-abelian gauge theories, however, it was notuntil 1986 that the first purely QCD calculationinvolving four external partons was carried out inthe work of Ellis and Sexton [52].

The first one-loop five-particle scattering am-plitude was then calculated in 1993 by LanceDixon, David Kosower and myself [53] for thecase of five-gluon scattering in QCD. This wasfollowed by calculations of the other five-pointQCD subprocesses [54], with the associated phys-

ical predictions of three-jet events at hadron col-liders appearing somewhat later [55,56]. A num-ber of other five-point calculations have also beencompleted. One example of a state-of-the-art five-point calculation was presented in a parallel ses-sion by Doreen Wackeroth [57], who described thecalculation of pp ! ttH at next-to-leading orderin QCD [58]. This process is a useful mode fordiscovering the Higgs boson as well as measure-ment of its properties. Other examples are NLOcalculations for e+e! ! 4 jets [59,60,61], Higgs+ 2 jets [62], and vector boson + 2 jet produc-tion [59,63], which is also important as a back-ground to the Tevatron Higgs search, if the jetsare tagged as coming from b quarks.

Beyond five-external particles, the only calcu-lations have been in special cases. By makinguse of advanced methods, for special helicity con-figurations of the particles, infinite sequences ofone-loop amplitudes with an arbitrary numberof external particles but special helicity configu-rations have been obtained in a variety of the-ories [39,40]. For the special case of maximalsupersymmetry, six-gluon scattering amplitudeshave been obtained for all helicities [40]. Therehas also been a recent calculation of a six-pointamplitude in the Yukawa model [64], as well as re-cent papers describing properties of six-point in-tegrals [65]. These examples suggest that that thetechnical know-how for computing general six-point amplitudes is available, though it may bea rather formidable task to carry it through. Ane!cient computer program for dealing with up tothree jets at hadron colliders now exists [56], sug-gesting that it would be possible add one morejet, once the relevant scattering amplitudes arecalculated. This would then give a much bet-ter theoretical handle on multi-jet production athadron colliders.

4.2. Status of Higher Loop Computations

Over the years, an intensive e"ort has goneinto calculating higher loop Feynman diagrams.A few samples of some impressive multi-loop cal-culations are:

• The anomalous magnetic moment of lep-tons, already described in Section 2.

D = (D�2)L+2

5March 28, 2013

It has been recognized since the earliest days of supergravity that counterterms would have to be invariant under local supersymmetry. The dangerous counterterms are those that do not vanish subject to the classical equations of motion.

Local supersymmetry implies that the pure-curvature part of such a D=4, 3-loop divergence candidate must be built from the square of the Bel-Robinson tensor

Deser, Kay & K.S.S 1977Z p�gTµnrsT µnrs , Tµnrs = Rµ

an

bRrasb + ⇤Rµa

nb ⇤Rrasb

6March 28, 2013

An important constraint on allowable counterterms is obtained whenever one has a linearly realised “off-shell” formulation of some portion of a theory’s supersymmetry. At loop orders L≥2, there is a standard non-renormalization theorem requiring counterterms to be written as full-superspace integrals with respect to the off-shell linearly realized supersymmetry.

For the maximal supergravity theory, it is anticipated that at least 1/2 of the 32 generators can be linearly realised in this way, thus requiring maximal supergravity counterterms to be written at least as ∫ d16θ superspace integrals.

7March 28, 2013

The 3-loop R⁴ maximal supergravity candidate counterterm has a structure under linearized supersymmetry that is very similar to that of an F⁴ N=4 super Yang-Mills invariant. Both of these are 1/2 BPS invariants, involving integration over just half the corresponding full superspaces:

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 counterterms exists and is consistent with all symmetries.

8

6 of SU(4)

70 of SU(8)DISG =Z

(d8qd8q)232848(W 4)232848

DISY M =Z

(d4qd4q)105 tr(f4)105 105

232848

fi j

Wi jkl

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6? 5?

BPS degree 0 0 12

14

18 0 1

4

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �4R4

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6 7

BPS degree 0 0 12

14

18 0 0

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �8R4

3

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6? 5?

BPS degree 0 0 12

14

18 0 1

4

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �4R4

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6 7

BPS degree 0 0 12

14

18 0 0

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �8R4

3

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 4 5Gen. form �12R4 �10R4 R4 �4R4 �6R4 �6R4 �4R4

Dimension D 10 8 7 6 5 4Loop order L 1 1 2 3 6? ⇥BPS degree 1

412

14

14

14

14

Gen. form �2F 4 F 4 �2F 4 �2F 4 �2F 4 finite

�ISYM =

�d4x(d4�d4�)105 tr(⇥

4)105 105 �

�ISG =

�d4x(d8�d8�)232848(W

4)232848 232848 �

2

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6? 5?

BPS degree 0 0 12

14

18 0 1

4

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �4R4

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6 7

BPS degree 0 0 12

14

18 0 0

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �8R4

3

Howe, K.S.S. & Townsend 1981Kallosh 1981

March 28, 2013

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 the super Yang-Mills and supergravity divergence onset.

plus 46 more topologies

Unitarity-based calculations

Max. SYM first divergences, current lowest possible orders (for integral spacetime dimensions).

Max. supergravity first divergences, current lowest possible orders (for integral spacetime dimensions).

Bern, Carrasco, Dixon, Johansson, Roiban et al. 2007 ... 2011

9

Blue: known divergences

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

Dimension D 10 8 7 6 5 4

Loop order L 1 1 2 3 6? 1BPS degree

14

12

14

14

14

14

Gen. form �2F 4 F 4 �2F 4 �2F 4 �2F 4finite

Dimension D 11 10 8 7 6 5 4

Loop order L 2 2 1 2 3 6? 5?

BPS degree 0 0

12

14

18 0

14

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �4R4

Dimension D 11 10 8 7 6 5 4

Loop order L 2 2 1 2 3 6 7

BPS degree 0 0

12

14

18 0 0

Gen. form �12R4 �10R4 R4 �6R4 �6R4 �12R4 �8R4

2

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 4 5Gen. form �12R4 �10R4 R4 �4R4 �6R4 �6R4 �4R4

Dimension D 10 8 7 6 5 4Loop order L 1 1 2 3 6? �BPS degree 1

412

14

14

14

14

Gen. form �2F 4 F 4 �2F 4 �2F 4 �2F 4 finite

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6? 5?

BPS degree 0 0 12

14

18 0 1

4

Gen. form �12R4 �10R4 R4 �4R4 �6R4 �12R4 �4R4

2

March 28, 2013

The construction of supersymmetric invariants is isomorphic to the construction of cohomologically nontrivial closed forms in superspace: is invariant (where is a pull-back to a section of the projection map down to the purely bosonic “body” subspace M0) if is a closed form in superspace, and it is nonvanishing only if is nontrivial.Using the BRST formalism, one can handle all gauge symmetries including space-time diffeomorphisms by the nilpotent BRST operator s. The invariance condition for is , where is the usual bosonic exterior derivative. Since and s anticommutes with , one obtains using Poincaré’s lemma , etc.

Algebraic Renormalization and EctoplasmDixon; Howe, Lindstrom & White; Piguet & Sorella; Hennaux; Stora;Baulieu & Bossard; Voronov 1992; Gates, Grisaru, Knut-Whelau, & Siegel 1998Berkovits and Howe 2008; Bossard, Howe & K.S.S. 2009

10

LD

LD

LD

d0

s2 = 0

I =�

M0��LD ��

d0

sLD + d0LD�1 = 0

sLD�1 + d0LD�2 = 0March 28, 2013

Solving the BRST Ward identities thus becomes a cohomological problem for the cocycle associated to a counterterm. One needs to study the cohomology of the nilpotent operator , whose components are (D-q) forms with ghost number q, i.e. (D-q) forms with q spinor indices. The spinor indices are totally symmetric since the supersymmetry ghost is commuting.

For gauge-invariant supersymmetric integrands, this establishes an isomorphism between the cohomology of closed forms in superspace (aka “ectoplasm”) and the construction of BRST-invariant counterterms.

� = s + d0 LD�q,q

11

Superspace cohomology:Bonora, Pasti & Tonin 1987

Cederwall, Nilsson & Tsimpis 2002Howe & Tsimpis 2003

Berkovits & Howe 2008

March 28, 2013

Counterterm cohomology then allows one to derive non-renormalization theorems: the cocycle structure of a candidate counterterm and its associated operators must match that of the classical action.

In maximal SYM, this leads to a non-renormalization theorem ruling out the F⁴ counterterm that was otherwise expected at L=4 in D=5.

Similar non-renormalization theorems exist in supergravity, but their study is complicated by local supersymmetry and the density character of counterterm integrands.

Cohomological non-renormalization theorem

12March 28, 2013

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

covariant (but non-manifestly Lorentz covariant) formalism. 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.

Duality invariance constraints

Marcus 1985

Bossard, Hillman & Nicolai 2010

13

Other approach to duality analysis from string amplitudes: Broedel & Dixon 2010Elvang & Kiermeier 2010; Beisert, Elvang, Freedman, Kiermaier, Morales & Stieberger 2010

March 28, 2013

In a curved superspace, an invariant is constructed from the top (pure “body”) component in a coordinate basis:

Referring this to a preferred “flat” basis and identifying components with vielbeins and gravitinos, one has, e.g. in D=4

Thus the “soul” components of the cocycle also contribute to the local supersymmetric covariantization.

Since the gravitinos do not transform under the D=4 E7 duality, the LABCD form components have to be separately duality invariant. 14

rigid E7(7), the measure will be E7(7) invariant whereas the integrand will necessarily transformnon-trivially with respect to E7(7). It would then follow that the ⌅6R4 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 higherdimensions (such as R4 in D = 8, ⌅4R4 in D = 7 and ⌅6R4 in D = 6), since the BPS invariantsare not unique in dimensions D > 5.

The non-existence of harmonic measures for the 1/2 and the 1/4 BPS invariants is not incontradiction 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 writtenas 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 a�orded by the ectoplasmformalism [28, 29, 30]. In D-dimensional spacetime, consider a closed super-D-form, LD, in thecorresponding superspace. The integral of the purely bosonic part of this form over spacetimeis then guaranteed to be supersymmetric by virtue of the closure property. Moreover, if LD isexact it will clearly give a total derivative so that we are really interested in the Dth superspacecohomology group. As we have seen in the preceding section, one cannot define a harmonicmeasure for every invariant, and in particular, not for the 1/2 and 1/4 BPS invariants in N = 8supergravity. However, according to the algebraic Poincare Lemma, 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 oddparts. Thus a (p, q)-form is a form with p even and q odd indices, totally antisymmetric on theformer and totally symmetric on the latter. The exterior derivative can likewise be decomposedinto parts with di�erent bi-degrees,

d = d0 + d1 + t0 + t1 , (13)

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

t20 = 0; d1t0 + t0d1 = 0; d21 + t0d0 + d0t0 = 0 . (14)

The first of these equations allows us to define t0-cohomology groups, Hp,qt [31], and the other two

allow us to introduce the spinorial derivative ds which maps Hp,qt to Hp,q+1

t by ds[⇥p,q] = [d1⇥p,q],where the brackets denote Ht cohomology classes, and which also squares to zero [32, 33].The point of this is that one can often generate closed super-D-forms from elements of thesecohomology groups.

In the context of curved superspace it is important to note that the invariant is constructedfrom the top component in a coordinate basis,

I =1

D!

�dDx ⇤mD...m1 EmD

AD · · ·Em1A1 LA1...AD(x, � = 0) . (15)

6

One transforms to a preferred basis by means of the supervielbein EMA. At ⇥ = 0 we can

identify Eam with the spacetime vielbein ema and Em

� with the gravitino field ⇤m� (where �

includes both space-time �, � and internal i indices for N = 8). In four dimensions, we thereforehave

I =1

24

⇤ �ea�e

b�e

c�e

d Labcd + 4ea�eb�e

c�⇤

�Labc� + 6ea�eb�⇤

��⇤

⇥ Lab�⇥

+4ea�⇤��⇤

⇥�⇤

⇤La�⇥ ⇤ + ⇤��⇤

⇥�⇤

⇤�⇤

⌅L�⇥ ⇤⌅

⇥. (16)

By definition, each component Labcd, Labc�, Lab�⇥ , La�⇥ ⇤ , L�⇥ ⇤⌅ is supercovariant at ⇥ = 0.This is a useful formula because one can directly read o� the invariant in components in thisbasis.

In N = 8 supergravity, all the non-trivial t0-cohomology classes lie in Ht0,4. Invariants are

therefore completely determined by their (0, 4) components L�⇥ ⇤⌅, and all non-trivial L0,4 sat-isfying [d1L0,4] = 0 in t0-cohomology define non-trivial invariants. Ht

0,4 is the set of functionsof fields in the symmetric tensor product of four 2 � 8 ⇥ 2 � 8 of SL(2,C) � SU(8) withoutSU(8) contractions (since such functions would then be t0-exact). Because of the reducibility ofthe representation, it will be convenient to decompose L�⇥ ⇤⌅ into components of degree (0, p, q)(p+ q = 4) with p 2� 8 and q 2� 8 symmetrised indices.

We will classify the elements of Ht0,4 into three generations.2 The first generation corresponds

to elements that lie in the antisymmetric product of four 2 � 8 ⇥ 2 � 8 of SL(2,C) � SU(8),and can therefore be directly related to the top component L4,0 through the action of thesuperderivatives. We will write M0,p,q for the corresponding components of a given L0,4. Theylie in the following irreducible representations of SL(2,C)� SU(8):

M0,4,0 : [0, 0|0200000]M0,3,1 : [1, 1|1100001]M0,2,2 : [2, 0|2000010]

M0,0,4 : [0, 0|0000020]M0,1,3 : [1, 1|1000011]M0,2,2 : [0, 2|0100002] .

(17)

In order to understand the constraints that these functions must satisfy in order for L0,4 tosatisfy the descent equation

[d1L0,4] = 0 , (18)

it is useful to look at the possible representations of d1L0,4 which define Ht0,5 cohomology classes

in general, without assuming any a priori constraint. We will split d1 = d1,0 + d0,1 according tothe irreducible representations of SL(2,C)� SU(8). One computes that

[d1,0M0,4,0] : [1, 0|1200000][d0,1M0,4,0] : [0, 1|0200001][d1,0M0,3,1] : [0, 1|0200001]⇥ [2, 0|2100001][d0,1M0,3,1] : [1, 0|1100010]⇥ [1, 2|1100002][d1,0M0,2,2] : [1, 0|1100010]⇥ [3, 0|3000010][d0,1M0,2,2] : [2, 1|2000011]

[d0,1M0,0,4] : [0, 1|0000021][d1,0M0,0,4] : [1, 0|1000020][d0,1M0,1,3] : [1, 0|1000020]⇥ [0, 2|1000012][d1,0M0,3,1] : [0, 1|0100011]⇥ [2, 1|2000011][d0,1M0,2,2] : [0, 1|0100011]⇥ [0, 3|0100003][d1,0M0,2,2] : [1, 2|1100002] .

(19)

2We will avoid discussing the elements of Ht0,4 of degree (0, 2, 2) in the [0, 0|0200020] representation, which do

not play any role.

7

One transforms to a preferred basis by means of the supervielbein EMA. At ⇥ = 0 we can

identify Eam with the spacetime vielbein ema and Em

� with the gravitino field ⇤m� (where �

includes both space-time �, � and internal i indices for N = 8). In four dimensions, we thereforehave

I =1

24

⇤ �ea�e

b�e

c�e

d Labcd + 4ea�eb�e

c�⇤

�Labc� + 6ea�eb�⇤

��⇤

⇥ Lab�⇥

+4ea�⇤��⇤

⇥�⇤

⇤La�⇥ ⇤ + ⇤��⇤

⇥�⇤

⇤�⇤

⌅L�⇥ ⇤⌅

⇥. (16)

By definition, each component Labcd, Labc�, Lab�⇥ , La�⇥ ⇤ , L�⇥ ⇤⌅ is supercovariant at ⇥ = 0.This is a useful formula because one can directly read o� the invariant in components in thisbasis.

In N = 8 supergravity, all the non-trivial t0-cohomology classes lie in Ht0,4. Invariants are

therefore completely determined by their (0, 4) components L�⇥ ⇤⌅, and all non-trivial L0,4 sat-isfying [d1L0,4] = 0 in t0-cohomology define non-trivial invariants. Ht

0,4 is the set of functionsof fields in the symmetric tensor product of four 2 � 8 ⇥ 2 � 8 of SL(2,C) � SU(8) withoutSU(8) contractions (since such functions would then be t0-exact). Because of the reducibility ofthe representation, it will be convenient to decompose L�⇥ ⇤⌅ into components of degree (0, p, q)(p+ q = 4) with p 2� 8 and q 2� 8 symmetrised indices.

We will classify the elements of Ht0,4 into three generations.2 The first generation corresponds

to elements that lie in the antisymmetric product of four 2 � 8 ⇥ 2 � 8 of SL(2,C) � SU(8),and can therefore be directly related to the top component L4,0 through the action of thesuperderivatives. We will write M0,p,q for the corresponding components of a given L0,4. Theylie in the following irreducible representations of SL(2,C)� SU(8):

M0,4,0 : [0, 0|0200000]M0,3,1 : [1, 1|1100001]M0,2,2 : [2, 0|2000010]

M0,0,4 : [0, 0|0000020]M0,1,3 : [1, 1|1000011]M0,2,2 : [0, 2|0100002] .

(17)

In order to understand the constraints that these functions must satisfy in order for L0,4 tosatisfy the descent equation

[d1L0,4] = 0 , (18)

it is useful to look at the possible representations of d1L0,4 which define Ht0,5 cohomology classes

in general, without assuming any a priori constraint. We will split d1 = d1,0 + d0,1 according tothe irreducible representations of SL(2,C)� SU(8). One computes that

[d1,0M0,4,0] : [1, 0|1200000][d0,1M0,4,0] : [0, 1|0200001][d1,0M0,3,1] : [0, 1|0200001]⇥ [2, 0|2100001][d0,1M0,3,1] : [1, 0|1100010]⇥ [1, 2|1100002][d1,0M0,2,2] : [1, 0|1100010]⇥ [3, 0|3000010][d0,1M0,2,2] : [2, 1|2000011]

[d0,1M0,0,4] : [0, 1|0000021][d1,0M0,0,4] : [1, 0|1000020][d0,1M0,1,3] : [1, 0|1000020]⇥ [0, 2|1000012][d1,0M0,3,1] : [0, 1|0100011]⇥ [2, 1|2000011][d0,1M0,2,2] : [0, 1|0100011]⇥ [0, 3|0100003][d1,0M0,2,2] : [1, 2|1100002] .

(19)

2We will avoid discussing the elements of Ht0,4 of degree (0, 2, 2) in the [0, 0|0200020] representation, which do

not play any role.

7

Supergravity Densities

March 28, 2013

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.

Although the pure-body (4,0) component 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) 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.

15

Labcd

Howe, K.S.S. & Townsend 1981

L↵���

Bossard, Howe & K.S.S. 2010

March 28, 2013

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: .

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: on-shell

L=7 and Vanishing Volume

Zd4xd32�E(x, �)

Bossard, Howe, K.S.S. & Vanhove 2011

16

Zd

4xd

32✓E(x, ✓) = 0

Kuzenko & Tartaglino-Mazzucchelli 2008

March 28, 2013

N=8 supergravity has a natural SU(8) R-symmetry group

under which the 8 gravitini transform in the 8 representation. In (8,1,1) harmonic superspace, one augments the normal superspace coordinates by an additional set of bosonic coordinates parametrizing the flag manifold

Contracting the usual superspace basis vectors with these and their inverses, one has

Then one works just with manifest U(1)xU(6)xU(1) covariance.

(xµ, �i�)

uIj I = 1; r = 2, . . . , 7; 8

17

E�I = ui

IE�i

E�I = uIiE

�i

Hartwell & Howe 1994

S(U(1)⇥ U(6)⇥ U(1))\SU(8)

March 28, 2013

Combining these with the dIJ vector fields on the harmonic flag manifold, one finds that the subset

is in involution:

One can then define Grassmann-analytic superfields annihilated by the dual superspace derivatives

Some non-vanishing curvatures are

where is Grassmann-analytic.

EA := {E1�, E� 8, d

1r, d

r8, d

18} , 2 � r � 7

{EA, EB} = CABC EC

D↵ 1 , D8↵

R1 1�⇥8, 1

= R1 8�⇥8, 8

= �B�⇥

B�⇥ = �1ij

⇥�� 8ij

18March 28, 2013

One can define normal coordinates

associated to the vector fields .Expanding the superspace Berezinian determinant in these, one finds the flow equation

Integrating, one finds the expansion of the superspace determinant in the four fermionic coordinates :

However, since this has only terms, integration over the four vanishes.

Normal coordinates for a 28+4 split

⇥A = {⇥� = ��µ⇤µi u

i1 , ⇥

� = ��µu8i ⇤

µ i, zr1, z8r, z

81}

EA

�↵⇥↵ lnE = �1

3B↵��

↵� � +1

18B↵�B↵↵�

↵�� �↵� �

�↵ = (�↵ , �↵)

E(x, �, �) = E(x)✓1� 1

6B↵��

↵� �◆

⇣↵

19

Kuzenko & Tartaglino-Mazzucchelli 2008

⇣2

March 28, 2013

1/8 BPS E7 invariant candidate notwithstandingDespite 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:

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.

I8 :=

Zdµ(8,1,1) B↵� B

↵�

20

Bossard, Howe, K.S.S. & Vanhove 2011

B�⇥ = �1ij

⇥�� 8ij

March 28, 2013

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.

Yet, the L=7 candidate counterterm of N=8 supergravity has a natural analogue here as a 1/4 BPS (4,1,1) G-analytic invariant:Expanding the content of this N=4 invariant at linearized level, one finds a leading R⁴ structure undressed by the complex scalar field: it is perfectly duality invariant, just like the 1/8 BPS candidate 7-loop N=8 counterterm.

The N=4 Supergravity L=3 surprise

Bern, Davies, Dennen & Huang 2012

I4 =

Zdµ(4,1,1) B↵� B

↵�

SL(2,R)/U(1)

Bossard, Howe, K.S.S. & Vanhove 2011

22

B↵� = �1↵��4

March 28, 2013

Analysis of the N=4 supergravity theory presents a particular difficulty, however: its SL(2,ℝ) duality symmetry is anomalous.

However, writing an infinitesimal sl(2,ℝ) transformation as

acting on the complex scalar τ=a+ie-2φ of the N=4 theory as δτ = e + 2hτ - fτ², one finds, upon detailed study of the anomaly’s cohomology up to L=3 loop order, that only the f generator is actually broken. So the parabolic subgroup generated by e and h remains unbroken to this order.Up to the 3-loop order, this is sufficient to require the available counterterms to be fully duality invariant.

Consequently, one is back to considering just the R⁴ type SL(2,ℝ) invariant counterterm

22

✓h ef �h

◆

I4 =

Zdµ(4,1,1)"

↵�"↵��1↵�

1��↵4��4

Marcus 1985

March 28, 2013

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.First, consider the cases where superspace volumes vanish on-shell, using techniques like those outlined above:

The full superspace volumes of all D=4 pure supergravities vanish, for any N supersymmetry extension.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.

23

Vanishing volumes and their consequences

March 28, 2013

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.

Note, however, that the field content of the D=5 and D=4 half-maximal theories are different: the D=5 theory dimensionally reduces to N=4, D=4 supergravity plus a single N=4, D=4 super-Maxwell multiplet.

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

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.

Bern, Davies, Dennen & Huang 2012

Half-maximal D=5, L=2

24

⌦↵�

Zdµ(4,1)⌦

↵�⌦��⇣�1↵�

1��

1��

1�

⌘

March 28, 2013

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

where Φ is the D=5 field-strength 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 .

Moreover, this candidate turn out to be just a rewriting of the above (4,1) G-analytic manifestly duality invariant candidate counterterm.

In this sense, the D=5 Δ=8 (4,1) R⁴ counterterm is of marginal F/D type.

25

I40 =

Zd16✓E�

Zd16✓E = 0

March 28, 2013

The D=4 (4,1,1) G-analytic counterterm has the same marginal F/D character.

The D=4, N=4 theory has as 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 and the N=4, Δ=8 (4,1,1) counterterm can equally well be written

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:

26

Zd16✓EK[⌧ ]

K[⌧ ] = �ln(Im[⌧ ])

� (E ln(Im[⌧ ])) = 2hE + fE(⌧ + ⌧)

March 28, 2013

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 requirement for duality invariance: not only must the integrated counterterm be invariant, but also the counter-Lagrangian superfield integrand must itself be duality invariant.Proof of this refined theorem requires introduction of the notion of a chain of superspace co-forms arising from the duality variation of the Lagrangian density. In order for the duality Ward identities to be satisfied, the whole chain of co-forms must be renormalised consistently as a single cohomology class.

27

Superspace nonrenormalization theorems:refinement of the duality invariance requirement

March 28, 2013

An analogy to the duality invariance requirement for counter-Lagrangian integrands can be found for N=(2,2) D=2 non-linear sigma models, based on chiral superfields Ta.The sigma-model action is given by the full superspace integral where the Lagrangian integrand is the target-space Kähler potential.Although the classical superspace Lagrangian integrand is not itself a globally defined scalar, the N=(2,2) nonrenormalization theorem requires all counterterms to have integrands that are globally defined scalars.Sigma models with isometries thus require counter-Lagrangian integrands that are isometrically invariant.

28

I� =

Zd

2xd

4✓K(Ta, Tb) K(Ta, Tb)

I

ct� =

Zd

2xd

4✓S(Ta, Tb)

S(Ta, Tb)

Nonrenormalization analogy: the N = (2,2) sigma model

Howe, Papadopoulos & K.S.S. 1986

March 28, 2013

Off-shell half-maximal supergravityFrom the point of view of field-theoretic nonrenormalization theorems, the 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 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:

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. Pure N=4 SG undoubtedly requires a harmonic superspace formulation. 29

Howe, Nicolai & Van Proeyen 1982

L10 =1

2Vabc�abc,defVdef � VabcD�abcDS � : D16, @D14, etc.

March 28, 2013

So far, things are under control for maximal supergravity from a purely field-theoretic analysis: what is prohibited does not occur, and what is not prohibited has occured, as far as one can see. So Murray Gell-Mann can relax.

As far as one knows, the first acceptable D=4 counterterm for maximal supergravity still occurs at L=7 loops (Δ = 16).

The current divergence expectations for maximal supergravity are consequently:

Current outlook

Blue: known divergences Green: anticipated divergences30

Dimension D 11 10 8 7 6 5 4Loop order L 2 2 1 2 3 6 7

BPS degree 0 0 12

14

18 0 1

8

Gen. form �12R4 �10R4 R4 �4R4 �6R4 �12R4 �8R4

D E11�D(11�D)(R) KD E11�D(11�D)(Z)10A R+ 1 110B 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

D � 2n(32�D � n)

⇥fn(�) = 0 (1)

3

March 28, 2013

The situation for half-maximal supergravity is somewhat fragile. Given the assumption that there exists a full 16-supercharge off-shell formalism, the marginal F/D character would be enough to rule out the Δ=8 R⁴ type counterterm, in accordance with the calculated results.

Inclusion of 16-supercharge vector multiplet matter into the calculated results would strengthen the tests of the hypothesized off-shell structure, as these systems should share their L=3, D=4 or L=2, D=5 finiteness with the pure half-maximal theory.

Clearly, the next strong test for “miracles” will occur for the Δ=10 ∂² R⁴ type anticipated divergences at D=4, L=4.

31

Agreement with string-based predictions of Tourkine & Vanhove 2012

March 28, 2013