COUNTERTERMS AND COHOMOLOGY Introduction Superspace NR thms Algebraic renormalisation - counterterms as cocycles - algebraic susy NR thm Superspace cohomology Max SYM G. Bossard, PH, K. Stelle (BHS) Hidden structures in field theory amplitudes, Copenhagen, August 12-14,

INTRODUCTION Discuss onset of UV divergences in MSYM & MSG from a purely field-theoretic viewpoint. Important to establish whether calculational results can be explained in this way, or not. Arguments based on string theory indicate that the onset of UV divergences may be at L=9 loops in D=4 MSG, whereas it would seem probable that straightforward field theory arguments cannot do better than L=7. Berkovits; Green, Russo, Vanhove Light-cone superfields may offer a different perspective Kallosh 2

New calculational results: MSG is finite at L=4 in D=5. MSYM: there is a double-trace invariant which is finite at L=3 in D=6. Bern, Carrasco, Dixon, Johansson, Roiban These results were unexpected from the point of view that one-half BPS counterterms are protected but not others (one-quarter BPS or non-BPS). BHS 3

In MSYM there are two one-half BPS invariants, corresponding to integrals over 8 odd coordinates (thetas): Next we have invariants with two derivatives as well. The single-trace one turns out to be non-BPS and so non-protected (it is a descendant of Konishi). The double-trace is divergent in D=7 at L=2 loops but finite in D=6 at L=3 loops. Marcus, Sagnotti Why is this? Invariants with four or more derivatives are non-BPS and hence non-protected. 4

SUPERSPACE NR THMS These are straightforward to understand and apply when there is an off-shell superfield formulation available. Eg, N=2 D=4 SYM/matter can be written in N=2, D=4 sspace from which it can be deduced that the only possible divergences arise at one loop. Grisaru, Siegel; PH, Stelle, Townsend But, for max susy, there do not seem to be any off-shell formulations preserving all susies. We can only preserve a fraction, f. In conventional superspace f is ½. The NR thm would then tell us that counterterms should be integrals over f x (16 or 32) thetas. They should also be invariant under the remaining susies as well as gauge transformations and R-symmetries. 5

MSYM: there is an off-shell version with N=3,D=4 susy. Galperin, Ivanov, Kalitsin, Ogievetsky, Sokatchev Naively permits 12-theta (1/4 BPS) counterterms. The problem seems to be in lifting this formulation to D>4 and the effect this has on R and Lorentz symmetries. Off-shell with ½ susy + 1: again we lose control over full Lorentz and R symmetries as well as the remaining susies. In fact, it is not so easy to construct ¼ BPS counterterms in this formalism. Baulieu, Berkovits, Bossard, Martin; BHS Not so easy to control all the symmetries in these formulations. 6

ALGEBRAIC RENORMALISATION Idea is to study the constraints imposed on the 1PI quantum action, Г, by symmetries and dimensional analysis, independently of choice of regulator. For max susy we can work in components. There are various technical problems: susy is non-linear, the algebra closes modulo field equations and gauge transformations, and gauge-fixing is not supersymmetric. These problems can be overcome by means of the BV extension of BRST methods. Need to introduce ghosts, inc for susy, sources for the transformations (anti-fields), etc. Dixon; PH, Lindstrom, White; Baulieu, Bossard, Sorella 7

The classical extended action, ∑, then satisfies the master equation, (∑,∑)=0, and the task is to construct a quantum extension, Γ, satisfying a similar equation. Linearisation leads to a nilpotent BRST operator, s. Any allowed counterterm will be invariant under s. We can relate the β-coefficients associated with the counterterms to the anomalous dimensions of the same expressions considered as operator insertions. The allowed counterterms will be those that can mix with the classical action. BHS 8

Counterterms are BRST invariants. They can be expressed as spacetime integrals of D-forms. Invariance under s gives The nilpotency of s implies and so on. So an invariant determines a cocycle of the operator In the gauge-invariant sector s is essentially the susy BRS operator, so the (D-q,q) term in the cocycle is a spacetime (D-q)-form with q spinor indices, totally symmetrised. So counterterm cocycles give rise to closed D-forms in superspace. 9

We can add each term in the cocycle into the action as part of an extended operator insertion associated with the counterterm. We do this as well for the action itself. The susy Ward identity then implies the algebraic NR thm: Any allowable counterterm must have the same cocycle structure as the action This problem can be analysed systematically using superspace cohomology. 10

SUPERSPACE COHOMOLOGY Flat superspace. Standard bases of invariant forms Dual to these are the derivatives Natural bi-grading of forms The exterior derivative d splits into with the indicated bi-degrees. 11 Bonora, Pasti, Tonin

For a (p,q) form we have Since d is nilpotent we have To study superspace cohomology we can start with the cohomology groups Define a new derivative by with 12

This enables us to define the spinorial cohomology groups We can solve for superspace cohomology groups in terms of spinorial cohomology. We shall see later that there is a relation to pure spinors and the associated cohomology groups. Note that this has nothing to do with topology – the coefficients of the groups we are interested in are restricted to be gauge-invariant functions of the physical fields. 13 Cederwall, Gran, Nilsson, Tsimpis; PH, Tsimpis

Invariants can be constructed from closed super D-forms in a process that has been dubbed ectoplasm. Gates, Grisaru, Knutt-Wehlau, Siegel On a supermanifold M with D-dimensional body M_0 invariants are constructed from closed super D-forms via where, for a general curved M, we are to integrate the purely bosonic component of the closed superform in a coordinate basis. In flat superspace this turns out to be the same thing as the (D,0) component in a preferred frame. In fact, we can identify The forms we are interested in can either be strictly invariant, i.e. all components of L_D are gauge invariant, or they can be of Chern- Simons type, in which case their integrals are invariant. 14

CS invariants can be constructed starting from a closed, invariant (D+1)-form, W, which can be written where Z is a potential D-form. If W can also be written as for some gauge-invariant D-form K then is a closed D-form whose integral will be gauge-invariant. A set of examples of this type of invariant is provided by brane actions. In this case M_0 is the bosonic worldvolume of the brane and M its superspace extension. In this way one can derive Green-Schwarz actions from the superembedding formalism. PH, Raetzel, Sezgin 15

16 To summarise: we want to solve the equation dL=0, for L a super D-form, modulo exact forms. If the lowest non-vanishing cpt (i.e. the one with most odd indices) is L_{p,q}, say, with p+q=D, it must satisfy and thus determines an element ofFurthermore, since L is closed, it must be d_s closed, so we need an element of We can now solve for all of the higher cpts of L in terms of L_{p,q} and its derivatives. If there are other non-trivial t_0-cohomology groups with fewer odd indices then there can be other solutions which would have different cocycle structures. Note that we also require that a candidate lowest cpt L_{p,q} should lead to a non-zero L_{D,0} since otherwise the invariant would be zero.

N=1 D=10 cohomology We start the analysis by looking at t_0 cohomology in D=10. The non-trivial possibilities are related to the existence of scalar branes (string and fivebrane) and the associated gamma-matrix identities For p=0, the cohomology is isomorphic to spaces of q-fold pure spinors. A pure spinor u obeys PH; Tonin; Berkovits For 0<p<6 the cohomology is given by pure spinors with additional antisymmetrised vector indices, and for p>5 it vanishes. E.g. is t_0-closed 17

To construct a Lagrangian form we need to start with a suitable t_0 cohomology group element. In D=10 it turns out that there is only one possible group, which has p=q=5. So the lowest component of any Lagrangian form is given by Berkovits, PH where, since L is closed, A basic example is given by a full superspace integral of a superfield S. Since a closed D-form gives rise to a closed (D-1)-form under dimensional reduction it follows that we know the cocycle structure of full superspace integrals (non-BPS invariants) in MSYM in any D. 18

The action itself is a CS invariant. It can be derived from the 11-form where H is closed. In flat space its only non-zero component is It is easy to show that W=dK, so that we do indeed get a CS invariant. The associated (0,3)-form M is given by the (0,3) component of the Chern-Simons three-form Q, dQ=Tr (FF). We can reduce this to any D greater or equal to 5 straightforwardly. The lowest non-zero cpt of L is L_{D-5,5}. We can therefore conclude that the structure of the cocycle associated with the action is the same as that of the cocycle associated with a full superspace integral, so that any invariant of this type is an allowable counterterm in the algebraic formalism. 19

One-half BPS invariants There are two of these, single and double-trace of F^4. However, there is not much difference between the two from the point of view of supersymmetry, and we shall focus on the double-trace one. In D=10 these are also CS invariants determined by where dH=0 with in flat superspace. However, when we reduce this to lower dimensions we have to redo the analysis because of the low rank of H. In other words, this invariant loses its CS nature, and changes its cocycle structure as a consequence. We shall focus on D=5,6,7, these being the cases of most interest. The double-trace ½ and ¼ BPS invariants can be written as bilinears of the supercurrent. 20

The supercurrent The MSYM field strength superfield is a scalar,, r=1,..n=(10-D). One can construct two scalar bilinears from it, the Konishi multiplet,, an unconstrained superfield, and the supercurrent, J is ultra-short ½ BPS (goes up to theta^4). The square of J in the totally symmetric, traceless rep of SO(n) is the ½ BPS multiplet we’re interested in. It contains the double-trace F^4 invariant. Let’s see how to write this as a Lagrangian form for the case N=2, D=6, where the R symmetry group can be taken to be SU(2)xSU(2). The multiplet is 21

The 1/2BPS constraints are:. where It is not difficult to show that this object does define a non-trivial element in (0,6) spinorial cohomology. The fact that it is non-trivial means that it is not an allowed counterterm as it has a different cocycle structure to the action (which starts at L_{1,5}). 22 The lowest cpt of the Lagrangian 6-form has (p,q)=(0,6)

One-quarter BPS invariants 23 The ¼ BPS invariant can also be written in terms of the square of J. The key point is that each version has a descendant which is similar to B but with the insertion of two contracted spacetime derivatives acting on the two factors of J. So there is a six-form whose lowest cpt is L_{0,6} as in the ½ BPS case, and we can examine whether this is cohomologically trivial or not, i.e. A similar situation obtains in D=7 where the R-symmetry is SU(2). In this case the ½ BPS invariant starts at L_{0,7}, and it is not difficult to show that this object is cohomologically trivial for the descendant of the ¼ BPS multiplet. Thus the double-trace d^2 F^4 invariant is an allowed counterterm in D=7.

If we reduce this 7-form to D=6 we obtain a closed 6-form which does indeed contain a d^2 F^4 invariant. This would apparently be unprotected. But it would not have the full SU(2) x SU(2) R-symmetry, only SU(2). So we have to check explicitly in D=6 whether the ½ BPS form of the Lagrangian 6-form for d^2 F^4 is trivial. The answer is that it seems that this is not the case. This is a (lengthy) calculation which is not so easy as one is trying to prove a negative. There does not seem to be any obvious reason why it is true. Clearly it is down to R-symmetry. If one identifies the two SU(2)s there are many more possibilities to consider in the cohomology. If this result is confirmed we would expect that the double-trace d^2 F^4 invariant in D=5 would also be protected, because the dimensionally reduced one with Sp(1) X Sp(1) symmetry is, but this needs to be verified explicitly. This is L=6 loops. 24

The algebraic NR thm for MSYM tells us that: Non-BPS invariants are not protected ½ BPS invariants (F^4) are protected The single-trace d^2 F^4 invariant is not protected (as it is non-BPS) The double-trace L=2 d^2 F^4 invariant is not protected in D=7 where it has SU(2) symmetry The double-trace L=3 d^2 F^4 invariant seems to be protected in N=2 D=6 with SU(2)xSU(2) Probable that it is protected in D=5, L=6 25

CONCLUSIONS Superspace NR thms are more difficult to understand for max susy than had been thought. Algebraic methods seem to be more powerful. There is an algebraic susy NR thm which states that allowable counterterms must have the same type of cocycle as the action. The cocycles can be investigated using superspace cohomology. 26

Algebraic methods show that ½ BPS invariants are protected in MSYM The ¼ BPS double-trace d^2 F^4 invariant is not protected in D=7 But, it does seem to be protected in D=6, due to R symmetry The dimensional reduction of this result would seem to suggest that it is protected in D=5 as well (6 loops). MSG under investigation 27