6.4 Including (1, 1) Branes
6.4.2 ABJM Theory
In the presence of (1,1) branes, some of the gauge groups gain Chern-Simons terms. This means that the IR fixed point can be written explicitly: one simply strikes out the Yang-Mills terms in those gauge groups with Chern-Simons terms. Performing this operation in the remaining gauge groups leaves them with no kinetic term at all, which is not very well-defined, so we will usually only do this if all gauge groups have a Chern-Simons term.
It turns out that the simplest example of this is ABJM theory. Namely, if one takes the con- figuration with a single N S5 brane and (1, k) brane, one finds the field content of ABJM theory, namely, aU(N)×U(N) gauge multiplet together with two bifundamental hypermultiplets. In the UV, the gauge field has a Yang Mills term, but since it also has a Chern-Simons term in all factors, one can simply remove the Yang-Mills term and find the IR fixed point to be precisely the action of ABJM theory.
Let us consider the effect of dualities on such a configuration. As above, we are forced to restrict to the case k= 1. Then an S-duality takes us to a configuration with a D5 and (1,1) brane. As argued above, T duality is trivial in this case, and this is manifestly the same as the contribution of a D5 and N S5 brane. Then, as noted above, a singleN S5 brane can always be removed, and one obtains the theory of a single D5 brane. This can be further dualized to a single N S5 brane, which can also be removed, and we are left with no five-branes at all. The result is simply N= 8 super Yang-Mills theory, and the chain of dualities has shown us that this should be equivalent at low energies to the ABJM theory, as noted in the previous chapter.
Our general arguments above prove the matching of the partition function of ABJM with the dual theory with an N S5 brane andD5 brane, which isN = 4U(N) Yang-Mills with an adjoint hypermultiplet and a fundamental hypermultiplet. This in turn should be dual to N= 8, as just argued, but this cannot be seen at the level of the partition function, since our arguments only applied to the case where at least oneN S5 brane is present at all times. In fact, we claim the naive matrix model one writes down for theN= 8 theory must be incorrect. This can be seen from the fact that, by N= 8 supersymmetry, theIR superconformalR-charge of the scalars in the adjoint
Figure 6.2: Sequence of dualities which lead from ABJM theory at level 1 toN= 8 SYM. HereN S5 branes are in red,D5 in blue, and (1,1) in purple.
hypermultiplet must be the same as that of the scalars in the vector multiplet, but we have assumed the former have R-charge 12 and the latter haveR-charge 1. Indeed, the naive matrix model does not even converge:
ZN=8=? Z
dNλ Q
i6=jsinhπ(λi−λj) Q
i6=jcoshπ(λi−λj)=∞ (6.30) Thus we cannot directly compare this theory to ABJM or itsN= 4 dual SYM dual. The nonrenor- malization theorems that usually apply to theories with at leastN= 4 supersymmetry break down here, ironically, because there is too much supersymmetry, and theN= 4SU(2)×SU(2) subgroup may rotate in an unspecified way inside the fullSO(8)R-symmetry as we flow to the IR.
Chapter 7
General R-Charges
Up until now, with the exception of pure Chern-Simons theory, we have considered only theories with extended supersymmetry, i.e., at least N = 3. There is no inherent reason why the local- ization calculation requires this, since in our derivation of the matrix model we used only N = 2 supersymmetry. However, we wrote a specific form for the superconformal transformations of the matter fields, and although this form is necessary in theories with extended supersymmetry, it is not the only one compatible withN= 2 supersymmetry. Generically, the matter may come in some other representation of the algebra – specifically, with an R-charge other than than 12 – and then theδ-exact terms above cannot be used. In this chapter, following [7, 5], we attempt to find these more general representations, and use them to localize arbitraryN= 2 theories. We will see that in addition to opening up the interesting realm ofN= 2 theories to investigation via localization, one is led to insights into RG flow that apply even to nonsupersymmetric three-dimensional theories.
7.1 Representations of N = 2 Superconformal Symmetry
In Chapters 2 and 3, we found representations of the superconformal algebra in terms of multiplets of fields on conformally flat manifolds. In the case of the chiral multiplet, these were motivated in part by the conformal covariance of the Laplacian and Dirac operator, as discussed in Appendix B.
These operators are only covariant provided one assignsφ a scaling dimension 12 and ψdimension 1. However, it is known that this is not, in general, the correct dimension in theIRfixed point.
For a simple example, consider the XY Z theory we discussed in the previous chapter. Since this has a superpotential XY Z, which is not renormalized and must have R-charge 2 in the IR, then given the symmetry under exchanging the fields, we see they must each be given R-charge 23. Then by general arguments of Chapter 2, this must also be the scaling dimension of the scalars in the chiral multiplets. This is not consistent with the actions for the scalars we have written above, which are only conformally invariant if we assign them scaling dimension 12. It is also not consistent with the superconformal transformationsδ we wrote above.
Before attempting to address this issue, we should note that for the vector multiplet, one typically does not run into this problem. This is because the operator ?F is a conserved current, by the Bianchi identity, and so in an interacting conformal theory it must have dimension 2, consistent with the superconformal transformations for the vector multiplet we discussed above. Actually, one should really only consider gauge-invariant operators like Tr(?F), but still, it seems reasonable that the vector multiplet does not receive corrections (in particular, it is real and uncharged under R-symmetry).
The transformations with generalR-charge ∆ were obtained in [7, 5]. In principle, they can be obtained by commuting the flat space, ordinary supersymmetries with conformal generators, whose action on fields of arbitrary dimension is known. We summarize them here:
δφ= 0, δφ†=ψ†
δψ= (−iγµDµφ−iσφ)−2i∆φ0, δψ†=F† (7.1)
δF =(−iγµDµψ−iλφ+iσψ) +i(2∆−1)0ψ δF†= 0
One can check that these give rise to the correct anticommutators, specifically, with the R-charge
∆.
In principle, one now simply needs to compute the newδ-exact actions using these supersymme- tries, and see how the 1-loop determinants are modified. This can be done following the procedure outlined in Chapter 4, and involves diagonalizing these new operators. However, there is a trick which slightly simplifies this procedure.