Eight Fermion Terms in the Effective Action of the ABJM Model

We study eight fermion terms in the effective action of the ABJM model. We show the non-renormalization of $v^2$ terms. After classifying all the possible eight fermion structures, we show that $\mathcal{N}=6$ supersymmetry determines all these terms completely up to an overall constant. This confirms the one loop non-renormalization of $v^4$ terms.

supercharges up to the order we are interested in. The eight fermion terms generically contain scalar fields. Since there are no corrections in supercharges, the supersymmetry transformations acting on these scalar fields in eight fermion terms are the only source for nine fermion terms and therefore they should vanish by themselves. In this way we determine the eight fermion terms modulo an overall constant. This implies that the |∂b| 4 terms and their superpartners are completely determined by supersymmetry. In fact since they come from one loop, our results naturally imply one-loop non-renormalization of the |∂b| 4 terms.

Non-renormalization of v Terms
We are interested in the effective action of the slowly moving probe brane. The target space coordinates of the branes are described by the diagonal components of the scalar fields. We put the source branes at the origin of C 4 /Z k and the probe brane at b A . The superpartners of these scalars, b A , are denoted as χ A . After integrating out the off-diagonal components of scalar, spinor and gauge fields, the only relevant fields remained are diagonal components of those fields. One combination of the remaining abelian gauge fields is decoupled from all matter fields and thus can be integrated out trivially. This gives the constraints on the other combination of the gauge fields to be pure gauge, and thus to be zero. The only remaining gauge symmetry is the global discrete one, Z k , identifying b A ∼ e 2πi/k b A and χ A ∼ e 2πi/k χ A . Therefore the effective action becomes a functional of b A and χ A .
The tree level supersymmetry transformations of these fields become where γ µ and γ I ,γ I denote the SO(2, 1) and the SO(6) gamma matrices, respectively. Various properties on these gamma matrices are summarized in Appendix. We always contract spinor indices from northwest to southeast, ψχ ≡ ψ α χ α , which gives ψχ = χψ and ψγ µ χ = −χγ µ ψ.
In general there could be loop corrections to the supersymmetry transformations. The generic form of the corrections can also be organized by weights. The lowest order terms in the effective action have weight two, which is the same as the classical action. Therefore the correction in the supersymmetry transformations at this order, if any, should have the same weight as the tree level supersymmetry transformations and thus generically given by where M I contain fermion bilinears. The supersymmetry algebra should remain closed under these modified transformations and thus from one can see that the second term should vanish. As shown in below, this condition demands that M J be zero and thus the supercharges do not get any corrections at this order. This guarantees that the |∂b| 2 terms do not get any quantum correction.
Proof: The condition which M J should satisfy is Multiplying byγ I CA , the equation becomes By symmetrizing the spinor indices and using the SO(6) Clifford algebra this becomes By multiplying δ IJ in the Eq. By multiplying γ J , we obtain that γ I M I = 0, which leads to M I A αβ = 0. ♣

3
In this section the possible fermion bilinears are classified. Note that χ †A and χ A transform as (2,4) and (2,4) under the three-dimensional Lorentz symmetry and the R-symmetry SO(2, 1) × SO(6) R , respectively. We can classify the fermion bilinears according to the irreducible representations of SO(2, 1) × SO(6) as follows.
• χ † χ † : Each irreducible representation corresponds to the following fermion bilinear form: Among these, the first and fourth terms identically vanish by the (anti-)symmetry of SO (6) gamma matrices.
• χχ: Each irreducible representation corresponds to the following fermion bilinear form: where the first and fourth terms vanish by the same reason as above.
• χ † χ: Each irreducible representation corresponds to the following fermion bilinear form: These fermion bilinears are the basic building blocks in the effective action. All the eight fermion terms which appear as the superpartners of |∂b| 4 should come from the combinations of the above fermion bilinears.

Structures of The Eight Fermion Terms
Eight fermion terms in the effective action should be a singlet under SO(2, 1) × SO (6). In particular, this implies that all the SO(6) vector indices should be contracted. Since they appear only through SO(6) gamma matrices, we can use the SO (6)  The requirements of being SU (4) and gauge singlet strongly restrict the possible form of eight fermion terms. Therefore the possible eight fermion terms, denoted collectively as f χ 8 , can be written as where (ACE) denotes the total symmetrization of A, C, E.
Now, let us classify all the possible eight fermion structrues for T l 0 , T A l 2 B and T AB l 4 CD , which correspond to terms containing zero, two and four scalars, respectively. The Fierz identity, can be used to replace both χ † χ † and χχ contractions by χ † χ contractions and vice versa. Here · denotes the contraction of spinor indices.
Firstly, it is clear from the above Fierz identity that there are only two independent structures in eight fermions with four scalars, which are given by It is a bit more complicated to find the independent structures with two scalars. Apparently, there are seven possible structures, It can be shown that they are related by three equations as Therefore, there are four independent structures with two scalars, which may be chosen as One may note that the contraction of two indices in eight fermion structures with four scalars gives rise to two independent ones with two scalars.
Similarly, there are five possible structures without scalars, which are related by two equations, We choose three independent structures as In summary, the effective action can have, at most, nine independent eight fermion structures.
In next section, by using the supersymmetry, we show that some of these terms can not appear while all the remaining terms should be related.

Determination of The Eight Fermion Terms
In general, the tree level supercharges of the classical action get quantum corrections which may be organized by weights. Generically, the effective action can be expanded in the increasing order of the weights, such that the weights of consecutive terms differ by two. Accordingly, the corrections in supercharges should be ordered in the same way. The tree level supersymmetry transformations acting on scalars in eight fermion terms give rise to nine fermion terms. On the other hand, in section 2, we showed that the supercharges do not get any corrections in the leading order and as a result the moduli space is flat. This implies that the nine fermion terms which arise from the supersymmetry variations of eight fermion terms should vanish by themselves. Therefore we require that In order to solve these equations, it is convenient to apply the operator O I α To simplify further, we introduce a fermion number operator for χ . This operator has the following commutation relation with the operator O : which leads, along with O(f χ 8 ) = 0, to the relation As a result, eight fermion terms should satisfy where ∆ = 4∂ A∂ A is an eight-dimensional Laplacian. These two equations, supplemented with simple dimensional counting, are enough to determine the eight fermion terms completely(up to an overall constant).
By expanding this eight dimensional Laplace equation in independent eight fermion structures, one obtains nine differential equations. Among these, the differential equations from the coefficient of T A 3B and T A 4B are given by