A New N=4 Membrane Action via Orbifold

We propose a new Lagrangian describing N=4 superconformal field theory in three dimensions. This theory is believed to describe interacting field theory on the worldvolume of a M2-brane on an orbifold, and is obtained as a Z_2-quotient of the theory proposed by Bagger and Lambert. Despite unusual Chan-Paton structures, we can take Z_2-orbifold by using SU(2)\times SU(2) bifundamental representations. We also analyze the moduli space of this theory and found three branches. With an assumption of a broken U(1) symmetry, the moduli space is consistent with that of the D2-brane in the strong coupling limit of Type IIA string theory if the gauge group is O(4). Our action has manifest Z_2-symmetry exchanging two R^4/Z_2's in M-theory, and this suggests a new non-perturbative duality between a O2^{-}-brane on orbifold R^4/Z_2 and a O2^{-}-brane with D6-branes.


Introduction
In [1], motivated by early attempts [2,3], Bagger and Lambert proposed a new Lagrangian description of three-dimensional maximally supersymmetric (N = 8) conformal field theory with manifest SO(8)-symmetry (see also [4][5][6]).The theory is believed to be realized on the worldvolume of multiple M2-branes in M-theory, and many aspects of the theory has been explored recently .
Despite their success, we have so far only a single example of interacting field theories on the worldvolume of membranes, the so-called A 4 -theory, which is interpreted as the worldvolume theory of two M2-branes in M-theory on R 8 /Z 2 [12,13].The original construction in [1] was based on new algebraic structures called Lie 3-algebras (and nonassociative algebra), and there was hope for some time that there might exist many other Lie 3-algebras.However, it later was conjectured [18] and later proven [19,20] that only the A 4 -theory is allowed in the framework of [4] under the condition of the positivity of the metric. 4Thus there is a pressing need to have more examples of Lagrangians describing theories on membranes.
In this paper we propose a new Lagrangian describing three-dimensional N = 4 superconformal gauge theory.Our theory is obtained as a Z 2 -quotient of Bagger-Lambert theory.This is non-trivial because the structure of Chan-Paton factor is unusual in Bagger-Lambert theory.Our study shows that SU(2) × SU(2) bifundamental representation [7], rather than the original SO(4) notation [1], is essential for our purposes.
Orbifolding also serves as a consistency check of the proposal that Bagger-Lambert theory describes theories on multiple M2-branes.For Z 2 -orbifolded Bagger-Lambert theory, we find three branches of the moduli space.For Coulomb branches, we assumed the breaking of the U(1) symmetry to its discrete subgroup Z m . 5The consistency with D2-branes picture [8] requires that such orbifolds should exist in the strong coupling limit and it should describe M-theory on R 8 /(Z 2 × Z 2 ). 6Actually in the case of m = 4, the moduli space for the D2-brane with O2 − -plane on the orbifold is consistent with that of Z 2 -orbifolded Bagger-Lambert theory.
Another motivation comes from the recent work of [28].Although our theory differs from that of [28], it also discusses three-dimensional Chern-Simons theories with N = 4 supersymmetry, which is similar to our theory in many respects.
The organization of this article is as follows.We begin in section 2 with a brief summary of Bagger-Lambert theory in SU(2) × SU(2) bifundamental representation.
Next we discuss in section 3 the Z 2 -quotient of Bagger-Lambert theory.Then in section 4 we discuss the moduli space of theory.Section 5 is devoted to conclusions and discussions.
In appendix A we summarize our notations of Γ-matrices.

Bagger-Lambert theory in bifundamental representation
In this section, in order to set up notations used in this paper, we briefly review the Bagger- Lambert theory [1] using the SU(2) × SU(2) bifundamental notation of [7].Although the original paper [1] uses the SO(4) notation, SU(2) × SU(2) notation is essential for our purposes.
The matter contents of the theory consists of eight scalar fields X I (I = 1, . . ., 8), 11-dimensional Majorana fermion Ψ, and two gauge fields A µ and Âµ .In bifundamental representation, the scalar fields X I and fermionic fields Ψ are represented by a 2 × 2 matrix and similarly for gauge fields Note that gauge fields are represented by traceless matrices, and their diagonal components are written as a µ and âµ , rather than a µ 3 and âµ 3 , respectively.The reality conditions for X I 's are given by and we also have the chirality condition for Ψ: In this notation, the Lagrangian of the Bagger-Lambert theory is given by where the covariant derivative is defined by The supersymmetry transformations, under which the action is invariant, are given by ) ) ) where the spinor ǫ has the opposite chirality from Ψ: Finally, in order to make the action invariant under large coordinate transformations, the parameter f should take the form where the level k is a positive integer.

Z 2 -action and its invariant sector
In this section we shall consider the Z 2 -quotient of the Bagger-Lambert theory.We consider a discrete group Z 2 acting on R 4 in the R 8 spatial directions transverse to M2branes.We therefore decompose the eight scalar fields For each field our Z 2 acts as follows: where γ is the regular representation of Z 2 given by This matrix γ is chosen so that it is consistent with the usual discussions of orbifolds [29] after the reduction to (the strong coupling limit of) D2-branes [8].For the fermionic field Ψ the quotient action is realized as the Γ 1234 := Γ 1 Γ 2 Γ 3 Γ 4 action.This corresponds to Z 2 -action on R 4 in R 8 , or π rotations in both 12 and 34 directions.The details are explained in the appendix.
For Z i , Y s and Ψ, the Z 2 -quotient acts simply as multiplications by ±1 on their diagonal (D) and off-diagonal (A) parts: The fermionic fields should be further decomposed into Γ 1234 eigenstates where are the projectors onto Γ 1234 = ±1.

Orbifold by Z 2
Now we would like to prove that the Z 2 -truncation as given by (3.1) gives a consistent theory with N = 4 supersymmetry.To begin with, we discuss conditions under which N = 4 supersymmetry is preserved after the Z 2 -truncation.
We first decompose the fields into the two types: the Z 2 -invariant fields and the other fields which will be projected out.The action of the orbifolded theory will be defined by from the original action S(I, N ).Then the symmetry δ of the original theory will become also a symmetry of the orbifolded theory if the following condition is satisfied: In such a case the symmetry of the orbifolded theory is generated by Indeed, from δS = 0 we can easily show that δ S = 0, (3.13) by expansion with respect to N .

Compatibility of Z 2 -orbifold with N = 4 supersymmetry
Let us now examine condition (3.11) to ensure that we have remaining N = 4 supersymmetry.From the definition of Z D and γ, Z D := (Z + γZγ)/2 and we find Thus, and the (3.11) implies that the surviving supersymmetry should satisfy a chirality condition We also find which will vanish with (3.16).
The supersymmetry transformations for Ψ D− and Ψ A+ are ) Thus we also find δN | N =0 = 0 for the fermionic fields if (3.16) is satisfied.It is also easy to check the compatibility condition for gauge fields.In this way we have proven that N = 4 supersymmetry is preserved after the truncation.

The Lagrangian and its remaining N = 4 supersymmetry
The surviving supersymmetry transformations are summarized as follows. ) ) In components, the supersymmetry transformations are ) ) The Lagrangian for Z 2 -orbifolded theory is where the covariant derivative D D is defined by (when acting on Y s D , for example) and the potential V (Z A , Y D ) is given by In these equations stands for summation over signed permutations with position of dagger fixed.For example, and (3.37) In terms of components, the Lagrangian is explicitly written down as follows.
The Chern-Simons gauge coupling k ′ of the Z 2 orbifolded theory is related with that of the original action as (3.39)

Discrete symmetries of the Lagrangian
By Z 2 -orbifolding, the gauge group of our theory is naively broken down to U(1) × U(1) generated by a µ and âµ .However, we have one discrete gauge symmetry Z 2 , which is generated by choosing iσ 2 from both SU(2) of the original SU(2)×SU(2) gauge symmetry, and thus the gauge symmetry after the orbifolding is given by U(1) × U(1) × Z 2 .This Z 2 symmetry acts as In addition to this gauged Z 2 -symmetry, we have two more global Z 2 -symmetries.The first is the parity invariance Naively we can use U(1) × U(1) gauge symmetry and fix one of the phases of y s 's.
Here we simply assume without justification that the U(1) gauge symmetry coupled to y is broken to a discrete subgroup Z m where m is a some integer number. 9This Z m acts on y s as y s → e 2πni/m y s .(4.5) We also have the gauged Z 2 -symmetry (3.40) Combining these, we have the dihedral group D m = Z 2 ⋉ Z m and the resulting moduli space is given by

.7)
9 There are subtleties in this argument.We cannot apply the mechanism in [12,13] via the dual photon, because both of the U (1) × U (1) gauge fields b µ := a µ − âµ and c µ := a µ + âµ couple to the scalar fields in the action (3.38) and the auxiliary fields cannot be introduced.Within the framework of the Bagger-Lambert theory, we could not justify this point explicitly.But we expect such mechanism happens, because the matching of the moduli spaces of M-theory and Type IIA for the each branches should be realized.and the unbroken gauge symmetry is U(1) V , which is generated by a µ + âµ .In the special case m = 4, we have

Phase (II): M2 at the other fixed locus
In this case, the solution for where z := z 2 + iz 1 .Due to the presence of Z 2 -symmetry (3.42), we find that the moduli space for phase (II) is isomorphic to that of phase (I): The unbroken gauge symmetry is U(1) A , 10 which is generated by a µ − âµ .
In this vacuum there are no residual symmetries in gauge fields in contrast to phases (I) and (II).Actually the action for scalars and gauge fields are given by For the generic point in the moduli space y s = 0, z i = 0, the minimum of this action is realized for a µ = âµ = 0. Then the moduli space for this case consists only of scalar fields.
As a result we find the moduli space M (III) for phase (III) is This result is independent of k ′ .

Comparison with Type IIA moduli space
We are now in a position to compare the moduli space of our theory obtained so far to that of D2-branes in the strong coupling limit of Type IIA string theory.If our theory really describes theories on membranes, then these two moduli spaces should match.This serves as a good consistency check of Bagger-Lambert theory and our Z 2 -orbifolding procedure.
At first sight the analyses in M-theory and Type IIA look similar, but at closer inspections of field contents in two theories are largely different and the match is far from trivial.
The discussion of Z 2 -orbifolding of O(4) gauge theory11 is analogous to the discussion above of the M-theory case.Using 4 × 4 matrix representations, take the regular representation γ to be and consider Z 2 -action as in (3.1): where the seven scalars are decomposed into four scalars Z i (i = 1, 2, 3, 4) and Y s ′ (s ′ = 5, 6, 7).Here we are taking the M-theory direction to be the 8-direction.By this Z 2action, the remaining fields are Y s ′ D , Z i A , Ψ D+ , Ψ A− and A Dµ , where suffixes D (and A) represents 2 × 2 block diagonal (block off-digonal) components.For example, gauge field A Dµ after the Z 2 -truncation is represented by where (up to irrelevant coefficients) in our previous notation in the M-theory, we have written a V µ = a µ + âµ and a A µ = a µ − âµ .After orbifolding, the gauge symmetry is given by SO(2) × SO(2) ≃ U(1) × U(1), plus discrete gauge symmetries which we will comment on in a moment.
The moduli space of this theory again consists of three branches: (ii) : (iii) : The corresponding configurations of D2-branes are almost the same as in M-theory case, namely as in Figure 1.The only difference is that we have only three Y s ′ directions, not four.We now analyze each phase in detail.
Phase (i): D2 at the fixed locus of the orbifold Z 2 In this phase, only the Y s ′ 's take non-zero value: where α s ′ and β s ′ are arbitrary real numbers.
At this phase, the gauge symmetry U(1) V ×U(1) A is completely preserved.This means in addition to scalars α s ′ and β s ′ , we have two periodic parameters σ V and σ A obtained by dualizing two gauge fields a A µ and a V µ .Thus we have R 3 × R 3 × S 1 × S 1 , parametrized by α s ′ , β s ′ , σ V and σ A .However, we still have to take care of discrete symmetries of O(4).
Namely, two discrete symmetries in SO( 4) gives two Z 2 -symmetries and while keeping other fields fixed.Further, discrete symmetry in O( 4) gives one more Z 2 -symmetry Combining all these three discrete Z 2 , the moduli space is given by When the coupling goes to infinite, S 1 decompactify12 and we have the correct moduli space ((R 4 /Z 2 ) × (R 4 /Z 2 ))/Z 2 , as expected 13 : In phase (ii), the scalars are given by where γ i and δ i are real numbers.The form of Z i 's are chosen so that Z i 's mutually commute, thereby minimizing the potential.On this phase, the gauge symmetry is completely broken 14 and we have no scalars coming from the gauge field.By taking care of discrete gauge transformations (4.22) and (4.25), we have three Z 2 -identifications (1) ) γ i ↔ δ i , and thus we have the moduli space In this case, the moduli space coincides with that of M-theory even before taking the strong gauge coupling limit.
Phase (iii): D2 at the generic point of the moduli space In this phase, both Y s ′ 's and Z a 's take non-zero value: 14 Gauge symmetry U (1) V (resp.U (1) A ) is restored, however, when γ i = δ i (resp.γ i = −δ i ).
theory on the worldvolume of a M2-brane placed on an orbifold R 8 /(Z 2 × Z 2 ), and is obtained as a Z 2 -orbifold of Bagger-Lambert theory in the SU(2) × SU(2) bifundamental representations.
We also analyzed the moduli space of our theory and found three branches.In the analysis of the Phase (I) and (II), we assumed some mechanism to make one of U (1) gauge symmetry be broken to the discrete subgroup Z m .Within the framework of the Bagger-Lambert theory, we could not justify this mechanism explicitly.But under this assumption, the matching of the moduli spaces of M-theory and Type IIA theory for each branches can be found especially for m = 4 in highly non-trivial way.In this discussion, the moduli space for the Type IIA theory is given by the Z 2 -orbifold of O(4) gauge theory, rather than SO(4) as in [12,13].Conversely speaking, the M2-brane theory on the Z 2 -orbifold should be defined as the strong coupling limit of the Type IIA brane configuration on Z 2 -orbifold, then the matching of each branches of moduli space supports our assumption and analysis in M-theory.
The interesting feature of our Lagrangian is the existence of Z 2 -symmetry (3.42), which exchanges two Z 2 -actions.In M-theory viewpoint this is natural and simplify exchanges two Z 2 -actions, but in Type IIA language this exchanges orbifold with orientifold, which is highly non-trivial.In our discussion, we have deleted 8-direction (i.e. one of the Y s -directions) to obtain Z 2 -orbifold of D2-O2 − system.If we instead reduce along Z i -directions, then we should have D6-D2-O2 − system without Z 2 -orbifold.Now the symmetry (3.42) implies a new duality between Z 2 -orbifold of O2 − and D6-O2 − .We call this new non-perturbative duality "O-duality". 15 16The existence of orientifold is crucial for the existence of this duality.As a possible check of this proposal, our moduli space in phase (I) should match with the instanton moduli space of SU(2)-instanton placed at an Z 2 -orbifold, and it would be interesting to explicitly verify this.
Finally, in this paper we have concentrated on a single example of Z 2 acting on R 4 .
We can consider more examples by considering Z 2 acting on R 2 , R 6 and R 8 , for example, and it would be interesting to study them.

. 28 )
Phase (ii): D2 on the orientifold In M-theory, moduli of Phase (I) and that of Phase (II) are automatically isomorphic, due to the presence of discrete Z 2 -symmetry (3.42).It is non-trivial, however, to verify the corresponding fact for Type IIA, because orbifold and orientifold are different in Type IIA.