M2-branes on M-folds
We argue that the moduli space for the Bagger-Lambert theory at level is , where is the dihedral group of order . We conjecture that the theory describes two M2-branes on a “M-fold”, in which a geometrical action of is combined with an action on the branes. For , this arises as the strong coupling limit of two D2-branes on an orientifold, whose worldvolume theory is the maximally supersymmetric gauge theory. Finally, in an appropriate large- limit we show that one recovers compactified M-theory and the M2-branes reduce to D2-branes.
A new class of conformal invariant, maximally supersymmetric field theories in dimensions has been found recently [1, 2]. These theories are based on “3-algebras” and include a non-dynamical gauge field with a Chern-Simons-like interaction. They have several striking properties including the absence of continuous marginal deformations. The motivation for studying these theories was to find a Lagrangian description of the conformally invariant fixed point of maximally supersymmetric Yang-Mills theories in dimensions, which is believed to describe the worldvolume dynamics of coincident membranes in M-theory.
While the 3-algebra theories share many features with the expected M2-brane theories, they also give rise to some puzzles. One is that only a single 3-algebra, denoted , is presently known, so an explicit theory exists for at best a small fixed number of membranes. It was proposed in Ref.  that this number is 3, which suggests the surprising possibility that the IR theory on 2 D2-branes is trivial. Also somewhat puzzling was how parity could be preserved when the gauge field has Chern-Simons interactions.
Some of these puzzles have been resolved in recent days [4, 5, 6]. For the 3-algebra, all these papers (as well as Ref. ) found that the theory could be recast as an gauge theory. In Ref.  it was further shown that giving one of the scalars a vev reduces the 3-algebra action to a strongly coupled supersymmetric Yang-Mills action by a novel Higgs mechanism. This renders one combination of the two Chern-Simons fields massive and the other one dynamical in consequence. In Refs. [5, 6] it was shown that the theory is parity-invariant if parity is taken to exchange the two ’s.
However, new puzzles emerged. Ref.  studied the moduli space of the theory and found that it does not appear to match expectations for either two or three M2-branes. Additionally the spectrum of chiral primary operators had some “missing” operators that should have been present for a multiple M2-brane interpretation to be correct. In this work it was also noted that the level of the is a free discrete parameter which, at large values, causes the theory to become weakly coupled – but a finite set of M2-branes should not have any weakly coupled limit. Also, although Ref.  found a result suggestive of compactification, it was not clear why going to the Coulomb branch should be related to a circle-compactified background.
In the present work we resolve some of the above puzzles. We conjecture that the Bagger-Lambert theory at level describes the worldvolume dynamics of two M2-branes on the orbifold, defined by the uplift to M-theory of two D2-branes on an orientifold. Equivalently, the level-one theory is the infrared fixed point of the maximally supersymmetric Yang-Mills theory in 2+1 dimensions. With this interpretation, we argue that the spectrum of chiral operators is as expected. For general , we argue that the moduli space is , where is the dihedral group of order . We conjecture that this corresponds to two M2-branes on a “M-fold”, in which a geometrical action of is combined with an action on the branes. Finally, we show that taking a large- limit at a point on moduli space where the branes are separated from the orbifold point, one recovers the worldvolume theory of D2-branes, as expected, since the orbifold locally becomes a cylinder.
We will work with the formulation of the theory in Ref. . The fields consist of two gauge fields, having Chern-Simons actions with opposite signs, and a set of 8 scalar fields along with 8 fermions. All the matter fields transform as bi-fundamentals of . The action is:
which is covariant under the action of the gauge transformations
In the above, where is the (integer) level of the two
Since our proposal involves orbifold 2-planes, let us briefly review
some relevant facts. There are three types of orientifold 2-planes in
type IIA string theory , denoted ,
and , that give rise to gauge groups respectively when D2-branes are brought near them. All
correspond to an inversion of 7 spatial directions transverse to the
orientifold plane, and all can be uplifted to M-theory. The uplifted
orientifold planes are really M-theory orbifolds rather than
orientifolds, in the sense that they do not reverse the orientation of
membranes or of the 3-form .
We can directly define the plane in M-theory as the orbifold
, where the action of is diag on
the four complex coordinates of . With this action the
supersymmetry near the plane is half-maximal and has 16 components
just like the supersymmetry on M2 branes. This will turn out to be the
case we understand best. For , the subgroup of
associated with the inversion of the is replaced
with in the definition of the moduli space. This
suggests that the level M-fold combines a geometrical action of
with an action on the branes. While we will not be able to
present a precise action of satisfying all the
requirements, we will discuss some possibilities in a subsequent
section. Unlike , the general case is not likely to
descend in a simple way to a type IIA orientifold since a
action with will presumably mix the M-circle with another
In the rest of this note, we present evidence for our conjecture that the theory whose Lagrangian is given in Eq. (1) describes two -branes at a orbifold (with the action of on the brane worldvolume fields appropriately defined).
When the original version of this paper was nearly complete, the paper  appeared, which has substantial overlap with our work. The original versions of our paper and of  differed in a few significant respects, but most of these differences have now been resolved in the revised versions. We thank David Tong and Neil Lambert for correspondence on these issues. Another very recent paper discussing multiple M2-branes is Ref. .
2 Moduli space
The moduli space for the theory was studied in Refs. [3, 6]. Here we will revisit this moduli space and argue that the complete moduli space at level is actually once the gauge fields are taken into account. Here, is the dihedral group, where the product is semidirect.
We begin with the action in Eq. (1). As noted in , generic scalar configurations for which the potential vanishes correspond (up to gauge transformations) to diagonal matrices , which we will parameterize by
Within the space of these diagonal configurations, there is a residual gauge symmetry, acting by simultaneous rotations on and by simultaneous complex conjugation. However, to describe the complete moduli space it will be important for us to take into account the gauge fields.
Generically, the diagonal configurations (4) break the gauge group down to , and the remaining components of the gauge field become massive by the Higgs mechanism. Also, expanding the potential about such configurations shows that physical scalar fluctuations which take us away from a diagonal configuration are all massive.
We now write the classical action describing the dynamics of the light fields on the moduli space. To do this, it will be convenient to include both the preserved gauge field and the gauge field associated with the that rotates . Together with the diagonal configuration (4), we take
with the normalization chosen so that and have gauge transformations
where and have period . This gives an action
We further define
so that is the gauge field associated with the preserved , and is associated with the that rotates .
The resulting action is
The gauge transformations are
Note that the last term in the action is gauge invariant because of the Bianchi identity for .
To this action, we can add a Lagrange multiplier term
and treat as an independent variable. The integral over
enforces the Bianchi identity. To be precise, we need to be
periodic with period . To see this, note that for monopole
configurations we can have,
For this, it is essential to note that is the sum of field strengths for two gauge fields which are normalized conventionally (so the gauge transformation is the derivative of an angle without any numerical factors). So rather than a standard delta function, we want a periodic delta function that allows these monopole configurations. This is ensured by a periodicity of .
Starting from the combined action
the equation of motion for gives
Using this, the full action reduces to
The gauge invariance transformation on translates to a gauge invariance transformation on
We can now fix our gauge to set . After doing this, we still have residual gauge transformations
which leave . The moduli space is therefore characterized by a set of eight complex numbers , with gauge transformations that take
Here, the action and the action don’t commute with each other for , and the combined group is the dihedral group . We conclude that the moduli space for level is
For , this is just
the moduli space of the superconformal theory that describes the
infrared physics of maximally SUSY Yang-Mills theory in 2+1
dimensions . In contrast, the superconformal theory
arising from gauge theory should have moduli space
For higher , we conjecture that this theory describes the low-energy physics of two M2-branes in M-theory with a generalized orbifold action on . We expect that the geometrical action is . However, the orbifold group must also act on the M2-brane fields, since the moduli space is not just .
The orbifold in question should preserve 16 supersymmetries and
maximal R-symmetry for all and have the desired action
Eq. (19) on moduli space. However, except for there
appears to be no known orbifold with this property.
where . Perhaps surprisingly, this action preserves as
many as 12 supersymmetries, or in 3 dimensions, and also
gives rise to an R-symmetry
[13, 14]. Even more intriguing, there are
two exceptions to this rule – the cases with . The former
obviously preserves , while the latter has also been
claimed to do the same . Though for general
this orbifold does not appear to meet all the requirements, it is
possible that it actually preserves more supersymmetry and R-symmetry
than is apparent for reasons that we do not yet
For the present, since we do not have a precise formulation of the theory whose moduli space we have found, we simply think of it as the theory of 2 M2-branes on an “M-fold,” and consider the theory at level to give a precise definition of the “M-fold”.
3 Chiral primary operators
In Ref. , it was pointed out that it is impossible to construct operators in the theory which lie in tensor representations of with an odd number of indices. This presented a puzzle for the interpretation of the theory as the worldvolume theory of a stack of M2-branes, since such theories are believed (at least for three or more M2-branes) to have a spectrum of chiral operators that includes these odd-indexed representations. We will now see that with our proposed interpretation, this is no longer a problem.
To see this most explicitly, let us focus on the case and consider the UV gauge theory from which the superconformal field theory flows. For the theory (or with ), the scalar fields are seven Hermitian matrices, and we can construct operators
that should become a subset of the chiral primary operators in the
infrared limit (the others are generated by the rotations that
are not manifest in the UV).
Generally, we expect that superconformal field theories that have the same moduli space should also have the same spectrum of chiral operators , so perhaps the discussion in this section is somewhat redundant. However, it is interesting to understand explicitly why the odd-indexed representations do not show up in the case.
4 Points on the moduli space
In this section, we discuss more explicitly the connection between points on the moduli space of the theory and configurations of M2-branes on an orbifold. We begin with the simplest case . Here, the conjecture is that the moduli space should coincide with the moduli space of two M2-branes on a orbifold. This arises in the strong coupling limit of type IIA string theory from a configuration of two D2-branes on an orientifold.
To understand the properties of such an orbifold, let us begin by considering D2-branes at an orientifold. The low-energy worldvolume theory for these is maximally supersymmetric gauge theory. The scalars in this theory are antisymmetric matrices, and configurations for which the scalar potential vanishes are gauge equivalent to
Residual gauge transformations preserving this form allow us to make the identifications
so the set of scalar field vevs with vanishing potential may be described by the space .
The full moduli space of the IR limit of the gauge theory is , which we can describe by two vectors in , subject to the identifications
We can interpret and as the locations of the two M2-branes. However, note that and are inequivalent, so the moduli space is not just a product of two ’s divided by the symmetric group, as one might naively expect. In this characterization, a special role is played by configurations where the branes are coincident. These are invariant under the transformations
and so lie at special points of the moduli space.
Now, going back to the theory for , we had derived the moduli space as the space of complex vectors up to gauge transformations
The first of these has no nontrivial fixed points, while the second has a fixed point for real vectors. Thus, for our choice of gauge, it is natural to make the associations
so that the fixed points of complex conjugation (equivalently, the special points on the moduli space preserving ) are identified with coincident branes. It may seem puzzling at first that there seem to be more configurations than those with real that preserve symmetry, namely any set of which lie in a line on the complex plane. However, for these configurations, we do not have a free abelian gauge field that can be dualized to a scalar, so these are all gauge-equivalent to the configurations with real.
So far the discussion has dealt with . It would be nice to carry out a similar analysis for higher , in particular to find the precise relation between our coordinates on the moduli space and the positions of the branes.
For , we can also offer a rather heuristic geometrical explanation for the origin of as follows (we expect this explanation could be made more precise with a better understanding of the “M-fold”). Suppose we bring two M2-branes to a orbifold. It is plausible that the result is, to start with, a theory on copies of the original branes, namely an quiver gauge theory. The quiver diagram is a -gon with the gauge fields at the vertices. The form of the action Eq. (1) is consistent with the presence of ’s, except that at the origin of moduli space the orbifold plane causes of these ’s to get identified with each other, so that their action is times the action of a single (level-1) Chern-Simons gauge field. Likewise, the other ’s get identified and their action is times that of another level-1 Chern-Simons gauge field, appearing in the action with a negative sign. Given the quiver interpretation, the associated symmetry group should be the set of all discrete transformations that map the quiver to itself. This includes cyclic rotations as well as reflections along any axis joining opposite vertices. By definition, this is the group of symmetries of a -gon, namely .
5 Large and compactification
In this section, we will see that the findings in Ref.  fit very naturally with our interpretation. In that paper it was found that expanding the action about a special point on the moduli space where gauge symmetry is preserved gives an action which is at leading order the maximally supersymmetric Yang-Mills theory. The extra comes from dualizing the scalar field that corresponds to multiplying all vevs by a constant. This is not really a free scalar but is approximately free at large distances from the orbifold, corresponding to the fact that the theory on two M2-branes effectively has a centre-of-mass mode when the branes are far away from the orbifold plane.
The procedure of  gives the Yang-Mills action plus an infinite series of higher dimension operators. While the latter can be decoupled in the limit , the Yang-Mills action simultaneously becomes strongly coupled in this limit. So there is no limit where one really has finitely coupled D2-branes. However, the the analysis of  was for level . Repeating it for general , we find the following. By rescaling , we easily see that the action Eq. (1) acquires an overall multiplicative factor of . Denoting this scaled action for the level- theory as , we have
Now in Ref.  the action was examined in the presence of a large vev (there, this vev was called and later ). It was shown (see Eq.(3.23) of that reference) that
where is the action for an Yang-Mills theory.
For the Lagrangian , we must define the Yang-Mills coupling by
Taking the limit with fixed, we see that the Yang-Mills part of the action has a finite coupling . However, successive terms scale to zero in this limit. Therefore in the limit we obtain precisely the D2-brane worldvolume theory with a tunable finite gauge coupling , and no higher dimension operators.
With our interpretation of the theory, this observation is exactly what we would expect. We have argued that points on the moduli space preserving correspond to taking two coincident M2-branes away from an orbifold fixed point. While a precise definition of the orbifold action is lacking, for this discussion it is sufficient to assume it leads to an opening angle that shrinks like as is the case for standard orbifolds. Now, in the limit where , the opening angle of the orbifold goes to zero, so at some point sufficiently far out on the moduli space, the local geometry approaches that of a cylinder . The scaling limit below Eq. (34) precisely takes the two M2-branes out into this cylindrical space, where they should behave like two D2-branes in type IIA string theory. So we expect a finitely coupled Yang-Mills theory – and that is exactly what we find.
In this paper, we have found that the moduli space for the Bagger-Lambert theory at level is , where is the dihedral group of order . Our interpretation is that the theory describes M2-branes on a “M-fold,” a generalization of the case defined by the uplift of the orientifold in string theory.
We feel compelled to mention that the superconformal theories defined as the IR fixed point of , , , , and all have moduli spaces of the form
where is respectively , , , , and
. Within our interpretation, only the identification of the level
theory with seems natural, however, it may be that the
level and cases happen to coincide with the infrared limit
of and maximally supersymmetric gauge theory
The discussion in the limit of large-order orbifolds bears a strong resemblance to the deconstruction approach to M5-branes discussed in Ref. . In section IIIB of that paper, a limit is taken where the order of the orbifold grows large and simultaneously the D-branes are moved far away from the orbifold so that effectively they end up propagating on a cylinder. It would be interesting to explore whether the corresponding limit in our paper is related to deconstruction and M5-branes.
A Chern-Simons Level Quantization
The purpose of this appendix is to review the quantization of the Chern-Simons level for the non-simply connected gauge groups, . None of the results are original but, particularly for the spin Chern-Simons case, they are not as well-known as they should be. We would like to thank Dan Freed for guiding us through the computation below.
As shown by Dijkgraaf and Witten , the level of a Chern-Simons theory with gauge group, , is specified by an element . If is one-dimensional, then is an integer multiple of the generator, and the choice of Chern-Simons action comes down to specifying that integer. When is not simply connected, it is convenient to write the normalization of the Chern-Simons action for relative to that of the Chern-Simons action for the simply connected covering group, . That is, the homomorphism induces a map , and we might wish to note that the generator of maps to some multiple of the generator of .
On a spin-manifold (which we certainly have, in our case, as we are interested in supersymmetric theories), one can define a refined version of Chern-Simons theory, called spin Chern-Simons . The precise definition of the action is slightly more subtle (it involves the choice of a spin structure on the 3-manifold), but the variation of the action, as one varies the gauge connection is the same as for conventional Chern-Simons. The only difference, from our perspective, is that the quantization condition on the level is somewhat relaxed.
The level for spin Chern-Simons is specified by an element, , of what Dan Freed calls  -cohomology, which combines information from the integer cohomology with some mod-2 information. In particular, for any space, , there is a long exact sequence
where the connecting homomorphism is the integer Bockstein, composed with the second Steenrod square.
Let’s apply this to the classifying spaces for , . For , , , with generator . , with generators and . , with generator . The above long exact sequence gives rise to short exact sequences
For , the map is multiplication by 2 ( for a bundle is always even), as is the map . Hence the map is an isomorphism. Thus, for , , the Chern-Simons coefficient must be even, while the spin Chern-Simons coefficient can be any integer.
For the map is multiplication by 4 (see, e.g., equation (4.11) of ). is still multiplication by 2, hence is multiplication by 2. Thus, for ordinary Chern-Simons, , whereas for spin Chern-Simons, .
For , the case of actual interest in this paper, the map has index-2. The map also has index-2. Hence is an isomorphism. Thus, for ordinary Chern Simons, the two levels satisfy , with . For spin Chern-Simons, can be any integers.
As it turns, for the Bagger-Lambert , we are interested in , , so the carry-away from this analysis is that any integer value of is allowed, and there is no distinction between the ordinary and spin Chern-Simons cases.
B Monopole Charge Quantization
In this appendix, we briefly explain why the quantization (13) of the monopole charge in this theory is such that the minimum charge is twice the one implied by Dirac’s quantization condition. It is well known that for ’t Hooft-Polyakov monopoles, the minimum charge is actually twice the Dirac value . In that case, all fields transform in the adjoint representation of the gauge group , so effectively the gauge group is .
In the theory we are considering, the gauge field
corresponding to the unbroken at generic points on the moduli
space sits inside the diagonal , and all the matter fields transform in the
adjoint of this , so the situation sounds similar to the ’t
Hooft-Polyakov case. However, while monopole configurations in the ’t
Hooft-Polyakov case are classified by
but it is not obvious that this is an isomorphism. In particular, if the generator of mapped to the square of the generator of , the BL theory would contain monopoles with the minimal Dirac charge. It turns out that there can be no such configurations, since the map (38) is onto, as we will now show using the following theorem .
Theorem: If is a fibration and if , , and , then taking as the base point of and of and as the base point of , we have the exact sequence:
For our purposes, we take
We can represent by pairs of matrices where
Take the fibration map to be . We can take so that is the subgroup of such that , which is . Now, a particular part of the exact sequence is
Since , exactness implies that the map is onto, as we wanted to show.
We would like to thank Jim Bryan and Shiraz Minwalla for helpful discussions. We thank Neil Lambert and David Tong for correspondence in sorting out the discrepancies between the original versions of our paper and of . J.D. would like to thank Dan Freed for topological guidance. The work of MVR has been supported in part by the Natural Sciences and Engineering Research Council of Canada, the Alfred P. Sloan Foundation, and the Canada Research Chairs programme. The work of J.D. was supported, in part, by NSF grant PHY-0455649.
- preprint: TIFR/TH/08-13
- The quantization of the Chern-Simons coefficient for a non-simply-connected gauge group, , is a little subtle. See Appendix A for details.
- In contrast, orientifold 4-planes in type IIA lift to orientifold 5-planes in M-theory [9, 10].
- By compactifying a direction transverse to one can relate it to a type IIA orbifold, however in this case it becomes an orbifold 1-plane and carries the charge of fundamental strings rather than D2-branes.
- In this theory, the minimum monopole charge is double the one implied by the Dirac quantization condition. We justify this in Appendix B.
- In general, the moduli space for gauge group with rank and Weyl group is .
- We thank Nima Arkani-Hamed, Neil Lambert and David Tong for their comments on this issue.
- This could be an M-theory analogue of the mechanism in Ref.  where a geometrical or supergravity analysis yields misleadingly low amounts of supersymmetry but additional stringy modes enhance the supersymmetry. In our system, the fact that acts on the M2-branes may also be relevant.
- Other trace structures give additional operators. For , such operators with an odd number of indices do vanish identically, but for this case, the moduli space is only eight dimensional.
- As we mentioned in Section 2, there is a plausible maximally supersymmetric orbifold  for precisely , suggesting a distinctive role for this case along with . The orbifold is related to the strong coupling limit of D2-branes at an orientifold. A more detailed discussion of this case may be found in .
- In general, monopole solutions are classified by where is the gauge group and is the unbroken subgroup.
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