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Supersymmetric M3-branes and G_2 Manifolds

We obtain a generalisation of the original complete Ricci-flat metric of G_2 holonomy on R^4\times S^3 to a family with a non-trivial parameter \lambda. For generic \lambda the solution is singular, but it is regular when \lambda={-1,0,+1}. The case \lambda=0 corresponds to the original G_2 metric, and \lambda ={-1,1} are related to this by an S_3 automorphism of the SU(2)^3 isometry group that acts on the S^3\times S^3 principal orbits. We then construct explicit supersymmetric M3-brane solutions in D=11 supergravity, where the transverse space is a deformation of this class of G_2 metrics. These are solutions of a system of first-order differential equations coming from a superpotential. We also find M3-branes in the deformed backgrounds of new G_2-holonomy metrics that include one found by A. Brandhuber, J. Gomis, S. Gubser and S. Gukov, and show that they also are supersymmetric.


Introduction
Recently the study of M-theory on spacs of G 2 holonomy has attracted considerable attention. In particular, it has been proposed that M-theory compactified on a certain singular seven-dimensional space with G 2 holonomy might be related to an N = 1, D = 4 gauge theory [1,2,3,4,5] that has no conformal symmetry. (See also the recent papers [6,7,8,9,10].) The quantum aspects of M-theory dynamics on spaces of G 2 holonomy can provide insights into non-perturbative aspects of four-dimensional N = 1 field theories, such as the preservation of global symmetries and phase transitions. For example, Ref. [5] provides an elegant exposition and study of these phenomena for the three manifolds of G 2 holonomy that were obtained in [11,12].
Studying the classical geometry of eleven-dimensional supergravity on spaces with G 2 holonomy provides a starting point for investigations of D = 4 vacua in M-theory. Recently, in [13], new configurations in M-theory were found, which describe M3-branes with a (3+1)-dimensional Poincaré invariance on the world-volume. These configurations arise as solutions of D = 11 supergravity in which the 4-form field is non-vanishing in the sevendimensional transverse space, which is deformation of the a Ricci-flat metric of G 2 holonomy.
The transverse spaces for M3-branes obtained in [13] were deformations of the three explicitly known metrics of G 2 holonomy that were found in [11,12]. The original metrics are all of cohomogeneity one: two have principal orbits that are CP 3 or SU (3)/(U (1)×U (1)), viewed as an S 2 bundle over S 4 or CP 2 respectively, and the third has principal orbits that are topologically S 3 × S 3 . The M3-branes on these deformed G 2 holonomy spaces arise from first-order differential equations derivable from a superpotential, which in fact are the integrability conditions for the existence of a Killing spinor. Thus the M3-branes obtained in [13] are supersymmetric.
A number of issues related to the properties of G 2 holonomy spaces invite further study.
For example, it is of interest to find generalisations of the original three complete metrics of G 2 holonomy that were obtained in [11,12]. Of particular interest is to find families of G 2 metrics on these manifolds that have non-trivial parameters (i.e. not merely the scale parameter that any Ricci-flat metric has). Such families of metric, when reduced on a circle could, for example, provide a connection between the six-dimensional metrics on the deformed and resolved six-dimensional conifolds [17], thus providing insights into the relation [2] between M-theory on these classes of G 2 holonomy metrics, and Type IIA string theory on the corresponding six-dimensional special holonomy spaces. 1 Recently, in Ref. 1 For 8-dimensional spaces of Spin (7) holonomy an explicit family of non-singular generalisations has been [16], a new class of G 2 holonomy metrics with such properties was constructed.
In section 2 we construct a new two-parameter family of metrics of G 2 holonomy, on the manifold of R 4 × S 3 topology. We do this by starting from a rather general ansatz for metrics of cohomogeneity one on manifolds whose principal orbits are S 3 bundles over S 3 , which is a generalisation of the ansatz for the original G 2 metrics of this topology in [11,12]. The general ansatz involves nine functions of the radial coordinate, which parameterise homogeneous squashings of the base and fibre 3-spheres. We have not found a superpotential formulation for the Einstein equations in this most general case, but by making a specialisation to a 3-function ansatz that is spherically symmetric in the base and the fibres, we have found a superpotential and hence first-order equations. We show that these equations are in fact the integrability conditions for the existence of a single covariantly-constant spinor, and hence for G 2 holonomy. The first-order equations can be solved explicitly, yielding Ricci-flat metrics with two parameters, one of which is the (trivial) scale size, while the other, which we call λ, is non-trivial and characterises genuinely inequivalent metrics. 2 For a generic value of the parameter λ these metrics are singular, but they are regular for three discrete values, namely λ = {−1, 0, +1}. The case λ = 0 corresponds to the original metric of G 2 in [11,12], while the cases λ = {−1, 1} are related to λ = 0 by an S 3 (permutation) automorphism of the SU (2) 3 isometry group. 3 The S 3 action restricts the parameter space of these metrics to a "fundamental domain" 0 ≤ λ ≤ 1 3 . However, when viewed as a reduction on a specific circle, M-theory on the three regular solutions is related to the type IIA theory on a deformed conifold and two types of resolved conifold respectively.
In section 3 we consider a different specialisation of the nine-function metric ansatz introduced in section 2, in which six metric functions remain. For this particular truncation a superpotential can be found, and we show that the resulting first-order equations are the integrability conditions for a covariantly-constant spinor, and hence G 2 holonomy. A specialisation in which two of the three directions on the base and fibre 3-spheres are treated on an equivalent footing yields the ansatz and first-order system introduced in [16]. We discuss found recently [14,15], which has such a non-trivial parameter. They provide a connection between N = 1 D = 3 vacua of M-theory, and Type IIA string theory on G2 holonomy manifolds with R 3 × S 4 topology. 2 There is also a third completely trivial parameter corresponding to a constant shift of the radial coordinate. 3 The discrete S3 coordinate transformations (for details see section 2.3) were found in [5] as a triality symmetry among the three regular G2 holonomy metrics with important consequences for the phase diagrams of the field theory associated with these three branches.
how this is related by dimensional reduction to the metric on the six-dimensional Stenzel manifold. A further specialisation of the six-function ansatz to a spherically-symmetric one with just two remaining functions yields a first-order system whose solution is the original metric of G 2 holonomy on the R 4 bundle over S 3 , expressed now in a manifestly Another aim of the paper is to construct further examples of supersymmetric M3-branes.
In section 4 we extend the methods introduced in [13] to obtain the equations of motion for M3-branes in the background of deformations of the new G 2 metrics that we constructed in section 2. We show that these equations admit a superpotential formulation, and we obtain the general M3-brane solution for these configurations. We also obtain the analogous system of equations for M3-branes in the background of deformations of the new G 2 ansatz introduced in [16], which we discussed in section 3. Again, we find a superpotential and associated first-order equations. In section 5 we show that these first-order equations are the integrability conditions for the existence of a Killing spinor. Thus the new M3-brane solutions are supersymmetric.
2 A class of metrics of G 2 holonomy

The ansatz and Einstein equations
An approach to looking for metrics of G 2 holonomy is to make a generalisation of the original G 2 metric found in [11,12] on R 4 × S 3 . We take as our starting point the metric where σ i and Σ i are left-invariant 1-forms on two SU (2) group manifolds, S 3 σ and S 3 Σ , and a i , b i and g i are functions of the radial coordinate t. The principal orbits are therefore S 3 bundles over S 3 , and since the bundle is topologically trivial, they have the topology where the two factors denote left-acting SU (2) transformations on S 3 σ and S 3 Σ respectively. The original metric ansatz used in obtaining the solution in [11,12] is obtained by taking a i = a, b i = b, and g i = − 1 2 . Note that in this "round" case the isometry group is enhanced to SU (2) σ L × SU (2) Σ L × SU (2) D R , where the third factor denotes the diagonal SU (2) subgroup of the product of right-acting SU (2) σ R and SU (2) Σ R transformations. There is a natural triality permutation symmetry S 3 , generated by the Z 2 "flop" described in [2] that interchanges the SU (2) σ L and SU (2) Σ L factors, and another Z 2 "flip" operation under which SU (2) group elements G σ and G Σ on the two 3-spheres are transformed according . These will be discussed in more detail in section 2.3. It is straightforward to obtain the conditions for Ricci-flatness of the metric (1). We find that they can be derived from the Lagrangian L = T − V , together with the constraint T + V = 0, where T = 1 2 g ij (dα i /dη) (dα j /dη), and the 9 × 9 metric on the "non-linear sigma model" for the functions in the ansatz is given by The 9 functions α i are defined by and the indices a and b in (2) range over the 6 × 6 block with 1 ≤ a ≤ 6 and 1 ≤ b ≤ 6.
The radial coordinate η is related to t by dt = ( i a i b i ) dη.
2.2 Specialisation to the "round" metric, and its general solution

Superpotential and first-order equations
There does not appear to exist a superpotential and associated first-order equations for the 9-function ansatz (1). However, we have found a superpotential for the case where we take all three directions equal, so that just three functions remain. It is convenient to write these as and so the ansatz (1) now reduces to Since in this case both the base and the fibre are round three spheres, giving th enhanced SU (2) 3 symmetry discussed in section 2.1, we shall refer to this as the "round G 2 manifold." The Lagrangian L = T − V describing the Ricci-flat equations for this is given by where α 1 ≡ log a, α 2 ≡ log b and α 3 ≡ g, η is defined by dt = a 3 c 3 dη, and The Lagrangian is of a "non-linear sigma model" type, where the metric g ij depends on the functions in the system. We find that there is a superpotential for this system, such that and The associated first-order equations, expressed in terms of the original radial variable t, are

Covariantly-constant spinor and G 2 holonomy
We now show that the first-order equations (10) are the integrability conditions for the existence of a covariantly-constant spinor. In turn, this demonstrates that the metrics (1) with functions given by (5) satisfying (10) have the special holonomy G 2 .
From the spin connection for the metric, we find that in the obvious choice of orthonormal basis e 0 = dt, e i = b σ i and eĩ = a (Σ i +(g− 1 2 ) σ i ), and spin frame, a covariantly-constant spinor η satisfying Dη ≡ dη + 1 4 ω ab Γ ab η = 0 is independent of the coordinates of the S 3 ×S 3 principal orbits, but it does depend on the radial coordinate t. After some algebra we find that if the first-order equations (10) are satisfied there is exactly one solution, which is given where where η 1 and η 2 are constant spinors, satisfying the constraints Here the explicit index values 1, 2 and 3 refer to the vielbein components e i in the S 3 base manifold, while 4, 5 and 6 refer to the vielbein components eĩ in the S 3 fibres.

The general solution to the first-order equations
We can obtain the general solution to the first-order equations (10) as follows. Defining a new radial coordinate w by dw = a K −1 dt, and new variables α, β and γ by a 2 = α, b 2 = β and g 2 = 1 4 γ, the first-order equations become which, after the further redefinition γ = ψ −4 , becomes From this we obtain (dψ/dw) 2 =k + 4ψ 2 + 4ψ −2 , and hence, after writing e 4w =k ρ and thenk = 8/λ, we arrive at the general solution where (Note that dF/dρ = 12ρ Y .) Thus the metric in this general solution is given by The metric is asymptotically conical (AC), with base the non-product Einstein metric on Note that if m = 0, the λ dependence of the solution drops out, and the metric becomes simply the Ricci-flat cone over the S 3 × S 3 base.
When |ρ| is large, the functions F and Y are both large and positive, with F ∼ 3ρ 4 , Y ∼ ρ 2 and g ∼ 0. Letting ρ = r 3 , we see that at large r, after dropping an overall constant scale factor 12(3) 1/3 , we shall have Thus in this region the metric is asymptotic to the cone over S 3 × S 3 .
As |ρ| reduces from infinity, the metric remains regular until the first singularity in the metric functions is encountered. Let us first check the function Y . This has zeros at ρ = m (1 ± λ). Substituting these values into F , we see that it is given by Thus leaving aside the special cases λ = 0 or λ = ±1 for now, we see from the fact that F is negative here that the first zero of F must have occurred before Y reached its first zero.
Thus when λ = ±1 or 0, it is the first zero of F , and not Y , that defines the inner endpoint ρ = ρ 0 of the range of the radial variable. This occurs at , we see that near ρ = ρ 0 the metric takes the form The metric is clearly singular at r = 0 (i.e. at ρ = ρ 0 ).
Having shown that the metric is singular for general values of the non-trivial parameter λ, we now consider the three special values, λ = 0 and λ = ±1, for which were excluded from the general analysis.

The non-singular solutions
There are three values of the non-trivial integration constant λ for which we find that the generically singular G 2 metric (18) is regular, namely λ = 0 and λ = ±1. We shall consider these one by one.
• λ = 0: In this case, we have If we define a new radial coordinate r = √ 3 (3ρ + m) 1/3 and scale parameter ℓ 3 = 4 √ 3 m, the metric (18) then becomes This is the previously-known complete metric of G 2 holonomy on the R 4 bundle over S 3 , as found in [11,12]. .
After making the redefinitions r = √ 3 (3ρ − 8m) 1/3 and ℓ 3 = −24 √ 3 m, the metric (18) becomes This metric is clearly regular, and indeed it can easily be seen to be equivalent to the original metric of [11,12]. This can be done by first "recompleting the square" so that the terms in the metric are instead organised with the structure

and then making the transformation
After doing this, the metric becomes precisely (22). Note that this Z 2 transformation is precisely the "flop" described in [2].
• λ = +1: In this case we have Under the same transformations r = √ 3 (3ρ − 8m) 1/3 and ℓ 3 = −24 √ 3 m as for the case λ = −1, the metric (18) becomes with This differs from λ = −1 case only in the sign of g, and the metric is again regular. However, if one repeats the previous procedure of recompleting the square and interchanging σ i and Σ i , one now finds that (28) simply maps into itself. It is, nevertheless, equivalent to the original metric (22), but the required coordinate transformation that explicitly implements this is more complicated. This will be discussed in the next subsection.

S 3 transformations of the round G 2 metrics
In this subsection, we shall show that there is an S 3 triality transformation of the metric (18) that maps it back into its original isotropic form, with transformed values of its parameters. 4 The three permutation transformations can be generated by two types of operation. The first is a generalisation of the one we already used in order to transform λ = −1 regular metric into the λ = 0 metric, which amounts to an exchange of the base and the fibre 3spheres. The second is a more complicated one, involving transformations of the coordinates on the 3-spheres. We shall discuss these in succession.
• The interchange of the 3-spheres This transformation follows from the "recompletion of squares" identity Since after exchanging σ i and Σ i this leaves the metric (18) in the same class of "round" metrics as (1), (5), and since (18) is the general solution of the first-order equations, it follows that the transformed metric must be contained within the general solutions (18), after appropriate transformations of the parameters. 5 It is easily seen from (31) that the effect of the σ i ↔ Σ i flop automorphism is to map the solutions as follows: Clearly this is an involution, giving a Z 2 transformation on the solutions.
A special case of the above is to take λ = −1, which maps toλ = 0. This is the transformation that we used in the previous subsection to show that (24) is equivalent to the original G 2 metric (22). On the other hand if we apply it to the case λ = 1, we see that 4 This triality symmetry among the regular G2 holonomy metrics was discussed in detail in [5]. M.C. would like to thank E. Witten for sharing with her the draft of [5] prior to the publication. 5 There are really three parameters in the general solutions, namely the non-trivial parameter λ, the scale parameter m, and a trivial constant shift of the radial coordinate that we have not made explicit in (18).
Our statement about the mapping of solutions into solutions must be understood to include the need to perform such a constant shift.
this is a fixed point of the transformation, which was already observed in our discussion of the metric (28).
• SU (2) transformations Let τ i be the Pauli matrices, and let U and V be 2 × 2 matrices for two SU (2) groups.
Then we can define the left-invariant 1-forms σ i and Σ i as follows: Now, make a coordinate transformation to new SU (2) matrices U and V , related to U and V as follows: automorphism of the round metrics that we mentioned in section 2.1.) It follows that we shall have We also define "tilded" left-invariant 1-forms Since we can write and so on, it now follows that In other words, the net effect of the transformation (34) is simply to send g −→ −g.
Expressed as a transformation of the parameters in the general solution (18), this therefore just amounts to while leaving the other parameters unchanged. Thus we see that the S 3 permutation group acting on the parameter λ is generated by λ → (λ + 1)/(3λ − 1) and λ → −λ. The fundamental domain of λ is therefore given by For a generic value of m, the solutions form triplets under the S 3 group. When m = 0, corresponding to the Ricci-flat cone over S 3 × S 3 , the triplet degenerates to a singlet.
A particular application of this transformation is to the metric (28), which corresponds to λ = +1. It is therefore mapped into the previous example (24), which corresponds to λ = −1. Indeed, we may note that as solutions of (18) these metrics differ only in the sign of g. This, therefore, provides the additional transformation that is needed in order to show that (28) is again equivalent to the original G 2 metric (22).
As another application of the general result (34), we may note that This enables us to rewrite the original G 2 metric (22) in a form that is manifestly invariant under the interchange of σ i and Σ i , namely As we shall see below, this basis is closely related to the one used in [19] for constructing the metric on the cotangent bundle of S 3 .

S 1 reduction to type IIA
Since the G 2 manifold is Ricci-flat with special holonomy, it provides a natural compactification (reduction) space for M-theory. The vacuum solution is just a metric product of four-dimensional Minkowski space-time and that of the G 2 manifold. Because of the discrete triality automorphism of the round G 2 manifold, it follows that there are three different ways of reducing it on S 1 , so that the principal orbits of the lower dimensional manifold are the T 1,1 = SO(4)/SO(2) coset space. 6 In order to do so, we note that the Σ i can be written in terms of Euler angles as Σ 1 = cos ψ dθ + sin ψ sin θ dϕ , Σ 2 = − sin ψ dθ + cos ψ sin θ dϕ , Σ 3 = dψ + cos θ dϕ .
One can then verify that [20] i where µ 1 = sin θ sin ψ , µ 2 = sin θ cos ψ , and D g µ i = dµ i + ǫ ijk (g − 1 2 ) σ j µ k . The λ = 0 solution therefore reduces to become a wrapped D6-brane in type IIA: The six-dimensional transverse space here has a similar structure to the deformed conoifold.
The λ = −1 solution becomes where g is given by (25). The six-dimensional transverse space has a structure similar to the resolved conifold. These two reductions were previously obtained in [2]. The λ = 0 reduction is singular from the ten-dimensional point of view, but the λ = −1 reduction is regular.
Employing the S 3 triality transformations, discussed in the previous subsections, we can obtain the third case, namely λ = 1. The form of the metric is the same as (47) but with g replaced by −g. It should be emphasised that from the type IIA point of view, the choice of sign for g has a non-trivial effect. The λ = 0 solution gives the same topology as the deformed conifold, whilst the two solutions λ = ±1 give the topology of the resolved conifold. These two resolutions arise from the fact that there can be a sign choice for the Fayet-Illiopoulos term for the field theory arising from type IIA string theory on a resolved conifold. 7

Squashed manifolds with G 2 holonomy
We can make a different truncation of the 9-function ansatz (1), in which six functions remain, which again allows us to contruct a superpotential whose first-order equations yield first integrals of the Ricci-flat Einstein equations. A further truncation in this example, to an ansatz with four remaining functions, gives the system considered first in [16]. 8 We shall 7 We thank E. Witten for making this observation. 8 We We take as our starting point the construction of the Stenzel metrics on T * S n+1 , using the formalism and notation of [19]. The metrics are of cohomogeneity one, with principal orbits that are SO(n + 2)/SO(n). Denoting the left-invariant 1-forms of SO(n + 2) by L AB , and decomposing under the SO(n) subgroup according to A = (1, 2, i), the following ansatz for Stenzel metrics was made in [19]: Note that the 1-forms L 1i , L 2i and L 12 are precisely the ones that live in the coset SO(n + 2)/SO(n).
Let us specialise now to n = 1, which gives a construction of the 6-dimensional Stenzel metric, with principal orbits SO(4)/SO (2). In this case, we can now give a natural extension of the ansatz to one for 7-dimensional metrics of cohomogeneity one, where the principal orbits are SO(4), by appending an additional term f 2 (L 34 ) 2 : (The factors of 4 are introduced for later convenience.) It is helpful now to exploit the local isomorphism between SO(4) and SU (2) × SU (2), which is made manifest by forming selfdual and anti-self-dual combinations of the L AB . A convenient choice for these combinations is It now follows from (48) that the σ i and Σ i are left-invariant 1-forms for the two SU (2) factors, satisfying In terms of these, the metric ansatz (50) becomes The principal orbits are SO(4) (or its double cover Spin(4)), but the isometry group may be larger in special cases. The form (53) is the same as the ansatz previously introduced in [16]. Indeed the above derivation of the ansatz (53) shows how the ansatz of [16] is related to the construction of the six-dimensional Stenzel metric [17,18,19], which is the deformed conifold. In particular, since L 34 = 1 2 (σ 3 + Σ 3 ), the reduction to six dimensions is achieved by omitting the last term in (53).
A further generalisation of the ansatz clearly suggests itself, namely to write where the functions a AB depend only on t. In the SU (2) × SU (2) basis for SO(4), this becomes By completing the square, it is easily seen that this ansatz is a special case of our original 9-function ansatz (1).
Note that this ansatz is subsumed in the (1) with the specialisation Calculating the curvature, we find that the conditions for Ricci-flatness can be derived from the Lagrangian L = T − V for the functions a i and b i , together with the constraint T + V = 0. Defining α i ≡ log a i and α i+3 ≡ log b i , the kinetic energy is given by T = where the new radial variable η is defined by dt = i (a i b i ) dη. We find that the potential energy V can be derived from a superpotential W , so that V = − 1 2 g ab (∂W/∂α a ) (∂W/∂α b ), with (Other sign choices can also arise, but these just correspond to relabellings of indices, and so they give equivalent results.) From the superpotential we can derive the first-order equations dα a /dη = g ab ∂W/∂α b .
Expressed back in terms of the original radial variable t, these arė We may note that a consistent truncation of these equations is to set a 2 = a 1 and b 2 = b 1 , leading to the simpler system of equationṡ .
These are the first-order equations found in [16]. 9 The solutions give rise to metrics with a Taub-NUT like squashing of the S 3 fibres (as well as a squashing of the S 3 base), with the circle corresponding to (σ 3 + Σ 3 ) tending to a constant radius at infinity. Thus the metrics are asymptotically locally conical (ALC).
We can make a further consistent truncation of the first-order equations (60), by setting By the standard methods, we can solve these by defining a new radial coordinate ρ such that dt = dρ/b, leading to a 2 = ρ and b 2 = 1 3 ρ + k ρ −1/2 . After setting ρ = r 2 , and k = −ℓ 3 /3, we get (dropping an overall factor of 12 from the front) where This is in fact the original G 2 metric on R 4 × S 3 , written in the Z 2 -symmetric form that we presented in (42).

Parallel spinor and G 2 holonomy
It is a straightforward matter to solve the equations Dη ≡ dη+ 1 2 ω ab Γ ab η = 0 for covariantlyconstant spinors, in the metrics (55). We find that there is precisely one such spinor, if and only is the six metric functions a i and b i satisfy the first-order equations (59) (or the discrete set of other possibilities corresponding to sign-convention choices). Thus the firstorder equations (59) coming from the superpotential have the interpretation of being the integrability conditions for the metrics (55) to admit a covariantly-constant spinor, and hence for them to have G 2 holonomy.
In the orthonormal frame with e 0 = dt, e i = a i (σ i − Σ i ) and eĩ = b i (σ i + Σ i ), with the obvious choice for the spin frame, we find that the covariantly-constant spinor η has constant components, and it satisfies the constraints Here the indices 1, 2 and 3 refer to the vielbein components e i , while 4, 5 and 6 refer to eĩ.
It is easy to verify that, up to overall scale, the conditions (64) specify the spinor uniquely.

Massless M3-branes 4.1 M3-branes on round G 2 manifolds
In [13], M3-brane were constructed on the background of the deformed G 2 manifolds, where the 4-form field strength of the eleven-dimensional supergravity was turned on. When the 4-form field strength is set to zero, the solution becomes the product of the Minkowski space-time and the regular G 2 manifold. In particular, we shall begin by considering the example studied in [13] that is based on the G 2 manifold of the R 4 bundle over S 3 . The metric for the associated massless M3-brane is [13] where We can now perform the two Z 2 transformations given in section 2.3, to obtain the previous M3-brane solution (65) written in the other two forms. Firstly, if we apply the σ i ↔ Σ i transformation as in (30), then we find that (65) becomes where H and Y are given in (66) and Instead, we can apply to (65) the SU (2) transformation as in (34). This gives the M3-brane in the form Indeed, (67) and (69) are equivalent, however they lead to very different solutions after reduction to type IIA theory on the circle corresponding to (σ 3 + Σ 3 ).
Since the regularity of the G 2 background in the above solutions is in any case lost when the M3-brane is introduced by turning on the 4-form field strength [13]) , it is not unreasonable to construct the more general M3-branes on deformations of the round G 2 manifolds of section 2.2. In analogy with [13], the metric ansatz will be given by where h i = Σ i + (g − 1 2 ) σ i . The ansatz for the 4-form field strength, and its resulting dual, are given by Thus we have The Bianchi identity and the equation motion for F 4 implies dF (4) = 0 : We find that the existence of a superpotential of such a system seems to require that the constant λ = 0. We shall set this constant to zero from now on. Thus we find that the 4-form ansatz is determined by one function only, namely f 3 . Thus, the total number of functions in our ansatz is 5, consisting of a, b, g, H and f 3 . After some algebra, we find that the Einstein equations and that for f 3 can be obtained from the Lagrangian L = T − V , where the kinetic term T can be expressed as and a dot denotes a derivative with respect to η defined by dt = a 3 b 3 H 4 dη. The metric g ij is somewhat complicated, but its inverse is simpler, which we present here The potential can be expressed in terms of a superpotential W , namely V = − 1 2 g ij ∂ i W ∂ j W , where W is given by where The first-order equations can now be straightforwardly obtained, and they are given by To solve them we begin by defining new hatted variableŝ and a new radial coordinate x by dx = H K 2 dρ. In terms of these, we find that the first-order equations (77) become We also find thatâ where κ is a constant of integration.
The general solution of these equations can be obtained as follows. To begin with, we note that the first three equations in (79) are precisely the same as (10), in terms of the new variables. Thus the solution forâ,b and g as functions of x will be identical that for a, b and g as functions of t for the Ricci-flat G 2 holonomy 7-metrics (6). 10 Thus in terms of a new radial variable r, defined by dx = F −1/6 dr, we obtain from (16), where From (81) we then obtainf Finally, solving the equation for H in (79), and fixing a constant of integration without loss of generality, we obtain In general the new Ricci-flat G 2 manifolds found in section 2 are singular where F = 0.
Here, we see that when the 4-form field strength is turned on, the associated M3-brane metric becomes singular before this point is reached, namely at the value ofr r for which F = 64κ 2 (and hence H = 0). Again, the function H falls off too rapidly as a function of the proper radial distance for the M3-brane to have non-zero charge and ADM mass.
It is worth noting that in this case, as in the previous M3-brane examples in [13], the integration constant κ appears in the 4-form field strength linearly, but in the metric quardratically. It follows that also in this case the metric remains real if κ is replaced by i κ. As discussed in [13], the replacement κ −→ i κ can be viewed as the replacement of a

M3-branes on squashed G 2 manifolds
In this section, we shall study the massless M3-brane whose transverse space is a deformation of the class of metrics of G 2 holonomy whose first-order equations, given in (60), have been constructed in [16]. These solutions will provide further examples of massless M3brane configurations that are of the same general pattern as those constructed in [13]. We therefore begin by making the ansatz where for the metric in the transverse space, the appropriate invariant ansatz for the 4-form is G (4) = u 1 e 1 ∧ e 2 ∧ e 4 ∧ e 5 + u 2 e 2 ∧ e 3 ∧ e 5 ∧ e 6 + u 2 e 3 ∧ e 1 ∧ e 6 ∧ e 4 +u 3 e 0 ∧ e 4 ∧ e 5 ∧ e 6 + u 4 e 0 ∧ e 1 ∧ e 2 ∧ e 6 +u 5 e 0 ∧ e 2 ∧ e 3 ∧ e 4 + u 5 e 0 ∧ e 3 ∧ e 1 ∧ e 5 .
It is straightforward to calculate F 2 ab in the vielbein basis, leading to F 2 00 = 6(u 2 3 + u 2 4 + 2u 2 5 ) , F 2 11 = F 2 22 = 6(u 2 1 + u 2 2 + u 2 4 + u 2 5 ) , F 2 33 = 6(2u 2 2 + 2u 2 5 ) , original second-order equations. In order to find a superpotential for the system, it seems to be necessary to take the integation constant λ in (93) to be zero. A second constant of integration also needs to be set to zero in order to find a superpotential, implying that f 1 and f 2 become algebraically related: Now our second-order differential equations involve six functions, namely a, b, c, f, H and f 2 . We find that their kinetic energy T = 1 2 g ij (dα i /dη) (dα j /dη) has a metric whose inverse is given by g ij = g ij 0 + g ij 1 , where and In fact an instructive way to study the Killing spinors is to reverse the logic, and to derive first-order equations by requiring the existence of such spinors. We can then show that these first-order equations are equivalent to those obtained in sections 4.1 and 4.2 via a construction of the superpotential. In the following two subsections we shall describe this procedure for the "round" and "squashed" M3-branes of sections 4.1 and 4.2 respectively.

Killing spinor in the "round" M3-brane
The gravitino transformation rule in D = 11 supergravity is given by It is useful to make a 4 + 7 split of the eleven-dimensional Dirac matrices: and to seek Killing spinors of the formǫ = ǫ⊗η. Since the field strength F (4) has components only in the 7-dimensional space transverse to the 3-brane, it follows from (102) that in the 3-brane world-volume directions the requirement δψ µ = 0 leads to ∂ µ ǫ = 0, γ 5 ǫ = ±ǫ, and the conditions respectively, where, as usual, the "0" index denotes the radial direction in the transverse space. The matrix that acts on η here has zero eigenvalues if H satisfies one of the following where u 1 , u 2 , u 3 and u 4 are the vielbein components of the field strength F (4) given in (71), i.e. with Since the matrix acting on η in (104) has just one zero eigenvalue if (107) is imposed, this means that requiring δψ µ = 0 determines the Killing spinor η in D = 7 completely, up to an overall multiplicative function of the 7-dimensional coordinates on the transverse space.
From the conditions δψ a = 0 in the six directions on the S 3 bundle over S 3 of the principal orbits, we then find that η must be independent of the six coordinates on the two 3-spheres.
These components of the gravitino variation also give rise to first-order equations. Since the full set of equations are somewhat involved and cumbersome to present, we shall not give them explicitly here, but simply record that they eventually turn out to be equivalent to the first-order equations for the M3-brane solutions that we obtained in section 4.1.
Finally, from the component of the gravitino variation in the radial direction, we can determine the radial dependence of the Killing spinor. The result for the case γ 5 ǫ = +ǫ is where , and η 1 and η 2 are covariantly constant spinors satisfying the constraint (13). For γ 5 ǫ = −ǫ, the associated spinor η in the transverse space is again given by (108), but now with f 3 replaced by −f 3 . The general solution for a Killing spinor can then be written as a linear combination of ǫ + ⊗ η + and ǫ − ⊗ η − , where the plus and minus subscripts refer to the two chirality choices under γ 5 . Thus in total there will be 4 real solutions, implying N = 1 supersymmetry in the world-volume of the M3-brane.

Killing spinor for the "squashed" M3-brane
We can follow a similar procedure for the "squashed" M3-brane configurations obtained in section 4.2. Again, one can derive first-order equations as integrability conditions for the existence of Killing spinors, and again it turns out that these coincide with the first-order equations for the M3-brane solutions. Thus the squashed M3-branes are also supersymmetric.
We find that the Killing spinors can again be expressed as a linear combination of terms ǫ + ⊗ η + and ǫ − ⊗ η − , where γ 5 ǫ ± = ±ǫ ± . The spinor η + in the transverse space is given by where and η 1 and η 2 are covariantly constant spinors satisfying the constraint (13). The spinor η − is given by the same expressions, but with f 2 sent to −f 2 . Thus again we have 4 real solutions in total, and hence N = 1 supersymmetry in the world-volume of the M3-brane.

Conclusions
The two main results of the paper are (i) the construction of a new class of metrics with G 2 holonomy, and (ii) construction of new massless M3-brane solutions whose transverse spaces are deformations of the new G 2 metrics. We then showed that these M3-branes are supersymmetric.
The new class of metrics with G 2 holonomy that we found can be viewed as generalisation of the original G 2 metrics of R 4 × S 3 topology [11,12], which have just a single (trivial) scale parameter, to a new family of two-parameter metrics of G 2 holonomy, on the manifold of the same R 4 × S 3 topology. This result was obtained by actually starting with a rather general ansatz for cohomogeneity one metrics whose principal orbits are S 3 bundles over S 3 . For the most general ansatz that we considered, which contains nine functions of the radial coordinate, we did not find any first-order system of equations as first integrals of the Einstein equations. However, for a specialisation to a 3-function ansatz, which is spherically symmetric both in the base and in the fibre, we were able to find first-order equations derivable from the superpotential. The explicit general solution yields metrics with two parameters (aside from the completely trivial constant translation of the radial coordinate), one corresponding to the (trivial) scale size, and the second, called λ, being non-trivial and characterising inequivalent metrics. While for a generic value of the parameter λ these metrics are singular, they become regular for λ = {−1, 0, +1}. The case λ = 0 corresponds to the original metric of G 2 holonomy in [11,12]. The solutions possess an S 3 automorphism that allows one to map λ = {−1, 1} to λ = 0, and that allows the continuum of values for the λ parameter to be restricted to a fundamental domain 0 ≤ λ ≤ 1 3 . The second set of results involved explicit solutions for M3-branes whose transverse spaces are deformations of the two-parameter metrics of G 2 holonomy obtained in setion 2.
We obtained these M3-brane solutions by first constructing a system of first-integrals for the equations of D = 11 supergravity, following from a superpotential. We also constructed first-order equations for M3-branes whose transverse spaces are deformations of the new metrics of G 2 holonomy whose first-order equations were obtained in [16]. Just like the M3-branes in [13], these new M3-branes have neither mass nor charge, and have a naked singularity. Since this singularity occurs outside the radius of the innermost "endpoint" of the original undeformed metric of G 2 holonomy, this means that the singularities in the generic Ricci-flat metrics in section 2 do not materially affect the singularity structure of the associated M3-branes. We then showed that the first-order equations in both cases are precisely the integrability conditions for the existence of a Killing spinor, and hence that the M3-branes are supersymmetric.

Note added
In an earlier version of this paper, and in [13], it was claimed that the M3-brane solutions were not supersymmetric, but instead were "pseudo-supersymmetric" with respect to a modified D = 11 supersymmetry transformation rule. This incorrect conclusion resulted from a systematic error in a computer program that we used for calculating the Killing spinors. We are grateful to Jim Liu for calculations that encouraged us to recheck the computer programs and discover the error.