Dirac equation on a G_2 manifold

We find a large family of solutions to the Dirac equation on a manifold of $G_2$ holonomy asymptotic to a cone over $S^3 \times S^3$, including all radial solutions. The behaviour of these solutions is studied as the manifold developes a conical singularity. None of the solutions found are both localised and square integrable at the origin. This result is consistent with the absence of chiral fermions in M-theory on the conifold over $S^3\times S^3$. The approach here is complementary to previous analyses using dualities and anomaly cancellation.


Introduction
Eleven dimensional supergravity cannot give a chiral four dimensional theory upon compactification on a smooth manifold [1]. One way to resolve this discrepancy with experiment is by compactifying on a manifold with boundary [2,3]. Recently, work has centred on obtaining chiral fermions by the alternative approach of compactifying on singular manifolds. As supergravity is no longer valid near the singular points, the arguments make essential use of M-theory. Given that M-theory remains unformulated, this means they work via dualities with string theories.
Approaches include duality with intersecting D6-branes and orientifolds of type IIA string theory [4,5,6] and fibrewise duality with Heterotic string theory compactified on certain Calabi-Yau manifolds [7]. The duality results are corroborated by Witten's analysis of anomaly cancellation at conical singularities [8].
A recurrent theme in these arguments is compactification on manifolds of G 2 holonomy ( [9] for mathematical background, [10] for a pedagogical introduction). This is unsurprising as supersymmetry plays an important rôle in duality arguments and G 2 holonomy is the condition on the internal seven dimensional manifold for compactification from eleven to four dimensions to preserve N = 1 supersymmetry. The preservation of N = 1 supersymmetry is phenomenologically appealing and G 2 manifolds were considered in early work on Kaluza-Klein supergravity [11,12]. The possibility of further obtaining chiral theories with non-abelian gauge symmetry via M-theory is even more phenomenologically interesting.
Parallel to the renewed interest in the physics of G 2 holonomy compactifications, there has been a lot of recent work on constructing and classifying G 2 metrics. All the examples known are cohomogeneity-one metrics on noncompact spaces that are asymptotic to generalised cones [13,14,15,16,17,18,19,20,21,22]. The metrics constructed are generalisations of the metics found a decade ago by [23,24]. They are asymptotic to cones over SU (3)/U (1) × U (1), CP 3 and S 3 × S 3 . No explicit metrics are known on compact G 2 manifolds because these cannot have continuous isometries except for trivial S 1 factors. The appearance of chiral fermions is a phenomenon localised near the conical singularity and so considering noncompact manifolds should not be problematic.
The original argument against obtaining chiral theories from eleven dimensions [1] is circumvented by the fact that the non-abelian gauge groups under which the chiral fermions are charged do not come from isometries of the internal manifold, but from massless branes wrapping collapsed homology cycles in the internal manifold [25,26]. It would be nice to see these new massless fermionic degrees of freedom emerge as homology cycles collapse directly within an eleven dimensional framework, without using dualities. Further, for the G 2 manifold asymptotic to a cone over S 3 × S 3 , call it X, the situation regarding chiral fermions is not yet understood, although other aspects of M-theory on this space have been studied in some detail (e.g. [4,27]).
There are various reasons for this: (a) The anomaly analysis of [8] is not applicable as for this space H 2 (X; U (1)) = 0. (b) The G 2 manifolds considered in [7] were R 3 bundles over a selfdual four-fold whilst X is an R 4 bundle over S 3 . (c) Whilst the cones over SU (3)/U (1) × U (1) and CP 3 were interpreted in [4] as the M-theory lift of intersecting D6-branes, the analogous interpretation for the S 3 × S 3 case is not clear.
The line of attack in this work will be to look for solutions to the massless Dirac equation on manifolds on G 2 holonomy and see how the behaviour of the solutions changes as the manifold developes a conical singularity. We use the Dirac equation and not the Rarita-Schwinger spin-3/2 equation because the degrees of freedom we are interested in do not originate from supergravity but from wrapped branes. It turns out that the G 2 manifold on which the Dirac equation is most tractable, allowing us to find all radial solutions and several non-radial solutions, is precisely the one asymptotic to a cone over S 3 × S 3 . Thus the information we gain through this approach is nicely complementary to the information already obtained through dualities. Solving the Dirac equation on other G 2 manifolds could also be interesting, but is not done here.
In for degrees of freedom corresponding to the chiral fermions. In §4 we discuss these results in the context of chiral fermions appearing in singular M-theory compactifications. We suggest that our results are consistent with the absence of chiral fermions arising in the singular limit of M-theory compactified on a G 2 holonomy manifold asymptotic to a cone over S 3 × S 3 .

Dirac equation and all radial solutions
We will work with the simplest G 2 holonomy metric asymptotic to a cone over S 3 × S 3 which has metric [23,24] where the σ i and the Σ i are left invariant one-forms on the two copies of S 3 , that is The range of r is r 0 ≤ r < ∞. It is useful to consider the vielbeinŝ We will use tangent space indices i, j, k = 1, 2, 3 andî,ĵ,k =1,2,3 = 4, 5, 6 and also a, b, c... = 0...6. Spacetime indices will be µ, ν....
The massless Dirac equation corresponding to the metric (1) can be read off from the calculation of the connection one-form in [24], where as usual Γ ab = 1 2 [Γ a , Γ b ] and e a are vector fields dual to the vielbeins (3), < e b , e a >= δ b a . We use a convention for the gamma matrics such that the Clifford algebra is To solve this equation we need to choose a representation for the Euclidean Clifford algebra in seven dimensions. For this algebra there is a representation in which all the matrics are real.
It is given by [28,29] (Γ a ) st = c ast + δ as δ t7 − δ at δ s7 , where the spinor indices s, t = 0...7 and c ast are zero if s = 7 or t = 7 and otherwise are the totally antisymmetric octonion structure constants, with In this representation, the Majorana or reality condition is particularly simple The Dirac operator (4) takes Majorana (real) spinors to Majorana spinors. Therefore by expressing an arbitrary spinor in terms of two Majorana spinors we may consider the Dirac equation for ψ (1) and ψ (2) separately.
For the remainder of this section we take ψ to be Majorana. We will thus obtain half of the solutions. However, the other half will be of exactly the same form but with a pure imaginary rather than real coefficient. Writing out ψ in components for i = 1, 2, 3. That is, we restrict the spinor to depend on r only.
To discuss the solutions, we need to distinguish the cases r 0 = 0 and r 0 = 0. In each case we

Solutions with r 0 = 0
The form of the determinant at the origin is now √ g ∼ r 6 as r → 0. The following table summarises the behaviour of the eight solutions.
The power-law asymptotics are in fact exact solutions in this case. We see that the asymptotic behaviour at infinity did not depend on r 0 , which is obvious from (2) anyway. Whilst the behaviour at the origin does change, it is qualitatively the same and still none of the solutions are both L 2 at the origin and localised at the origin. The possibility of the Lichnerowicz theorem not holding existed in this case because the manifold is singular.

Non-radial solutions
Some non-radial solutions may be obtained by using relationships between eigenfuntions of different differential operators. This is possible because of the existence of covariantly constant spinors on special holonomy manifolds [9]. The idea is developed in [30] for four dimensional hyperKähler metrics and in [24] for massive eigenfunctions on G 2 and Spin (7) metrics. The massless case we need is slighty more subtle. We will find solutions for the massles Dirac equation from solutions to the massless scalar equation. Suppose that φ satisfies ∆φ = −∇ a ∇ a φ = 0, (12) and that η is the covariantly constant spinor, Dη = 0. Then One can also obtain solutions of the Dirac equation from solutions of the massless vector equation [24], but this is no easier to solve than the Dirac equation itself and so is not useful to us here.
The first thing we need to calculate is the covariantly constant spinor. This can be done [24,31] by imposing the projection conditions and then requiring η to have constant norm η † η = 1 and to be Majorana as in (7). This first condition is consistent because D(η † η) = 0 and the reality condition is necessary because the supersymmetry generator must be real for degrees of freedom to match up in the supergravity theory. All this determines η up to a sign, which can be chosen. One obtains η T = 1 2 (1, 0, −1, 0, 0, 1, 0, 1) .
So the covariantly constant spinor has constant components. This will be generically true for cohomogeneity one metrics. The immediate consequence is that all r dependence of ψ will come from the scalar solution φ. By substitution it is easy to check that Dη = 0 as required (cf. the constant solutions of §2.1 and §2.2 above).
The massless Klein-Gordon equation on the manifold can be written where e µ a are the inverse vielbeins. This expression for the Hodge-de Rham operator is well suited for exploiting the symmetries of cohomogeneity-one metrics (1). It has been used to study the Schrödinger equation on the Atiya-Hitchen metric in [32] and on the Eguchi-Hanson metric in [33]. For the metric (1), the equation becomes where J i are the vector fields dual to Σ i and the J ′ i are dual to σ i . It follows that these vectors satisfy the algebra This means that as differential operators, Further, their eigenvalues are well known from elementary angular momentum theory. Acting on the simultaneous eigenfunctions Φ mn st we have where m, n ∈ 1 2 Z + ∪ {0} and {s, t} ∈ {m, n} + Z with −m ≤ s ≤ m and −n ≤ t ≤ n. Note that the half integers are allowed because the group in question is SU (2) and not SO (3). We are not interested in the precise functional form of the angular eigenfunctions. Details and references can be found in [33]. One now decomposes the function into modes of the form φ = Φ(r)Φ mn st . The equation (17) becomes This equation has four regular singular points, at r 3 = r 3 0 and at infinity. It does in fact have a solution in terms of hypergeometric functions, but these are only valid for r < r 0 whilst the range of r is r ≥ r 0 . This reduction to a single ordinary differential equation was only possible because of the relatively large symmetry of the metric (1). More general G 2 metrics with less symmetry would have resulted in a Klein-Gordon equation in which the differential operators were not all simultaneously diagonalisable.
The behaviour at infinity is independent of r 0 and we have that as r → ∞ where Q = m(m + 1) + n(n + 1) + st ≥ 0. In the radial case, Q = 0, the two possible behaviours are just the | ψ |∼ r 0 and | ψ |∼ r −6 solutions which we found in the previous section. For the second solution we use | ψ |∼ ∇ r φ ∼ ∂ r φ as discussed above. The case of Q = 0 will be discussed below. The behaviour at the origin depends on whether r 0 is nonzero.
3.1 Solutions with r 0 = 0 The behaviour asr → 0 for the non radial modes depends on whether the integer n is nonzero.
If n = 0 then asr → 0 The power ofr is nonzero and therefore | ψ |∼r 1/2 ∂rφ ∼r −1 2 2± √ 1+4n(n+1) . All of these solutions either are not localised or are not square integrable at the origin. Recall that the metric goes as √ g ∼r. The effect of having angular dependence on these solutions is to increase the degree of divergence at the origin of the localised solutions and increase the degree to which the non-localised solutions increase away from the origin.
And if n = 0 then asr → 0 These correspond to spinors behaving like | ψ |∼r 0 and | ψ |∼r −3/2 . The first does not decay and the second is not L 2 at the origin. This set of solutions does not change its radial behaviour as the angular dependence changes. The behaviour found is again consistent with the results of the previous section.

Solutions with r 0 = 0
The solution as r → 0 is the same as in the r → ∞ limit of (20). We now consider the effect of angular dependence. All of the solutions are nonconstant in r and so Again, all of these solutions either are not localised or are not square integrable at the origin, because Q ≥ 3/4. The metric now goes as √ g ∼ r 6 . Again, the effect of including angular dependence is to make the solutions divergent at the origin more divergent and the solutions that grow away from the origin grow faster.

Discussion
We have found solutions that are constant, solutions decaying away from the origin and solutions growing away from the origin. None of these satisified the two properties of being both localised and square integrable at the origin. The first of these is expected for degrees of freedom emerging as a homology cycle collapses and the second is necessary for the solution to make sense quantum mechanically. Allowing angular depedence worsens the situation in all the solutions we found.
The results admit three possible interpretations. Firstly, that the degrees of freedom corresponding to chiral fermions localised near the singularity correspond to nonradial solutions of the Dirac equation that we have not found. Secondly, that solving the Dirac equation is not the way to exhibit these degrees of freedom in any case, or that a modified Dirac equation including gauge connections is required. Thirdly, that the G 2 manifold asymptotic to a cone over S 3 × S 3 does not give chiral fermions in the singular limit, unlike the cones over SU (3)/U (1) × U (1) and The third of these possibilities is the most interesting. It links nicely with the results from dualities and anomaly cancellation discussed in the introduction. It also fits with the picture of extra massless modes coming from collapsed membranes and the fact that the G 2 metric asymptotic to a cone over S 3 × S 3 has zero second homology group. Solving the Dirac equation on other G 2 metrics that are known to give chiral fermions appears to be more challenging, but could lend further support to this possibility. Another interesting question is to consider the effect of including gauge field backgrounds in the Dirac equation.
Many thanks to Gary Gibbons for useful suggestions. Thanks also to Ruben Portugues, James Sparks and Tibra Ali for discussions. The author is funded by the Sims scholarship.