Intersecting M2- and M5-branes

If an M2-brane intersects an M5-brane the canonical Wess-Zumino action is plagued by a Dirac-anomaly, i.e. a non-integer change of the action under a change of Dirac-brane. We show that this anomaly can be eliminated at the expense of a gravitational anomaly supported on the intersection manifold. Eventually we check that the last one is cancelled by the anomaly produced by the fermions present. This provides a quantum consistency check of these intersecting configurations.


Introduction and Summary
An M2-and an M5-brane form an electromagnetically dual pair of branes in eleven dimensions, and since 3 + 6 − 11 is a negative number their worldvolumes have generically an empty intersection. However, for exceptional brane configurations it can happen that their intersection is non empty. Their intersection manifold Σ ≡ M2 ∩ M5 can then be a manifold of dimensions d = 0, 1, 2 or 3. An analysis of the quantum-consistency of such intersections, a special case of so called non-transversal intersections between two generic manifolds [1,2], is the main topic of this letter. The are two types of quantum inconsistencies we will have to worry about: 1) gravitational ABBJ-anomalies and 2) Dirac-anomalies i.e. (non-integer) changes of the action under a change of the Diracbrane. As we will see these two types of anomalies are intimately related.
The relevance for M-theory of the exceptional configurations considered in this letter, stems from the fact that eleven-dimensional supergravity admits indeed classical susy-preserving solutions, that can be interpreted as an M2-brane intersecting with an M5-brane [3]- [8]. These solutions are typically localized only in the common transverse directions of the two branes, i.e. the currents of the branes are δ-functions only in the common transverse coordinates. There exist also susy-preserving (implicit) solutions where one of the two branes is fully localized and the other is localized only in the common transverse directions [5,9]. In absence of a complete classification of all possible solutions, the existence of solutions where both branes are fully localized is still an open question.
Nevertheless, in this letter we assume that from a quantum point of view both branes are fully localized. The fundamental reason for this assumption is that since M2 and M5 are dual objects, only if both currents are δ-functions on the corresponding worldvolumes there exists a consistent minimal coupling of the branes to each other, because only in this case the charge is locally integer and, therefore, the Dirac-brane unobservable, [10,11]. The secondary reason is that only branes with a δ-like support represent a universal type of charge distribution. The situation is similar to the D = 4 Julia-Zee dyons [12]. These dyons represent semi-classical solutions of the Georgi-Glashow model, whose magnetic charge is fully localized (point-like localization) while their electric charge is smeared out. Nevertheless, at the level of second quantized quantum field theory the Julia-Zee dyons appear with fully localized magnetic and electric charges [13].
While this point deserves clearly further investigation, here we take a pragmatic point of view and assume that both branes, and therefore also their possible intersection, carry well-defined worldvolumes and that the associated currents are δ-functions supported on those worldvolumes.
For generic configurations (Σ = ∅) the two branes are at a finite non-vanishing distance and their classical dynamics is trivially free from relative short-distance singularities, i.e. singularities due to their mutual interaction. In this case a Dirac-brane can (must) be used to describe their dynamics, and the minimal-coupling Wess-Zumino term describing the mutual interaction is independent of the Dirac-brane mod 2π, if Dirac's quantization condition holds.
If, on the other hand, the configuration is exceptional (Σ = ∅) the two branes stay at zero distance. In this case the mutual interaction is plagued by short-distance singularities and we will see that the canonical minimal W Z-term becomes Dirac-brane-dependent, i.e. it carries a (non-integer) Dirac-anomaly. In this letter we show that for such configurations the recently developed Chern-kernel approach [11,14,15] allows to write a new (manifestly) Dirac-brane-independent W Z-term. However, this new W Z-term turns out to be plagued by an inflow gravitational anomaly -supported on Σ -if d = 0 or 2, while it is anomaly free if d = 1 or 3. A crucial ingredient for its construction is the new "descent-identity" (3.1).
Eventually we show that for d = 0, 2 the inflow gravitational anomaly on Σ is cancelled by the quantum anomaly produced by the fermions living on it. This means that the total classical + quantum effective action is 1) free from gravitational anomalies and 2) Diracbrane-independent.
The canonical and new W Z-terms differ by a local counterterm supported on Σ, that maps the Dirac-anomaly in the gravitational anomaly, whose construction will be outlined in the concluding section.

Chern-kernels and Dirac-branes
In presence of a closed M2-and a closed M5-brane the Bianchi identity and equation of motion for the four-form fieldstrength of eleven-dimensional Supergravity amount to 1 where J 8 (J 5 ) is the δ-function supported Poincarè-dual form of the electric (magnetic) brane worldvolume M2 (M5), i.e. its current; h = db + B| M 5 , where b is the chiral twoform on M5 and B is the potential for H. The brane tensions are set to T M 2 = T M 5 = 2π. In absence of the M2-brane (J 8 = 0) the basic ingredient for the construction of a consistent W Z-term for this system (see (2.6)) is the Chern-kernel. We recall now briefly the essential features of this construction [15], concentrating on the main properties of the Chern-kernel, the details of the resulting W Z-term itself -S A W Z -being unessential for what follows.
To write an action for the system above one must first solve (2.1) in terms of a potential, introducing a four-form antiderivative K for the magnetic current J 5 , This solution is subject to the transformations (called Q-transformations in the following) These transformations leave the curvatures H and h invariant, and in writing an action one must ensure that this invariance -Q-invariance -remains preserved. As shown in [11,15], the r.h.s. of (2.2) becomes a well-defined closed form and there exists a Q-invariant action if one chooses as solution for dK = J 5 a four-form Chernkernel, where the y a (a = 1, · · · , 5) are normal coordinates on M5,ŷ a = y a /| y|, Dŷ is the covariant differential w.r.t the normal bundle SO(5)-connection A, and F = dA + AA is its curvature. This kernel, although being invariant near M5, is not unique but subjected to the Q-transformations (2.4) [11]. For what follows it is important to notice that K is singular on the whole M5, becauseŷ a does not admit limit when y a goes to 0, while the three-form Q is regular and has, actually, vanishing pullback on M5. The four-form K can be seen as a kind of generalized Coulomb-like field (inverse-power-like singularities), or also as an angular form [18]. We recall also that in normal coordinates the M5-brane current reads J 5 = dy 1 · · · dy 5 δ 5 (y).
In absence of the M2-brane one can then write down a W Z-term giving rise to the eq. of motion (2.2). It is convenient to write it as an integral over a twelve-dimensional manifold M 12 whose boundary is the eleven-dimensional target space M 11 , of a closed and Q-invariant twelve-form 2 , where 7 is the second Pontrjagin form of the normal bundle of M5. We remember that the property dL A 12 = 0 ensures that, in absence of topological obstructions, S A W Z does not depend on the particular M 12 chosen. Writing it in this way the W Z is manifestly Q-invariant, depending only on H and h, and in [15] it has been shown that L A 12 is a closed form and that S A W Z cancels the residual normal bundle SO(5)-anomaly localized on the M5-brane, [19].
As we observed already, for what follows the detailed form of L A 12 is irrelevant; what is crucial is that its consistency relies heavily on the presence of the Chern-kernel and on the corresponding solution H = dB + K of the Bianchi-identity dH = J 5 .
In presence of an M2-brane, J 8 = 0, one can write the W Z-term as where L B 12 must describe 1) the minimal interaction of the M2-brane with supergravity and 2) the mutual interaction between M2 and M5. The first interaction is canonical and corresponds to a contribution to L B 12 given by dBJ 8 = d(BJ 8 ); the presence of the second is needed because dBJ 8 , although being closed, is not Q-invariant. A Q-invariant completion could be achieved by adding the mutual interaction term KJ 8 , leading to dBJ 8 + KJ 8 = HJ 8 , but this is no longer a closed form. Eventually one should have where the missing terms have to be 1) Q-invariant, 2) B-independent and 3) such that L B 12 becomes a closed form. Our main problem consists therefore in figuring out to what the missing terms correspond to.
If M2 and M5 do not intersect there is, of course, a standard procedure for writing down the missing terms above, that involves an (electric) Dirac-brane for M2 i.e. a foursurface D 4 whose boundary is M2, ∂D 4 = M2. Denoting the δ-function on D 4 with W 7 , a seven-form, we have and one can perform the completion changes by an integer multiple of 2π, since H has integer integrals over any closed fourmanifold that does not intersect M5 3 . From a twelve-dimensional point of view independence of the Dirac-brane is manifest since the Dirac-brane-dependent term J 5 W 7 has integer integrals over arbitrary (closed or open) manifolds.
On the other hand, if the intersection manifold Σ is non empty the W Z-action D 4 H becomes Dirac-brane-dependent. Indeed, under a change of Dirac-brane it changes by where S 4 is a closed four-manifold. But since M2 intersects M5 also S 4 intersects M5 and therefore part of the flux of K stays in S 4 and part stays outside. This means that S 4 K is no longer integer, and it represents a Dirac-anomaly. 3 For an alternative argument for Dirac-brane-independence, based on integer forms, see [20].
From a twelve-dimensional point of view, the term J 5 W 7 can no longer be used to make L B 12 a closed form, because if Σ is non empty then the product J 5 W 7 contains squares of δ-functions (δ(x)δ(x)) and becomes ill-defined; this is a consequence of the non-vanishing intersection of the normal bundles of the two branes, see below.
The canonical Dirac-brane construction must therefore be abandoned if Σ = ∅. In this case, since L B 12 must be closed, the first step to find out what the missing terms in (2.8) may be, consists in computing d(HJ 8 ) = d(KJ 8 ). Now, the product KJ 8 and its differential in the sense of distributions are well defined even if Σ = ∅ (for d = 3) 4 , but the point is that one is not allowed to apply Leibnitz's rule to compute it. In fact, the result obtained using naively this rule, i.e. d(KJ 8 ) = J 5 J 8 , contains squares of δ-functions -for the same reasons as above -and it is ill-defined.
On the other hand, as we will show in the next section, the result of the evaluation of d(KJ 8 ) in the sense of distributions is well-defined, and it has a simple interpretation if expressed in terms of normal bundles.

Intersecting branes and normal bundles
Suppose that Σ = M2 ∩ M5 is a closed manifold with dimension d = 0, 1, 2 or 3, and introduce its current J which is a closed (11 − d)-form. For this case we will show that the unknown terms in (2.8) can be deduced from a new kind of "descent-identity" -as such formulated in thirteen dimensions -that involves the normal bundles of M2 and M5.
The normal bundles of M2, M5 and Σ, denoted by N M 2 , N M 5 and N Σ , carry respectively fibers of dimensions 8, 5 and 11 − d. On Σ the bundles of M2 and M5 intersect to a bundle N = N M 2 ∩ N M 5 , whose fiber is of dimension n = 5 + 8 − (11 − d) = d + 2, with structure group SO(n). For example, if the intersection is just a point -a (−1)-branethen n = 2; and if M2 ⊂ M5 then n = 5 because in this case N = N M 5 .
If n is even we can define the Euler-form χ of the bundle N , a form of degree n; if n is odd we take χ to be zero by definition. The Euler-forms of interest are then where T rs is the curvature of N . Our descent notations are χ = dχ (0) , δχ (0) = dχ (1) .
In going to thirteen dimensions we want to keep the degrees of the currents J 8 , J 5 , J unchanged. This implies that the worldvolumes of M2, M5 and Σ have to be extended respectively to five-, eight,-and (d+2)-dimensional manifolds. This keeps the dimensions of the normal bundles, in particular the dimension of N and hence the degree of χ, unchanged. In absence of topological obstructions such extensions are always possible [15], and they were implicitly understood in the twelve-dimensional construction of the previous section.
The result of the computation we referred to above amounts then to a descent-identity between thirteen-forms, involving the δ-function on Σ and the Euler-form of N , The proof is given in the appendix. It is obvious that d(KJ 8 ) must be supported on Σ and hence proportional to J; the proportionality factor χ follows then essentially for invariance reasons. It is understood, as said above, that for d = 1 one takes χ = 0. If M2 ⊂ M5 (d = 3) the product KJ 8 is ill-defined; this case is in some sense trivial and it is solved separately below. Our descent-identity is, actually, a local realization of the corresponding cohomological relation presented in [2].

Wess-Zumino action and anomaly cancellation
Given the above identity it is easy to complete the twelve-form (2.8) to make it a Qinvariant and closed form (d = 3): where we introduced a standard Chern-Simons form through χ = dχ (0) . The Wess-Zumino actions for the four possible intersection manifolds of M5 and M2 can then be written eventually as From the list (4.2) one sees that for d = 0, 2 the W Z-action is plagued by a gravitational anomaly supported on the intersection manifold Σ, δ S B W Z = −2π Σ χ (1) , corresponding to the inflow anomaly-polynomial − 2π χ, (4.3) while for d = 1, 3 it is anomaly free.
On the other hand on Σ there are also fermions living, coming from the common reduction of the 32-component spinors ϑ α , living on M5 and on M2. If d is even, these fermions are a section of the chiral spinor bundle lifted from T (Σ) ⊕ N , where T (Σ) is the tangent bundle to Σ, and N is the intersection of the normal bundels, as above. For such fermions the anomaly can be computed as in [2], and the resulting polynomial reads where ch indicates the Chern character, S ± N is the spin bundle lifted from N with ± chirality, andÂ is the roof genus. From this polynomial one has to extract the two-form part for d = 0, and the four-form part for d = 2. Since the Euler form is already a form of degree two and four respectively, the roof genera above contribute both with unity and the anomaly polynomial reduces precisely to 2πχ, cancelling the inflow.
This represents a quantum consistency check of the intersecting M2/M5 configurations considered in this paper.

Concluding remarks
The anomaly cancellation mechanism presented in this letter has a transparent meaning for d = 2, where Σ is the worldvolume of a closed string.
For d = 0 the intersection manifold Σ is just a point P and represents an instanton. In this case the bundle N is a two-plane centered on P , with structure group SO(2) and abelian connection W rs , and the inflow anomaly −2π Σ χ (1) reduces to −Λ(P ), where δW rs = d(ε rs Λ). This anomaly has a clear meaning since the normal bundle transformations of N correspond just to rotations around P in the two-plane centered at P , and Λ(P ) is the variation of the polar angle ϕ of that plane, δϕ = Λ(P ). What is more obscure is the meaning of (chiral) fermionic degrees of freedom on an instanton and the appearance of the corresponding quantum anomaly. In lack of this insight, above we took simply advantage from the fact that the index formula (4.4) makes sense also for d = 0.
Our strategy for constructing a consistent interaction between intersecting M2-and M5-branes assumes that the branes intersect "strictly", i.e. that they stay strictly at zero distance. An alternative strategy for describing intersecting branes would be to introduce a framing regularization, where the branes are moved at a finite distance ε from each other. For each finite ε the branes are non-intersecting and one could introduce consistently a Dirac-brane to describe their interaction, as explained in section two, see (2.10). However, for ε → 0 this W Z-term, although remaining finite, would become Dirac-brane-dependent, as explained in the text.
Keeping then the branes strictly intersecting, there are two ways for writing a classical action. The first is S B W Z given in (4.2): it is (manifestly) Dirac-brane-independent but carries a gravitational anomaly (for d = 0, 2). The second is S B W Z given in (2.10): it is (manifestly) free from gravitational anomalies but it is plagued by a Dirac-anomaly. This means that there exists a local (in the sense of "Wess-Zumino") counterterm that maps the Dirac-anomaly in a gravitational anomaly and vice versa. Its implicit construction goes along the following lines. Starting point is an identity similar to (3.1), for some (d + 1)-form Φ, which can be proven using -for example -the regularizations given in the appendix of [11]; again, one is not allowed to use naively Leibnitz's rule. For d = 3 the term KJ 8 in this identity has to be replaced by 0. The form Φ, supported on Σ, is diffeomorphism invariant, but depends on W 7 i.e. on the Dirac-brane D 4 . Using (3.1) together with (5.1) one gets for some d-form ω on Σ; for d odd Φ is thus a closed form. It is then immediately seen that The counterterm we searched for is Σ ω and it is supported on Σ, as one may have expected. This proves in particular that the Dirac-anomaly itself is supported on Σ, in agreement with the fact that if Σ = ∅ then there is no Dirac-anomaly at all. The configurations we have considered in this letter are exceptional in that, a priori, a small perturbation makes the two branes again non-intersecting. The stability of these configurations can, however, be inferred from the existence of their classical-solution (semi-localized) counterparts of D = 11 Sugra, mentioned in the introduction, whose stability is guaranteed by supersymmetry. There exist indeed solutions for d = 3, preserving 1/2 susy [3], and solutions for d = 2, preserving 1/4 susy [4,6]. To our knowledge no solutions for d = 1 or d = 0 are yet known. The results of the present paper, indicating that intersecting M2/M5 configurations are quantum consistent for any value of d, suggest that also for d = 0, 1 supersymmetric classical solutions may exist. A dimensional reduction of the complete interacting system Sugra+M2+M5 to ten dimensions, analogous to the one of [21], may help to answer this question.

Appendix: proof of the descent-identity
Since J 8 is the δ-function on M2, the first step in proving (3.1) consists in evaluating K restricted to M2. After that one can compute Since away from M5 K is a closed form, the pullback K| M 2 is closed a part from (possible) δ-function contributions supported on Σ. This means that it is sufficient to evaluate K| M 2 in the vicinity of Σ. On Σ the normal bundle of M5 has n coordinates in common with the normal bundle of M2 -precisely the ones of N -so we can split the normal coordinates of M5 as y a = (y r , y i ), (r = 1, · · · , n), (i = n + 1, · · · , 5). Near Σ we have y r = 0, while the y i become 3 − d = 5 − n normal coordinates for Σ with respect to M2. Correspondingly the SO(n)-connection W of N is embedded in the SO(5)-connection A according to A rs = W rs , T = dW + W W .
We perform now the evaluation of K| M 2 near Σ, i.e. for y r = 0 and keeping only terms that can give δ-function contributions when applying the differential, for each case separately. d = 0. In this case Σ is a point in D = 11, corresponding to a two-dimensional surface in D = 13, and its current is an eleven-form J = J 11 . The fiber of N is two-dimensional (n = 2), with structure-group SO(2) and Euler-form χ 2 (T ). The identity to be proved is therefore d(KJ 8 ) = J 11 χ 2 (T ). (6.2) δ-function contributions localized at Σ are supported in y i = 0, (i = 3, 4, 5) and they can arise from the angular form since dK 0 = d 3 y δ 3 (y). It is then easy to evaluate (2.5) for y r = 0, (r = 1, 2) and to extract the contribution proportional to K 0 , Since one has d[(δ ij − 3ŷ iŷj )K 0 ] = 0, when taking the differential only the first term contributes with a δ-function and one gets d (K| M 2 ) = d 3 y δ 3 (y) χ 2 (T ).
(6.2) follows then from (6.1) and from d 3 y δ 3 (y) J 8 = J 11 . d = 1. In this case Σ is a worldline in D = 11 and its current is a ten-form J = J 10 . The fiber of N is three-dimensional (n = 3) and its Euler-form vanishes. The descentidentity reduces therefore to d(KJ 8 ) = 0.
As above one should extract from K| M 2 , taken at y r = 0, (r = 1, 2, 3), contributions proportional to the angular form, that is now K 0 = 1 2π ε ijŷi dŷ j , (i, j = 4, 5), dK 0 = d 2 yδ 2 (y). But since K contains only odd powers ofŷ, and K 0 is even inŷ, in K| M 2 the angular form K 0 appears always multiplied by odd powers ofŷ, and taking the differential the current d 2 y δ 2 (y) can never show up. This implies that d (K| M 2 ) = 0. d = 2. In this case Σ is a two-surface in D = 11, and its current is a nine-form J = J 9 . The fiber of N is four-dimensional (n = 4) with structure-group SO(4) and Euler-form χ 4 (T ). The identity becomes then d(KJ 8 ) = J 9 χ 4 (T ).
One concludes then as in the case d = 0. d = 3. In this case we have n = 5, J = J 8 and χ = 0, and the r.h.s. of (3.1) vanishes. On the other hand, since Σ = M2 ⊂ M5, the pullback K| M 2 would require to evaluate K for y a → 0. But this limit depends on the directionŷ a = V a (σ) one chooses to approach M2 at each point σ, and K| M 2 is ill-defined. Notice however, that if one performs the limit along an arbitrary but fixed vector field V a (σ), then the resulting four-form (K| M 2 ) V can be shown to be closed. But such a definition saves the identity (3.1) only formally, because K| M 2 , and hence KJ 8 , acquires a dependence on an unphysical vector field, and it can not be used in the W Z-action.