Some aspects of the role of p-branes in non-perturbative superstring theory and M-theory are reviewed. It is then shown how the Chern-Simons terms in D=10 and D=11 supergravity theories determine which branes can end on which, i.e. the ‘brane-boundary rules’.
Extended objects, known as ‘branes’, currently play an essential role in our understanding of the non-perturbative dynamics underlying ten-dimensional (D=10) superstring theories and the 11-dimensional (D=11) M-theory (see  for a recent review). In the context of the effective D=10 or D=11 supergravity theory a ‘p-brane’ is a solution of the field equations representing a p-dimensional extended source for an abelian -form gauge potential with -form field strength . As such, the -brane carries a charge
where is the Hodge dual in the D-dimensional spacetime and the integral is over a -sphere encircling the brane, as shown schematically in the figure below:
In the case of a static infinite planar p-brane this formula is readily understood as a direct generalization of the case, i.e. a point particle in electrodynamics, with the -dimensional ‘transverse space’ (spanned by vectors orthogonal to the -dimensional worldvolume) taking the place of space. In the case of a closed p-brane, static or otherwise, the charge can be understood (after suitable normalization) as the linking number of the p-brane with the -sphere in the -dimensional space.
Examples for are provided by the D=10 heterotic strings, for which
where is the 3-form field strength for the 2-form gauge potential from the massless Neveu-Schwarz (NS) sector of the string spectrum. Further D=10 examples are provided by the type II superstrings, with the difference that is now the 2-form of origin in type II superstring theory. A D=11 example is the charge
carried by a supermembrane, where is the 4-form field strength for the 3-form potential of D=11 supergravity.
The statement that a -brane carries a charge of the above type can be rephrased as a statement about interaction terms in the effective worldvolume action governing the low-energy dynamics of the object. Consider, for example, type II and heterotic strings. Let be the worldsheet coordinates and let describe the immersion of the worldsheet in the D=10 spacetime. Then the worldsheet action in a background with a non-vanishing 2-form will include the term
where is the alternating tensor density on the worldsheet. Thus the string is a source for and, since the coupling is ‘minimal’, it will contribute to the charge defined above. Similarly, in a D=11 background with non-vanishing 3-form potential , the membrane action includes the term 
where describes the immersion of the supermembrane’s worldvolume in the D=11 spacetime, and are the worldvolume coordinates. This minimal interaction implies that the membrane is a source for with non-vanishing charge .
The actions and are actually related by double-dimensional reduction, as are the full supermembrane and IIA superstring actions . The dimensional reduction to D=10 involves setting where is the coordinate of the compact 11th dimension, and taking all fields to be independent of . From the worldvolume perspective this amounts to a special choice of background for which is a Killing vector field. Double-dimensional reduction is then achieved by setting where is the coordinate of a compact direction of the membrane, and then setting and , which is the ansatz appropriate to a membrane that wraps around the 11th dimension. The action then becomes after the identification , where indicates contraction with the vector field .
A coupling to of the form (4) is possible only for oriented strings. Of the five D=10 superstring theories all are theories of oriented strings except the type I theory. Thus, the type I string does not couple minimally to . Instead, it couples non-minimally. In the Lorentz-covariant GS formalism in which the worldsheet fermions, , are in a spinor representation of the D=10 Lorentz group, the worldsheet interaction Lagrangian is
Because of the ‘derivative’ coupling of the string to through its field strength , the charge carried by the type I string vanishes. As the above interaction shows, the type I string theory origin of is in the sector rather than the sector. This example illustrates a general feature of string theory: charges are not carried by the fundamental string. If there is anything that carries the charge in type I string theory it must be non-perturbative. It is now known that there is such a non-perturbative object in type I string theory [4, 5, 6]; it is just the heterotic string! This is one of the key pieces of evidence in favour of the proposed ‘duality’, i.e. non-perturbative equivalence, of the type I and heterotic string theories. Another is the fact that the two effective supergravity theories are equivalent, being related to each other by a field redefinition that takes , where is the dilaton [7, 8]. Since the vacuum expectation value is the string coupling constant this means that the weak coupling limit of one theory is the strong coupling limit of the other.
An important consequence of the charge carried by a -brane is that it leads to a BPS-type bound on the p-volume tension, of the form , where is some constant characteristic of the particular supergravity theory, the choice of vacuum solution of this theory, and the value of . If one considers the class of static solutions with -fold translational symmetry then a bound of the above form follows from the requirement that there be no naked singularities. This bound is saturated by the solution that is ‘extreme’ in the sense of General Relativity, i.e. for which the event horizon is a degenerate Killing horizon111When a dilaton is present this is true only for an appropriate definition of the metric.. However, these considerations are clearly insufficient to show that the p-brane tension actually is bounded in this way because the physically relevant class of solutions is the much larger one for which only an appropriate asymptotic behaviour is imposed. Remarkably, the attempt to establish a BPS-type bound succeeds if and only if the theory is either a supergravity theory, or a consistent truncation of one 222In contrast, the proof of positivity of the ADM mass of asymptotically-flat spacetimes is not subject to this restriction since, for example, it is valid for arbitrary D.. In particular, the presence of various Chern-Simons terms in the Lagrangians of D=10 and D=11 supergravity theories is crucial to the existence of a BPS-type bound on the tensions of the -brane solutions of these theories. This is so even when, as is usually the case, these Chern-Simons (CS) terms play no role in the -brane solutions themselves in the sense that they are equally solutions of the (non-supersymmetric) truncated theory in which the CS terms are omitted. These facts hint at a more important role for the supergravity CS terms in determining the properties of -branes than has hitherto been appreciated. This observation provided the principal motivation for this article, as will become clear.
Although the charge has only a magnitude, it is associated with an object whose spatial orientation is determined by a -form of fixed magnitude. Thus, a -brane is naturally associated with a -form charge of magnitude . Indeed, the supersymmetrization of terms of the form (4) or (5) leads to a type of super-Wess-Zumino term that implies a modification of the standard supersymmetry algebra to one of the (schematic) form [10, 11]
where is an antisymmetrized product of Dirac matrices and is a -form charge whose magnitude is given by the coefficient of the Wess-Zumino term. For this is the well-known modification that includes as a central charge. More generally, may be identified as the magnitude of , and an extension of the arguments used in the case [12, 13] shows that the supersymmetry algebra (7) implies the BPS-type bound on the p-brane tension . It also shows that the ‘extreme’ -brane solutions of supergravity theories which saturate the bound must preserve some of the supersymmetry, and the fraction preserved is always 1/2 for -brane solutions in D=10 and D=11333There are other solutions which preserve less than half the supersymmetry, and which have an interpretation as -branes in , but these can always be viewed as composites (e.g. intersections) of -branes in D=10 or D=11. We shall not need to consider such solutions here.. The heterotic and type II superstrings and the D=11 supermembrane are examples not only of charged -branes but also of extreme charged -branes. This follows from the ‘-symmetry’ of their Lorentz covariant and spacetime supersymmetric worldsheet/worldvolume actions (see  for a review). The BPS-saturated -branes are important in the context of the non-perturbative dynamics of superstring theories or M-theory for essentially the same reasons that BPS-saturated solitons are important in D=4 field theories. In fact, most of the the latter can be understood as originating in D=10 or D=11 BPS-saturated -branes. For these reasons, the BPS-saturated p-branes are the ones of most interest and will be the only ones considered here. It should therefore be understood in what follows that by ‘brane’ we mean ‘BPS-saturated brane’.
One of the lessons of recent years has been that much can be learned about the non-perturbative dynamics of superstring theories from the effective D=10 supergravity theories. One example of this is the fact that there exist p-brane solutions of type II supergravity theories which are charged, in the sense explained above, with respect to the -form gauge fields from the sector of the corresponding string theory. By supposing these branes to be present in the non-perturbative string theory one can understand how otherwise distinct superstring theories might be dual versions of the same underlying theory. The basic idea is that branes can ‘improve’ string theory in the same way that strings ‘improve’ Kaluza-Klein (KK) theory. For example, the -compactified IIA and IIB supergravity theories have an identical massless D=9 spectrum but are different as Kaluza-Klein theories because their massive modes differ. The corresponding string theories are the same, however, because the inclusion of the string winding modes restores the equivalence of the massive spectra. Similarly, the -compactified type IIA superstring theory and the -compactified heterotic string theory have an identical massless D=6 spectrum, but since they differ in their perturbative massive spectra they are inequivalent as perturbative string theories. However, the non-perturbative massive spectrum of the IIA superstring includes ‘wrapping’ modes of 2-branes around 2-cycles of . The inclusion of these leads to the same massive BPS spectrum in the two theories, and there is now strong evidence of a complete equivalence [7, 16, 17, 18]. This evidence rests, in part, on the fact that branes now have a remarkably simple description  in string theory as D-branes, or D--branes if we wish to specify the value of ; the worldvolume of a D--brane is simply a (p+1)-dimensional hyperplane defined by imposing Dirichlet boundary conditions at the boundaries of open string worldsheets.
The distinctions between the various kinds of type II p-brane, such as whether they are of or type, are not intrinsic but are rather artefacts (albeit very useful ones) of perturbation theory. Non-perturbatively, all are on an equal footing since any one can be found from any other one by a combination of ‘dualities’. This feature is apparent in the IIA or IIB effective supergravity theories which treat all p-form gauge fields ‘democratically’. Their string theory origin is nevertheless apparent from the supergravity solutions if the latter are expressed in terms of the string metric, instead of the canonical, or ‘Einstein’, metric. One then finds that there are three categories of p-brane, ‘fundamental’ (F), ‘Dirichlet’ (D) and ‘solitonic’ (S) according to the dependence of the p-volume tension on the string coupling constant . Specifically,
Note that according to this classification only strings can be ‘fundamental’. This is hardly surprising in view of the fact that we are discussing the dependence of the tension in terms of the string metric, but it seems to be a reflection of a more general observation [7, 20] that a sensible perturbation theory can be found only for particles or strings. Similarly only 5-branes can be ‘solitonic’. This is a reflection of the electric/magnetic duality between strings and 5-branes in D=10 and the fact that the magnetic dual of a fundamental object is a solitonic one. It does not follow from this that all strings are fundamental and all 5-branes solitonic. This is nicely illustrated by the string solution of N=1 supergravity representing the heterotic string. This solution is ‘fundamental’ as a solution of the effective supergravity theory of the heterotic string, as it must be of course, but it is a D-string when viewed as a solution of the effective supergravity theory of the type I string. Thus, a single supergravity solution can have two quite different string theory interpretations.
The M-theory branes, or ‘M-branes’, consist of only the D=11 membrane and its magnetic dual, a fivebrane. We saw earlier that the classical IIA superstring action is related to that of the D=11 supermembrane by double-dimensional reduction. This was widely considered to be merely a ‘coincidence’, somewhat analogous to the fact that IIA supergravity happens to be the dimensional reduction of D=11 supergravity; after all, the quantum superstring theory has D=10 as its critical dimension. However, the critical dimension emerges from a calculation in perturbative string theory. It is still possible that the non-perturbative theory really is 11-dimensional, but if this is so the KK spectrum of the -compactified D=11 supergravity must appear in the non-perturbative IIA superstring spectrum. It was pointed out in [21, 7] that the extreme black holes of IIA supergravity, now regarded as the effective field theory realization of D-0-branes, are candidates for this non-perturbative KK spectrum. This means that the IIA superstring really is an -wrapped D=11 supermembrane, but it does not then follow that the supermembrane is also ‘fundamental’ because this adjective is meaningful only in the context of a specific perturbation theory. For example, the heterotic string is ‘fundamental’ at weak coupling but as the coupling increases it transmutes into the D-string of the type I theory. Another example is the IIB string which is ‘fundamental’ at weak coupling but which transmutes into the D-string of a dual IIB theory at strong coupling [22, 7]. In the IIA case the strong coupling limit is a decompactification limit in which the D=11 Lorentz invariance is restored and the effective D=10 IIA supergravity is replaced by D=11 supergravity . The ‘fundamental’ IIA superstring transmutes, in this limit, into the unwrapped D=11 membrane of M-theory but, because of the absence of a dilaton, there is no analogue of string perturbation theory in D=11 and so there is no analogous basis for deciding whether or not the membrane is ‘fundamental’. Nevertheless, as we shall shortly see, there is an intrinsic asymmetry between M-theory membranes and fivebranes which suggests a fundamental role for the membrane in some as yet unknown sense444This is also suggested by the ‘n=2 heterotic string’ approach to M-theory ..
Given that the heterotic string appears as a D-brane in type I string theory one might wonder whether the type I string should make an appearance somewhere in the non-perturbative SO(32) heterotic string theory. As we have seen, however, the type I string carries no charge, so its description in the effective supergravity theory would have to be as a non-extreme, or ‘black’, string. Infinite uncharged black strings have been shown to be unstable against perturbations that have the tendency to break the string into small segments  (whereas extreme strings are stable because they saturate a BPS-type bound). This is exactly what one expects from string theory since a closed type I string can break, i.e. type I string theory is a theory of both closed and open strings. The reason that this is possible for type I strings, but not for heterotic or type II strings, is precisely that the type I string carries no charge. To see this, suppose that a string carrying a non-zero charge were to have an endpoint. One could then ‘slide off’ the 7-sphere encircling the string and contract it to a point. Provided that the integral defining is homotopy invariant, which it will be if , the charge must then vanish, in contradiction to the initial assumption. We conclude that the only breakable strings are those for which . Thus type II and heterotic strings cannot break. Clearly, similar arguments applied to -branes carrying non-zero charge lead to the conclusion that they too cannot break.
By ‘break’ we mean to imply that the -brane boundary created in this process is ‘free’ in the sense that its dynamics is determined entirely by the -brane of which it is the boundary. An ‘unbreakable’ p-brane may nevertheless be open if its boundary is tethered to some other object because there may then be an obstruction to sliding the -sphere off the end of the p-brane. Examples of such obstructions are the D-branes on which type II superstrings can end. One way to understand how this is consistent with conservation of the charge is to consider the D-brane’s effective worldvolume action, which governs its low-energy dynamics. The field content of this action is found from the massless sector of an open type II superstring with mixed Neumann/Dirichlet boundary conditions at the ends. These fields are essentially the same as those of the open type I string without Chan-Paton factors with the difference that they depend only on the D-brane’s worldvolume coordinates (see e.g. ). In particular, these worldvolume fields include an abelian 1-form potential . The bosonic sector of the effective worldvolume action, in a general background, can be deduced from the requirement of conformal invariance of the type II string action for a worldsheet with a boundary . This effective worldvolume action is found to contain the term
where the integral is over the -dimensional worldvolume and it is to be understood that the spacetime 2-form is pulled back to . This shows that the D-brane is a source of . If we modify the equation in order to include this source we find, by integration, that
where is defined as before in (1). The integral on the right hand side of (10) is over a -sphere in the D-brane surrounding the string’s endpoint and is the worldvolume Hodge dual. This result can be interpreted as the statement that the charge of the string can be ‘transferred’ to an electric charge of a particle on the D-brane, so charge conservation is compatible with the existence of an open string provided that its endpoints are identified with charged particles living on a D-brane.
A similar analysis can be applied to the D=11 membrane which, we recall, is an electric-type source for the 3-form gauge potential of D=11 supergravity. In this case, the D=11 fivebrane has a worldvolume action containing the terms [27, 28]
where is the 3-form field strength for a worldvolume 2-form potential , and it is again to be understood that is the pullback of the spacetime field to the worldvolume. Actually, the worldvolume 3-form is self-dual, but this condition must be imposed after variation of the action (11). The second term in this action is needed for consistency of the self-duality condition with the field equation. Apart from this subtlety, we see from its worldvolume action that the fivebrane is a source for . Its inclusion in the field equation for leads, after integration, to the equation
which can be interpreted as the statement that the membrane charge can be transferred to a charge carried by a self-dual string within the fivebrane. This string is just the boundary of an open membrane. Thus, the D=11 fivebrane is the M-theory equivalent of a D-brane [27, 29].
The above analysis can be generalized  to determine whether a -brane can end on a -brane, as follows. One first determines the worldvolume field content of the -brane. If this includes a -form gauge field , and if the spacetime fields include a -form gauge potential , then one can postulate a coupling of the form in the -brane’s effective worldvolume action. This leads to the -brane appearing as a source for such that
where the integral in the -brane is over a -sphere surrounding the -brane boundary of the p-brane. Thus, the -brane charge can be transferred to the electric charge of the -brane boundary living in the -brane. That is, charge conservation now permits the -brane to be open provided its boundary lies in a -brane. The cases discussed above clearly fit this pattern, but there are drawbacks to this approach. Firstly, it is indirect because one must first determine the worldvolume field content of all relevant branes. Secondly, it is ad hoc because, in general, the worldvolume coupling is postulated rather than derived. The subtleties alluded to above in the construction of the fivebrane action show that this is not a trivial matter. In fact, even the bosonic fivebrane action is not yet fully known and until it is one cannot be completely certain that the wanted terms in this action really are present.
In this contribution I will present a new, and extremely simple, method for the determination of when -branes may have boundaries on -branes. Essentially, one can read off from the Chern-Simons terms in the supergravity action whether any given -brane can have a boundary and, if so, in what -brane the boundary must lie. As such, the method provides a further example of how much can be learned about the non-perturbative dynamics of superstring theories, or M-theory, from nothing more than the effective supergravity theory. I have called the method ‘brane surgery’ because of a notional similarity to the way in which manifolds can be ‘glued’ together by the mathematical procedure known as ‘surgery’, but it is not intended that the term should be understood here in its technical sense.
It is pleasure to dedicate this contribution to the memory of Claude Itzykson, who would surely have apreciated the remarkable confluence of ideas that has marked recent advances in the theory that is still, misleadingly, called ‘string theory’.
2 IIB brane boundaries
I shall explain the ‘brane surgery’ method initially in the context of the IIB theory. Both IIA and IIB supergravity have in common the bosonic fields () from the sector, all of which have already made an appearance above. The remaining bosonic fields come from the sector. The (massless) fields of the IIB theory are
i.e. a pseudoscalar , another 2-form gauge potential and a 4-form gauge potential with a self-dual 5-form field strength . The self-duality condition makes the construction of an action problematic but, as with the self-duality condition on the D=11 fivebrane’s worldvolume field strength , one can choose to impose this condition after varying the action. When the IIB action is understood in this way it contains the CS term555The conventions can be chosen such that the coefficient is as given.
where , as before, and . This CS term modifies the , and field equations.
Consider first the equation. This becomes
where is the self-dual 5-form field strength for . This can be rewritten as
Since is no longer a closed form its integral over a 7-sphere will no longer be homotopy invariant. Clearly, the well-defined, homotopy invariant, charge associated with the fundamental IIB string is not as defined in (1) but rather
Let us again suppose that the IIB string has an endpoint. Far away from this endpoint we can ignore all fields other than , to a good approximation, so that . How good this aproximation is actually depends on the ratio of the radius of the 7-sphere to the its distance from the end of the string, and it can be made arbitrarily good by increasing for fixed . This shows, in particular, that for the fundamental IIB string. Let us now ‘slide’ the 7-sphere along the string towards the endpoint. If the term could be entirely ignored we would be back in the situation described previously in which we arrived at a contradiction, so we are forced to suppose that an endpoint is associated with a non-vanishing value of . Nevertheless, the approximate equality of to , which ignores the term, can be maintained, to the same precision, if the 7-sphere surrounding the string is contracted as we approach the endpoint so as to keep the ratio constant. Then, as so also , until at the 7-sphere is contracted to the endpoint itself. We can then deform the 7-sphere into the product so that now receives its entire contribution from the term as follows:
The integration region is illustrated schematically by the figure below:
Observe that the integral is just the definition of the charge carried by a 3-brane, so the IIB string has its endpoint on a 3-brane; the integration surface lies within the 3-brane and surrounds the string endpoint. Let us choose . If we further suppose that within the 3-brane, which is reasonable in the absence of any D-string source for this field, then is a closed 2-form which we may write, locally, as for some 1-form . Then
Effectively, is a field living on the worldvolume of the 3-brane. Clearly, it cannot be globally defined because the right hand side of (20) is a magnetic charge on the 3-brane associated with the vector potential .
Now consider the equation. Taking the CS term (15) into account we have
By the same reasoning as before we deduce that the D-string can end on a 3-brane. Charge conservation is satisfied because the D-string charge can be expressed as
Since there is no fundamental string source in the problem we may suppose that , so that now is a closed 2-form which we may write, locally, as . For we now have
so the D-string charge has been transferred to a magnetic charge of the 1-form potential on the 3-brane’s worldvolume.
It must be regarded as a weakness of the above analysis that it does not supply the relation between and , although we know that there must be one because both supersymmetry and an analysis of the small fluctuations about the 3-brane solution show that there is only one worldvolume 1-form potential. In fact, and are dual in the sense that
where we recall that indicates the worldvolume Hodge dual. Using this relation, (23) becomes
i.e. the endpoint of the IIB string on the 3-brane is an electric charge associated with . We thereby recover the D-brane picture for the IIB 3-brane; the fact that the D-string can end on the magnetic charge associated with is then a consequence of the strong/weak coupling duality in IIB superstring theory interchanging the fundamental string with the D-string. It will be seen from the examples to follow that the need to impose a condition of the type (24) is a general feature, which is not explained by the ‘brane surgery’ method. However, the method does determine whether a given p-brane can have a boundary and, if so, the possible q-branes in which the boundary must lie.
As a further illustration we now observe that whereas (16) was previously rewritten as (17), we could instead rewrite it as
Thus an equivalent definition of is
Proceeding as before, but now deforming the into the product , we can express as
We recognise the first integral as the D-5-brane charge . Setting and , we conclude that
where is a locally-defined 3-form field on the 5-brane worldvolume, which can be traded for a 1-form potential via the relation
We conclude that the CS term allows the fundamental IIB string to end on a 5-brane as well as on a 3-brane, and that the end of the string is electrically charged with respect to a 1-form potential living on the 5-brane’s worldvolume. This is just the usual picture of the D-5-brane. Interchanging the roles of and leads to the further possibility of the D-string ending on the solitonic 5-brane.
We have not yet exhausted the implications of the CS term (15) because we have still to consider how it affects the equation of motion. We find that666The same equation follows, given the self-duality of , from the ‘modified’ Bianchi identity for .
This means that the 3-brane charge should be modified to
This reduces to the previously-defined 3-brane charge if the 5-sphere surrounds a 3-brane sufficiently far from the boundary. As before the 5-sphere can be slid towards, and contracted onto, the boundary, after which it emerges as the product . Setting again we arrive at the expression
for the 3-brane charge. The singularity involved in this deformation of the 7-sphere is now the 2-brane boundary of the 3-brane within a D-5-brane, since we recognise the first integral on the right hand side of (34) as . Setting we learn that the 3-brane charge can be transferred to a magnetic charge of a D-5-brane worldvolume 1-form potential , defined by a 2-sphere in the D-5-brane surrounding the 2-brane boundary. The main point in all this is that a 3-brane can have a boundary in a D-5-brane, as pointed out in . In fact, this possibility follows by T-duality from the previous results: the configuration of a D-string ending on a D-3-brane is mapped to a D-3-brane ending on a D-5-brane by T-duality in two directions orthogonal to both the D-string and the D-3-brane. By interchanging the roles of and in the above analysis one sees that a 3-brane can also end on a solitonic 5-brane.
We have seen that the CS term (15) allows a IIB string to end on a D-3-brane or a D-5-brane, but we know from string theory that it can also end on a D-string or a D-7-brane. As we shall see shortly, these possibilities are consequences of the fact that the kinetic term for actually has the form
There is no obvious relation to CS terms yet, but if we perform a duality transformation to replace the 2-form by its 6-form dual with 7-form field strength , so that on shell
then one finds that the dualized action contains the CS term
Clearly, this modifies the equation so that, following the steps explained previously, we end up with an expression
The first integral can be identified, using (36), as the D-string charge. The final ‘integral’ over is just the difference between the value of on either side of the string boundary on the D-string; by the same logic as before we may assume that , locally, but allow the constant to be different on either side. Thus, the charge on the fundamental IIB string is transformed into the topological charge of a type of ‘kink’ on the D-string.
Alternatively, we can deform to , so that
The first integral is the charge associated with the D-7-brane. This charge can be non-zero because of the periodic identification of implied by the conjectured invariance of IIB superstring theory . For , and setting for 5-form potential (since we may assume that ), we have
Defining the 1-form on the 7-brane’s worldvolume by
we can rewrite (40) as
We conclude that the IIB string may end on an electric charge in a 7-brane. This is just the description of the D-7-brane.
3 IIA boundaries
The ‘brane surgery’ method should now be clear. We shall now apply it to IIA supergravity, for which the gauge potentials are
i.e. a 1-form and a 3-form . We might start by considering the CS term
where is the 4-form field strength of . Consideration of this term leads to the conclusion that (i) a IIA string can end on a 4-brane, and (ii) a 2-brane can end on either a 4-brane or a 5-brane. Since the CS term (44) is so obviously related to the similar one in D=11 to be considered below we shall pass over the details. The fact that the IIA string can also end on either a 2-brane or a 6-brane follows from the fact that the field strength has a ‘modified’ Bianchi identity
where is the field strength of (this has a Kaluza-Klein origin in D=11). We can dualize to convert this modified Bianchi into a CS term of the form777This dualization is inessential to the result, but it allows a convenient uniformity in the description of the method.
where the 6-form is, on-shell, the Hodge dual of . This modifies the equation to
We may therefore take the modified charge to be
Now, by the identical reasoning used in the IIB case, we first deform the 7-sphere so as to arrive at the formula
We then identify the first integral as the charge of a membrane. We then set and for some scalar defined locally on the worldvolume of the membrane to conclude that the IIA string can end on a membrane, with the string’s charge now being transferred to the magnetic-type charge
of a particle on the membrane . This charge can be non-zero if is periodically identified. Clearly, from the KK origin of , we should interpret as the coordinate of a hidden 11th dimension. Defining the worldvolume 1-form by
we recover [31, 27, 32] the usual description of the IIA D-2-brane, in which the end of the string on the membrane carries the electric charge
Returning to (47) we can alternatively define the modified string charge to be
where is the 5-form potential associated with , i.e. . Since
is the 6-brane charge, similar reasoning to that above, but now setting , leads to the conclusion that a IIA string can also end on a 6-brane and that the string charge is transferred to the 6-brane magnetic charge
which can be rewritten in the expected electric charge form
by introducing the worldvolume 1-form potential dual to .
The remaining IIA D-branes are the 0-brane and the 8-brane. The possibility of a IIA string ending on a 0-brane is not found by the ‘brane surgery’ method for the good reason that it is actually forbidden by charge conservation unless the 0-brane is the endpoint of two or more strings. Thus, a modification of the method will be needed to deal with this case. Neither is it it clear how the method can cope with the IIA 8-brane, because of the non-generic peculiarities of this case.
Leaving aside these limitations of the method, there are further consequences to be deduced from the CS term (46). We have still to consider its effect on the equation of motion. Actually it is easier to return to the modified Bianchi identity (45), which we can rewrite as
This shows that the homotopy-invariant magnetic 4-brane charge is actually
By the now familiar reasoning we deform the 4-sphere and set to arrive at
We recognise the first integral as the charge of a 6-brane. The second integral is the magnetic charge associated with a 3-brane within the 6-brane. The 3-brane is of course the 4-brane’s boundary. Thus a 4-brane can end on a 6-brane. This is not unexpected because it follows by T-duality from the fact that a IIB 3-brane can end on a D-5-brane.
We could as well have rewritten the modified Bianchi identity (45) as
in which case a similar line of reasoning, but setting , and so , leads to the expression
The first integral is the magnetic 5-brane charge , so we deduce that a 4-brane can also end on a (solitonic) 5-brane. The 3-brane boundary in the 5-brane is a magnetic source for the scalar field . The KK origin of suggests a D=11 interpretation of this possibility. It is surely closely related to the fact that two D=11 fivebranes can intersect on a 3-brane , since by wrapping one of the 5-branes (but not the other one) around the 11th dimension we arrive at a D-4-brane intersecting a solitonic 5-brane in a 3-brane. This is not quite the same as a D-4-brane ending on a 5-brane, but the intersection could be viewed as two 4-branes which happen to end on a common 3-brane boundary in the 5-brane. This illustrates a close connection between the ‘brane boundary’ rules and the ‘brane intersection rules’, which will not be discussed here.
4 M-brane boundaries
Finally, we turn to M-theory, or rather D=11 supergravity and its p-brane solutions. The bosonic fields of D=11 supergravity are the 11-metric and a 3-form gauge potential with 4-form field strength . The Bianchi identity for is
from which we may immediately conclude that the D=11 fivebrane must be closed. The same is not true of the D=11 membrane, however, because there is a CS term in the action of the form
which leads to the following field equation888An additional singular 5-brane source term was included in  leading to a rather different interpretation of the significance of the term. We note that since the 5-brane is actually a completely non-singular solution of the D=11 field equations  it should not be necessary to include it as a source.
We see that the well-defined membrane charge is actually
Now consider a membrane with a boundary. Contract the 7-sphere to the boundary and deform it to the product so that the entire contibution to is given by
The first integral is the charge associated with a fivebrane. Set . We may also set to zero the components of ‘parallel’ to the fivebrane, so that in the second integral. We then have
which is the magnetic charge of the string boundary of the membrane in the fivebrane.
In fact, the 3-form field-strength (or rather in a general background) is self-dual but we do not learn this fact from the ‘brane surgery’ method. As for the IIB 3-brane, where we saw that the worldvolume 1-forms and are related by Hodge duality of their 2-form field strengths, this information must be gleaned from a different analysis. The similarity between these constraints on the worldvolume gauge fields suggests that a deeper understanding of the phenomenon should be possible.
In this contribution I have discussed the rules governing ‘brane boundaries’ in superstring and M-theory and shown that they follow from consideration of interactions in the corresponding effective supergravity theories. It should be appreciated that brane boundaries constitute a subset of possible ‘brane interactions’, which include intersecting branes and branes of varying topologies. A reasonably complete picture is now emerging of the static aspects of brane interactions, but little is known at present about the dynamic aspects, i.e. the analogue of the splitting and joining interaction in string theory. This problem is presumably bound up with the problem of finding an intrinsic definition of M-theory, which may well require a substantially new conceptual framework. Hopefully, the current focus on branes will prove to be of some help in this daunting task.
Acknowledgements: I am grateful to George Papadopoulos for helpful discussions.
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