RU9568
CALT682025
hepth/9510086
The Power of M Theory
John H. Schwarz
Rutgers University, Piscataway, NJ 088550849 USA
and
California Institute of Technology, Pasadena, CA 91125 USA^{1}^{1}1Permanent address.
Abstract
A proposed duality between type IIB superstring theory on and a conjectured 11D fundamental theory (“M theory”) on is investigated. Simple heuristic reasoning leads to a consistent picture relating the various branes and their tensions in each theory. Identifying the M theory on with type IIA superstring theory on , in a similar fashion, leads to various relations among the branes of the IIA theory.
1 Introduction
Recent results indicate that if one assumes the existence of a fundamental theory in eleven dimensions (let’s call it the ‘M theory’^{2}^{2}2This name was suggested by E. Witten.), this provides a powerful heuristic basis for understanding various results in string theory. For example, type II superstrings can be understood as arising from a supermembrane in eleven dimensions [1] by wrapping one dimension of a toroidal supermembrane on a circle of the spatial geometry [2, 3, 4, 5]. Similarly, when the spatial geometry contains a , one can obtain a heterotic string by wrapping a fivebrane with the topology of on the [6, 7]. This provides a very simple heuristic for understanding ‘stringstring duality’ between type IIA and heterotic strings in six dimensions [8, 9, 10, 6, 11, 12]. One simply considers the M theory on . This obviously contains both type II strings and heterotic strings, arising by the two wrappings just described. Moreover, since the membrane and 5brane are electricmagnetic duals in 11 dimensions, the two strings are dual in six dimensions, and so it is natural that the strongcoupling expansion of one corresponds to the weakcoupling expansion of the other. The remarkable thing about this kind of reasoning is that it works even though we don’t understand how to formulate the M theory as a quantum theory. It is tempting to say that the success of the heuristic arguments that have been given previously, and those that will be given here, suggest that there really is a welldefined quantum M theory even when perturbative analysis is not applicable. The only thing that now appears to be special about strings is the possibility of defining a perturbation expansion. In other respects, all branes seem to be more or less equal [13, 14].
Recently, I have analyzed heuristic relationships between Type II strings and the M theory [15]. The approach was to compare the 9D spectrum of the M theory on with the IIB theory on . A nice correspondence was obtained between states arising from the supermembrane of the M theory and the strings of the IIB theory. The purpose of this paper is to extend the analysis to include higher branes of both theories, and to see what can be learned from imposing the natural identifications.
Let us begin by briefly recalling the results obtained in [15]. We compared the M theory compactified on a torus of area in the canonical 11D metric with the IIB theory compactified on a circle of radius (and circumference ) in the canonical 10D IIB metric . The canonical IIB metric is the convenient choice, because it is invariant under the group of IIB supergravity. By matching the 9D spectra of the two models (especially for BPS saturated states), the modular parameter of the torus was identified with the modulus , which is the vev of the complex scalar field of the IIB theory. This identification supports the conjectured nonperturbative duality symmetry of the IIB theory. (This was also noted by Aspinwall [16].)
A second result was that the IIB theory has an infinite spectrum of strings, which forms an multiplet. The strings, labelled by a pair of relatively prime integers , were constructed as solutions of the lowenergy 10D IIB supergravity theory using results in Refs. [17, 18]. They have an covariant spectrum of tensions given by
(1) 
where is a constant with dimensions of masssquared, which defines the scale of the theory, and^{3}^{3}3Equation (2) was given incorrectly in the original versions of my previous papers [15]. Also, and were called and , and was called . A more systematic notation is now desirable.
(2) 
Note that strings with , those carrying RR charge, have tensions that, for small string coupling , scale as . The usual string, on the other hand, has . In the string metric, these become and , respectively.
The mass spectrum of point particles (zerobranes) in nine dimensions obtained from the two different viewpoints were brought into agreement (for BPS saturated states, in particular) by identifying winding modes of the family of type IIB strings on the circle with KaluzaKlein modes of the torus and by identifying KaluzaKlein modes of the circle with wrappings of the supermembrane (2brane) on the torus.^{4}^{4}4The rule that gave sensible results was to allow the membrane to cover the torus any number of times (counting orientation), and to identify all the different ways of doing as equivalent. For other problems (such as Strominger’s conifold transitions [19]) a different rule is required. As yet, a single principle that gives the correct rule for all such problems is not known. I am grateful to A. Strominger and A. Sen for correspondence concerning this issue. The 2brane of the M theory has a tension (mass per unit area) in the 11D metric denoted . If one introduces a parameter to relate the two metrics (), then one finds the following relations [15]
(3) 
(4) 
Both sides of eq. (3) are dimensionless numbers, which are metric independent, characterizing the size of the compact spaces. Note that, since and are fixed constants, eq. (3) implies that .
Strings (1branes) in nine dimensions were also matched. A toroidal 2brane with one of its cycles wrapped on the spatial twotorus was identified with a type IIB string. When the wrapped cycle of the 2brane is mapped to the homology class of the spatial torus and taken to have minimal length , this gives a spectrum of string tensions in the 11D metric . Converting to the IIB metric by precisely reproduces the previous formula for in eq. (1), which therefore supports the proposed interpretation.
2 More Consequences of M/IIB Duality
Having matched 9D point particles (0branes) and strings (1branes) obtained from the IIB and M theory pictures, let us now explore what additional information can be obtained by also matching branes with in nine dimensions.^{5}^{5}5For useful background on branes see Refs. [20, 21, 22]. It should be emphasized that even though we use extremely simple classical reasoning, it ought to be precise (assuming the existence of an M theory), because we only consider branes whose tensions are related to their charges by saturation of a BPS bound. This means that the relations that are obtained should not receive perturbative or nonperturbative quantum corrections. This assumes that the supersymmetry remains unbroken, which is certainly believed to be the case.
We begin with . In the M theory the 2brane in 9D is the same one as in 11D. In the IIB description it is obtained by wrapping an factor in the topology of a selfdual 3brane once around the spatial circle. Denoting the 3brane tension by , its wrapping gives a 2brane with tension . Converting to the 11D metric and identifying the two 2branes gives the relation
(5) 
Using eqs. (3) and (4) to eliminate and leaves the relation
(6) 
The remarkable thing about this result is that it is a relation that pertains entirely to the IIB theory, even though it was deduced from a comparison of the IIB theory and the M theory. It should also be noted that the tension is independent of the string coupling constant, which implies that in the string metric it scales as .
Next we consider 3branes in nine dimensions. The only way they can arise in the M theory is from wrapping a 5brane of suitable topology (once) on the spatial torus. In the IIB theory the only 3brane is the one already present in ten dimensions. Identifying the tensions of these two 3branes gives the relation
(7) 
Eliminating and substituting eq. (6) gives
(8) 
This result pertains entirely to the M theory. Section 3 of ref. [12] analyzed the implication of the Dirac quantization rule [23] for the charges of the 2brane and 5brane in the M theory. It was concluded that (in my notation) should be an integer. The present analysis says that it is . Indeed, I believe that eq. (8) corresponds to the minimum product of electric and magnetic charges allowed by the quantization condition. It is amusing that simple classical reasoning leads to a nontrivial quantum result.
Next we compare 4branes in nine dimensions. The IIB theory has an infinite family of 5branes. These are labeled by a pair of relatively prime integers , just as the IIB strings are. The reason is that they carry a pair of magnetic charges that are dual to the pair of electric charges carried by the strings. Let us denote the tensions of these 5branes in the IIB metric by . Wrapping each of them once around the spatial circle gives a family of 4branes in nine dimensions with tensions . In the M theory we can obtain 4branes in nine dimensions by considering 5branes with an factor in their topology and mapping the to a cycle of the spatial torus. Just as when we wrapped the 2brane this way, we assume that the cycle is as short as possible, i.e., its length is . Identifying the two families of 4branes obtained in this way gives the relation
(9) 
Substituting the relations [15]
(10) 
and
(11) 
and using eq. (8) gives
(12) 
This relation pertains entirely to the IIB theory. Since 5brane charges are dual to 1brane charges, they transform contragrediently under . This means that in this case is a magnetic RR charge and is a magnetic NSNS charge. Thus 5branes with pure RR charge have a tension that scales as and ones with any NSNS charge have tensions that scale as . Converting to the string metric, these give and , respectively. Of course, is the characteristic behavior of ordinary solitons, whereas is the remarkable intermediate behavior that is characteristic of all branes carrying RR charge. It is gratifying that these expected properties emerge from matching M theory and IIB theory branes.
We have now related all 1brane, 3brane, and 5brane tensions of the IIB theory in ten dimensions, so that they are determined by a single scale. We have also related the 2brane and 5brane tensions of the M theory in eleven dimensions, so they are also given by a single scale. The two sets of scales can only be related to one another after compactification, however, as the only meaningful comparison is provided by eqs. (3) and (4).
All that remains to complete this part of the story, is to compare 5branes in nine dimensions. Here something a little different happens. As is wellknown, compactification on a space with isometries (such as we are considering), so that the complete manifold is , give rise to massless vectors in dimensions. Electric charges that couple to these vectors correspond to Kaluza–Klein momenta and are carried by pointlike 0branes. The dual magnetic objects are branes. This mechanism therefore contributes “Kaluza–Klein 5branes” in nine dimensions. However, which 5branes are the Kaluza–Klein ones depends on whether we consider the M theory or the IIB theory. The original 5brane of the M theory corresponds to the unique Kaluza–Klein 5brane of the IIB theory, and the family of 5branes of the IIB theory corresponds to the Kaluza–Klein 5branes of the M theory. The point is that there are three vector fields in nine dimensions which transform as a singlet and a doublet of the group. The singlet arises à la Kaluza–Klein in the IIB theory and from the threeform gauge field in the M theory. Similarly, the doublet arises from the doublet of twoform gauge fields in the IIB theory and à la Kaluza–Klein in the M theory.
We can now use the identifications described above to deduce the tensions of Kaluza–Klein 5branes in nine dimensions. The KK 5brane of the IIB theory is identified with the fundamental 5brane of the M theory, which implies that its tension is . Combining this with eq. (11) gives
(13) 
Note that this diverges as , as is expected for a Kaluza–Klein magnetic brane. Similarly the multiplet of KK 5branes obtained from the M theory must have tensions that match the 5branes of the 10D IIB theory. This implies that . Substituting eqs. (4) and (12) gives
(14) 
This also diverges as , as is expected. As a final comment, we note that if all tensions are rescaled by a factor of (in other words, equations are rewritten in terms of ), then all the relations we have obtained in eqs. (3) – (14) have a numerical coefficient of unity.
3 The IIA Theory
The analysis given above is easily extended to the IIA theory in ten dimensions. The IIA theory is simply interpreted [4, 5] as the M theory on . Let be the circumference of the circle in the 11D metric . The string metric of the IIA theory is given by , where is the dilaton of the IIA theory. The IIA string coupling constant is given by the vev of . These facts immediately allow us to deduce the tensions of IIA branes for . The results are
(15) 
(16) 
(17) 
(18) 
Since and are constants, eq. (15) gives the scaling rule [5, 15]. Substituting eqs. (15) and (8) into eqs. (17) and (18) gives
(19) 
(20) 
Again we have found the expected scaling behaviors: for the 2brane and 4brane, which carry RR charge, and for the NSNS solitonic 5brane. Combining eqs. (19) and (20) gives
(21) 
This shows that the quantization condition for the corresponding charges is satisfied with the same (minimal) value in each case.
The IIA theory also contains an infinite spectrum of BPS saturated 0branes (aka ‘black holes’) and a dual 6brane, which are of Kaluza–Klein origin like those discussed earlier in nine dimensions. Since the Kaluza–Klein vector field is in the RR sector, the tensions of these should be proportional to , as was demonstrated for the 0branes in [5].
4 PBranes With
The IIB theory has a 7brane, which carries magnetic charge. The way to understand this is that transforms under just like the axion in the 4D N=4 theory. It has a PecceiQuinn translational symmetry (broken to discrete shifts by quantum effects), which means that it is a 0form gauge field. As a consequence, the theory can be recast in terms of a dual 8form potential. Whether or not one does that, the classical supergravity equations have a 7brane solution, which is covered by the general analysis of [20], though that paper only considered . Thus the 7brane in ten dimensions is analogous to a string in four dimensions. Let us call the tension of the IIB 7brane .
The existence of the 7brane in the 10D IIB theory suggests that after compactification on a circle, the resulting 9D theory has a 7brane and a 6brane. If so, these need to be understood in terms of the M theory. The 6brane does not raise any new issues, since it is already present in the 10D IIA theory. It does, however, reinforce our confidence in the existence of the IIB 7brane. A 9D 7brane, on the other hand, certainly would require something new in the M theory. What could it be? To get a 7brane after compactification on a torus requires either a 7brane, an 8brane, or a 9brane in the 11D M theory. However, the cases of and can be ruled out immediately. They require the existence of a massless vector or scalar particle, respectively, in the 11D spectrum, and neither of these is present. The 9brane, on the other hand, would couple to a 10form potential with an 11form field strength, which does not describe a propagating mode and therefore cannot be so easily excluded. Let us therefore consider the possibility that such a 9brane with tension really exists and trace through its consequences in the same spirit as the preceding discussions.
First we match the 7brane obtained by wrapping the hypothetical 9brane of the M theory on the spatial torus to the 7brane obtained from the IIB theory. This gives the relation
(22) 
Substituting eq. (4) gives
(23) 
This formula is not consistent with our assumptions. A consistent picture would require to be independent of or , but we have found that . Also, the 8brane and 9brane of the IIA theory implied by a 9brane in the M theory do not have the expected properties. I’m not certain what to make of all this, but it is tempting to conclude that there is no brane in the M theory. Then, to avoid a paradox for 9D 7branes, we must argue that they are not actually present. I suspect that the usual methods for obtaining BPS saturated branes in dimensions from periodic arrays of them in dimensions break down for , because the fields are not sufficiently controlled at infinity, and therefore there is no 7brane in nine dimensions. Another reason to be suspicious of a 9D 7brane is that a brane in dimensions is generically associated with a cosmological term, but straightforward compactification of the IIB theory on a circle does not give one.
In a recent paper [24], Polchinski has argued for the existence of a 9brane in the 10D IIB theory and an 8brane in the 10D IIA theory, both of which carry RR charges. (He also did a lot of other interesting things.) It ought to be possible to explore whether the existence of these objects is compatible with the reasoning of this paper, but it is unclear to me what the appropriate rules are for handling such objects.
5 Conclusion
We have shown that by assuming the existence of a quantum ‘M theory’ in eleven dimensions one can derive a number of nontrivial relations among various perturbative and nonperturbative structures of string theory. Specifically, we have investigated what can be learned from identifying M theory on with type IIB superstring theory on and matching (BPS saturated) branes in nine dimensions. Similarly, we identified the M theory on with type IIA superstring theory on and matched branes in ten dimensions. Even though quantum M theory surely has no perturbative definition in 11D Minkowski space, these results make it more plausible that a nonperturbative quantum theory does exist. Of course, this viewpoint has been advocated by others – most notably Duff and Townsend – for many years.
Clearly, it would be interesting to explore other identifications like the ones described here. The natural candidate to consider next, which is expected to work in a relatively straightforward way, is a comparison of the M theory on with the heterotic string theory on . There is a rich variety of branes that need to be matched in seven dimensions. In particular, the M theory 5brane wrapped on the surface should be identified with the heterotic string itself.
The M theory on , where is a Calabi–Yau space, should be equivalent to the type IIA superstring theory on . Kachru and Vafa have discussed examples for which there is a good candidate for a dual description based on the heterotic string theory on [25]. A new element, not encountered in the previous examples, is that while there is plausibly a connected moduli space of models that is probed in this way, only part of it is accessed from the M theory viewpoint and a different (but overlapping) part from the heterotic string theory viewpoint. Perhaps this means that we still need to find a theory that is more fundamental than either the heterotic string theory or the putative M theory.
6 Acknowledgments
I wish to acknowledge discussions with R. Leigh, N. Seiberg, S. Shenker, L. Susskind, and E. Witten. I also wish to thank the Rutgers string theory group for its hospitality. This work was supported in part by the U.S. Dept. of Energy under Grant No. DEFG0392ER40701.
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