29 November 2001

Physics Letters B 521 (2001) 383–390 www.elsevier.com/locate/npe

Supersymmetric index of the M-theory 5- and little theory

Giulio Bonelli

Spinoza Institute, University of Utrecht, Leuvenlaan 4, 3584, CE Utrecht, The Netherlands Received 12 July 2001; received in revised form 25 September 2001; accepted 4 October 2001 Editor: P.V. Landshoff

Abstract

We propose a six-dimensional framework to calculate the supersymmetric index of M-theory 5- wrapped on a six- 2 manifold with product topology M4 × T ,whereM4 is a holomorphic 4-cycle in a Calabi–Yau three-fold. This is obtained by zero-modes counting of the self-dual tensor contribution plus “little” string states and correctly reproduces the known results which can be obtained by shrinking or blowing the T 2 volume parameter. We also extract the geometric of the multi M5-brane system and infer the generic structure of the supersymmetric index for more general geometries.  2001 Elsevier Science B.V. All rights reserved.

1. Introduction 5-branes. This strong characterization of M-theory has passed several consistency checks [1,3,4,6]. Specifi- A proper definition of M-theory as a nonperturba- cally, little has been extensively studied tive framework for superstring theories is still an open on tori and K3, while much less is known problem. It is strongly shaded among the others by the about it in more general cases. Anyway it turns out [7] lack of understanding the very structure of the world- that the appropriate nature of the degrees of freedom volume theory of the M5-branes. which complete in the UV the interacting theory is As far as this sector of the theory is concerned, its stringy and this further enforces the above proposal. totally decoupled phase has been named little string The picture that is coming out and that we have in theory [1–3]. Little string theory (for a review, see also mind is the following. Suppose one is dealing with [4,5] and references therein) is still anyway poorly un- a bounce of M5-branes. In the low energy approx- derstood. Its low energy limit is accepted to be a the- imation, if they can be separated neatly one from ory of (0, 2) self-dual tensor multiplets which lacks each other, each brane hosts a self-dual tensor mul- both locality and a Lagrangean formulation and can be tiplet theory describing its effective degrees of free- therefore studied by now just with constructive meth- dom. They come as zero-modes of membranes ending ods. The UV complete theory has been conjectured [1] on them. Suppose now that we take two M5-branes to be a closed string theory in 6 dimensions describing close-by. In this case, at some distance, the effective the boundary states of the membranes ending on the theory is expected to develop interacting terms corre- sponding to an analog of gauge symmetry enhance- ment U(1) × U(1) → U(2), but it turns out that these E-mail address: [email protected] (G. Bonelli). kind of structure—i.e., a higher rank generalization

0370-2693/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0370-2693(01)01172-8 384 G. Bonelli / Physics Letters B 521 (2001) 383–390 of nonabelian gauge symmetry—does not exists in a some points about the multi 5-brane result, we com- local sense. Notice that this fact is natural because pletely determine the structure of the geometric mod- the boundary of the membranes stretched between the uli space of the multi five-brane bound states and in- M5-branes are stringy objects while a local field the- fer the generic structure of the supersymmetric index ory interaction would describe particle vertices. This for more general geometries. This is done by combin- situation has to be compared with the other situation in ing together the information encoded in the supersym- which D-branes come together: in this case the bound- metric index formula, superstring dualities and results ary of the strings stretched between the D-branes is about the structure of (0, 2) theories. As an important composed of point-like objects whose field theory de- by-product we check the appearance of extra massless scription turns out to be given by the nonabelian gauge string states corresponding to the interaction between degrees of freedom via the usual Chan–Paton con- M5-branes as they approach each others. Section 5 is struction. The outcome of this analysis (see the cited dedicated to the discussion about open questions. literature for more complete treatment) is that the ef- fective interacting theory of M5-branes has to be de- 2 scribed as a string theory in six dimensions. The aim 2. The M5-branes index on T × M4 of this Letter is to check this picture with a supersym- metric index calculation. This means that we will start The geometric set-up that we refer to is the follow- 2 from a known result for this object (in a particular geo- ing [8,14]. We consider M-theory on W = Y6 × T × 3 metrical set-up) which has been already obtained by R ,whereY6 is a Calabi–Yau threefold of general other methods and we will recalculate it from a six- holonomy. Let M4 be a supersymmetric simply con- dimensional string theory point of view. This will be nected four-cycle in Y6 whichwetaketobearepre- done within an on-shell model. The off-shell formula- sentative of a very ample divisor. Notice that M4 is au- tion of the six-dimensional string theory is out of reach tomatically equipped with a Kaehler form ω induced in the present Letter and an open problem. from Y6 and is simply connected. We consider then N 2 In this Letter, in fact, we use the analysis performed M5-branes wrapped around C = T × M4. in [8] by proposing a six-dimensional framework for Very few is known about the full world-volume the- the BPS state counting of the M5-brane which cor- ory describing a bunch of parallel M5-branes, but from rectly reproduces the results for the supersymmetric what we believe to be true in M-theory, we can extract index which have been obtained there in a dimension- already some information about it. It can be shown ally reduced framework. This interpretation gives also that in this specific geometrical set-up [8] the poten- some hints on the structure of the moduli space of tial anomalies which tend to ruin gauge invariance preserving solutions of the M5-brane of the world-volume theory are absent and that it is world volume theory which we check to be compatible then meaningful to define a supersymmetric index for with nonperturbative superstring dualities results. In the above 5-branes bound states by extending the ap- the framework that we propose, the complete counting proach in [9]. As it is well known, the supersymmet- of these configurations amounts of two sectors which ric index is independent on smooth continue parame- are the set of fluxes of the self-dual tensor multiplet ters and as a consequence we have that this counting plus certain little string BPS saturated configurations. of supersymmetry preserving states have to coincide This Letter is organized as follows. In Section 2 we in the large and small T 2 volume. In [8] this calcula- will briefly review the results obtained in [8] for the tion was performed and this equivalence was shown supersymmetric index of certain M5-branes configu- to be effective. In particular, in the large T 2 volume rations in a specific geometrical setup. In Section 3 we calculated the supersymmetric index of the rele- we will explore a six-dimensional point of view about vant two-dimensional σ -model with target the moduli the M5-brane supersymmetric index which correctly space of susy-preserving configurations of the corre- reproduces the previous results by a combined argu- sponding twisted N = 4SYMtheoryonM4.Thiswas ment coming from the analysis of the self-dual 2-form shown to consist of the Hilbert scheme of holomorphic potential as counted in [12,13] and the little strings coverings of M4 in Y6. On each stratum, characterized to which we referred above. In Section 4 we discuss by the total rank of the covering N and by the topo- G. Bonelli / Physics Letters B 521 (2001) 383–390 385 logical numbers of the associated spectral surface Σ, 3. The single M5-brane case we calculate the supersymmetric index as 3.1. The low energy contribution (Imτ)d/2  ¯  E = − F σ/2 L0 ¯ L0 TrRR ( 1) FR q q , (2.1) Vd Let us start with the single M5-brane case. The bosonic spectrum of the low energy world-volume where d is the number of noncompact scalar bosons, theory of this 5-brane is given by a 2-form V with Vd their zero-mode volume and (0,σ) are the super- self-dual curvature and five real bosons taking values E symmetries of the model. By general arguments, is in the normal bundle NC induced by the structure of − − + a ( d/2, d/2) (0,σ/2) modular form. the embedding as TW |C = TC ⊕ NC . Passing to the On the other side of the equivalence, our explicit holomorphic part and to the determinants and using calculation, done by making use of the lifting tech- the properties of Y6, it follows that the five transverse nique [10,23] (see also [11]), for the 4-dimensional bosons are respectively, three noncompact real scalars gave, for the generic irreducible sector φ and one complex section Φ of K = Λ−2T (1,0), i M4 M4 relative to the partition N = na ·a andtoagivenir- a which is the canonical line bundle of M4. reducible holomorphic covering of M4 in Y6 of rank a, The (partially) twisted chiral (0, 2) supersymmetry completes the spectrum. It is given by a doublet of  θ Σ E = Λ a +x complex anti-commuting fields which are (2, 0) forms na ,a Hna , (2.2) ηχΣa in six dimensions and a doublet of complex anti- ε commuting fields which are scalars in six dimensions. where Hn is the Hecke operator of order n, ε is a Notice that these fermionic content reduces to the two label for the square-roots of the canonical line bundle relevant fermionic spectra in the large and small T 2 (spin structures) on M4 with respect to a given one volume limits respectively. O ⊗ 1/2 O2 = =[O⊗a+1] as ε K with ε 1, x ε shifts In principle there would be two multiplicative con- correspondingly the lattice of integer periods ΛΣa on tributions to the supersymmetric index. One given by 2 H (Σa,R).Theθ-function on the lattice Λ is defined zero modes counting and another given by the one as loop determinants. In our case, anyhow, we are deal- ing with a (partially) twisted version of the (0, 2) su-  1 (m,∗m−m) 1 (m,∗m+m) persymmetric theory which we expect to be of a topo- θ (q, q)¯ = q 4 q¯ 4 (2.3) Λ logical type (see for example [15] for the link between ∈ m Λ twisting and topological six-dimensional QFTs). This and is a modular form of weight (b−/2,b+/2). η(q) is suggests that the oscillatory contribution to the index the Dedekind η-function and χ = 2 + bΣa is the Euler is 1 and we will take this point of view. Actually, by 2 adapting the results in [17] to the relevant quadratic number of the spectral surface Σa . In particular (2.2) for the single 5-brane reads Lagrangean of the type considered in [15], it can be 4 1 checked to happen if M4 is a T -orbifold. Therefore, in the subsequent analysis we will consider the zero- θ (q, q)¯ E = Λ modes contribution. χ (2.4) η The contribution to the partition function coming from a single self-dual tensor can be analyzed follow- which is a modular form of weight (b−/2,b+/2) + ing the results of [12,13]. It consists of a θ-function of (−χ/2, 0) = (−1 − b+/2,b+/2). All this agrees with the lattice of the self-dual harmonic three forms. The the result obtained by the explicit evaluation of the θ-function is not completely specified because of the supersymmetric index (2.1). In particular, we have σ = 2b+ + 2 = 4b(2,0) + 4 right fermions and d = + (2,0) = + + 3 2b 2 b noncompact scalar bosons by 1 Notice that this result should extend to a generic simply E dimensional reduction giving to be a modular form connected Kaehler M4 by an extension of the methods worked out of total weight (−1−b+/2,b+/2) as we just obtained. in [13,17]. 386 G. Bonelli / Physics Letters B 521 (2001) 383–390   α possible inequivalent choices of its characteristics β . recalling the relation Notice that the technique developed in [13] does not in  [ ]=− (2) (1) (2) −1[ ]− (2) ∗[ ] fact single out a particular value for the characteristic a Ω Ω Ω b Ω b as an ambiguity in the choice of the relevant holomor- which holds on every Riemann surface [16] with phic factor. We will find that, in the case at hand, a period matrix Ω = Ω(1) + iΩ(2) and the property simple choice is automatically made by the require- Q2 = 1, we get ment of reconstructing the supersymmetric index that   −1 we have reviewed in the previous section. E (6) = −Q ⊗ Ω(2)Ω(1) Ω(2) E(6)  The relevant θ-function (as calculated in [13]) is + ⊗ (2) ∗ (6)   1 Ω E .    α + 0 + + + θ Z00 = eiπ((k α)Z (k α) 2(k α)β), Comparing with the general definition we finally read β  k − Z0 =−Q ⊗ Ω(2)Ω(1) Ω(2) 1 + i1 ⊗ Ω(2). where Z0 is a period matrix of the relevant six- = 2 = = manifold cohomology that we specify in the follow- In our specific case, since Σ T and Ω τ (1) + (2) ing. Let {E(6), E (6)} be a symplectic basis of harmonic τ iτ is the modulus of the torus, we have simply 3-forms on the six-manifold at hand such that, in ma- that trix notation, 0 =− (1) + (2) Z τ Q iτ 1. E(6)E(6) = 0, E (6)E(6) = 1, Now we calculate the relevant θ-function from the zero modes of the self-dual form in six dimension (6) (6) = choosing the zero characteristic candidate E E 0.       0 0 0 iπkZ0k ∗ ∗ Θ Z = θ Z 0 = e . We can expand E (6) = X0E(6) + Y 0 E(6),where 0 is the Hodge operator. Then Z0 is defined as Z0 = k X0 + iY0. Defining m = k · e(4), we calculate kZ0k =−τ (1)(m, In our case the world volume is in the product form m) + iτ(2)(m,∗ m) and we rewrite × = 2   Σ M4,whereΣ T , and therefore, being M4 1 ∗ − 1 ∗ + 0 = 4 (m, m m) ¯ 4 (m, m m) simply connected, we have Θ Z q q , m∈Λ H 3(Σ × M ) = H 1(Σ) ⊗ H 2(M ). (3.5) 4 4 where q = e2iπτ, which is equal to (2.3) as it has been This means that we can expand {E(6), E (6)} in terms calculated from the reduced dimensional perspectives. of a symplectic basis {[a], [b]} for H1(Σ),where Let us notice that the choice of the null characteris- tics candidate coincides with that of [17] where it has [a][a]=0, [a][b]=1, been shown, in the contest of calculating the self-dual tensor partition function on T 6, to be the only possi- Σ Σ ble contribution leading to a fully modular invariant [b][b]=0, result. Σ 3.2. The little strings contribution and an orthonormal basis {e(4)} for H (M ), i.e., ∗ 2 4 e(4)e(4) = 1. In terms of the previous objects we M4 To count the full spectrum of the theory a second have sector is still lacking. In fact, the 5-brane theory is completed in the UV by the little string theory which E(6) = e(4) ⊗[b] and E (6) = Qe(4) ⊗[a], has BPS saturates strings which eventually have to be where Q is the intersection matrix on M4 given by kept into account in the calculation of the complete Q = e(4)e(4). We calculate ∗E(6) =−Qe(4) ⊗ supersymmetric index. Even if a full off-shell model M4 ∗[b],where∗[b] is in the two-dimensional sense. By for this six-dimensional string theory is not available G. Bonelli / Physics Letters B 521 (2001) 383–390 387 at the moment, we will propose an on-shell simple tion. This can be done in our case since the world 2 calculation scheme for the supersymmetric index. volume is in the product form T × M4 by placing The little string theory model for the world vol- the light-cone coordinates along the T 2. Generalizing ume theory of the M5-brane is built by identifying the construction in [1] the twisted superalgebra can the boundary states of the membranes ending of the be obtained from the fermionic zero-modes and the 5-branes with closed strings configurations whose tar- brane supercharges whose anti-commutation relations get space is the 5-brane world volume itself. These can be given with a central charge matrix modeled on strings are naturally coupled to the self-dual tensor as the extended intersection matrix Z = H ⊕ Q,where = 01 the Poincaré dual of their world-sheet acts as a source H 10. Here the two factors corresponds to two- for it. This in fact guarantees the gauge invariance of form fluxes and momentum/winding degrees of free- the effective action for the 5-brane/membrane system. dom. Therefore we can switch on a set of charges in To calculate the contribution to the supersymmetric in- the lattice Γb++1,b−+1 = Γ1,1 ⊕Λb+,b− and, due to the dex we can proceed as follows. As it is given by a one to one correspondence between charges and fluxes trace on the string Hilbert space, it corresponds to a we can calculate the contribution from the BPS strings. one-loop string path integral. Moreover, as it is usual In fact, since each BPS saturated string corresponds in these index calculations, the semiclassical approx- to a chiral scalar bosonic mode, these contribute with imation is exact. Now, since M4 is simply connected, a factor of 1/η(τ) for each possible flux that we can the only contributions to this path integral can arise turn on, which is the dimension of the above lattice 2 + + = from string world sheets wrapping the T target itself. that is b+ b− 2 χM4 . Therefore we get a to- χ Therefore the configuration space of n of these string tal further multiplicative contribution of (1/η(τ)) M4 world-sheets will be given by the symmetric product which is exactly the same contribution that we have n (M4) /Sn whose points parametrize the transverse po- found before. sitions. Notice that here we are assuming that since Multiplying this last factor with that coming from there exists only one kind of membranes ending on the low energy degrees of freedom the total supersym- χ the M5-brane, there is a single type of BPS strings to metric index is given by θΛ/η as given by the dimen- be counted. Now, since the supersymmetric index cal- sionally reduced calculations (2.4). culated the Euler characteristics of the configuration space, we claim that the full contribution from these string BPS configurations is given by   4. The multiple M5-brane case −χM /24 n n q 4 q χ M4 /Sn n As far as the multi M5-brane case is concerned, − n bodd −χ /24 (1 q ) −χ = q M4 = η(q) M4 , we do not have still a direct six-dimensional way to (1 − qn)beven n>0 perform a precise counting similar to the one that we have done for the single M5-brane case since it is still −χ /24 where we fixed a global multiplicative factor q M4 not known very much about the relevant world-volume because of modularity requirements and we used well theory. Anyhow, we have some constructive arguments known results from [18]. which explain the structure of the multi 5-brane index We can compare this result with a natural general- calculation that we review in Section 2. ization of the construction done in [1] for toroidal and It is natural to read from these results 3 that the mod- K3 compactifications to our case. In fact, although a uli space of susy preserving N M5-branes wrapped on covariant quantization scheme does not exists for six- a supersymmetric Kaehler six manifold is given by the dimensional , one can consider its space of rank N holomorphic coverings of the mani- 2 light cone quantization as a tentative good defini- fold itself plus some (nonlocal) analog of the gauge bundle on them. 2 As far as the cancellation problem in a covariant quantization scheme it is concerned, it is likely to be solved as indicated in [2]. 3 See an extended discussion about this point in Section 4 of [8]. 388 G. Bonelli / Physics Letters B 521 (2001) 383–390

This can be checked by the following arguments. A possible way out is given anyhow by the pullback Let us restrict to the case in which the base six mani- procedure once the covering map from the base cycle fold admits a free S1 action and a S1 fibration. These to the multi 5-brane world-volume is given. Notice that are the cases where the known formulations for the the pull back map fails exactly at the branching locus self-dual two form [19,20] can be formulated with- of the covering, where in the typical D-brane cases out entering further problems 4 and therefore where the full nonabelian structure of the underlying vector the (0, 2) little string theory admits a better defined bundle becomes crucial, and therefore we cannot refer low energy limit (moreover there are no problems with to any well defined local geometrical structure on the anomaly [12] since the Euler characteristic vanishes base cycle to be understood to be the pull back of a automatically). We compactify further one of the flat gerbe on the covering cycle. We conclude therefore transverse directions on a circle that we take to be the that a nonlocal completeness of the pull back map M-theory eleven coordinate and map the M5-branes should be accounted for by the UV nonlocal structure to NS5-branes in IIA. Under these circumstances, the of the theory. (0, 2) theory can be mapped by fiber-wise T-duality to In the case in which the world volume manifold is 2 the (1, 1) little string theory which represents the de- in the product form T × M4, with M4 Kaehler and coupled limit of the NS5-branes of IIB. By the S self- simply connected, formula (2.2) applies to the super- duality of type IIB we map this system to an equiva- symmetric index. The T 2 holomorphic self-covering lent system of D5-branes in type IIB. In fact, D5-brane is unbranched and the little strings contribution is ex- bound states are described, in the low energy approx- plicitly exposed and we interpret it also to encode the imation, by susy-preserving configurations of the di- above mentioned effect. In fact it enters in the form χ mensional reduction of the N = 1U(N) SYM10 to the (1/η) Σa ,whereΣa is a rank a holomorphic covering susy-cycle C on which they are wrapped on. In par- of M4 (spectral manifold). The explicit dependence on ticular this means that the spectrum of the transverse the branching locus Ba appears just because bosonic fields define an holomorphic covering of rank χ = a · χ − χ . N in the total space of the normal bundle of C in the Σa M4 Ba ambient ten-dimensional manifold (which is the ambi- We can compare positively all these arguments with ent manifold itself). We naturally expect that the UV the approach developed in [7] where BPS stringy completeness of the S-dual (1, 1) theory still contains representations of the (0, 2) algebra are obtained this BPS geometric moduli space naturally and there- and the natural role of string degrees of freedom in fore also its fiber-wise T-dual (0, 2) little string theory encoding the very structure of the interacting M5- that we started from. brane world-volume theory is enlighten. From this perspective one could try to speculate Let us notice moreover also the following conse- about the possible structure of the analog of the gauge quence of the construction given in the previous sec- bundle structure. In the case of a single M5-brane, it is tion, which appears once we compare it with the geo- given by a higher-dimensional analog of the Abelian metrical moduli space which we calculated in [8]. gauge bundle structure realized within the 1-form Once we consider this moduli space from the point valued Cech cohomology. 5 As it seems by now, there of view of the little string theory BPS states, we find does not exist any straightforward nonabelian analog these strings to wrap along the T 2 and join/split in 6 for it (at least in a local framework generalization ). points located at the branching locus of the M4 cov- ering. Notice that this behavior is typical of the ma- trix strings [22,23] and is coherent with the possibil- 4 That is because in both the cases nonzero well defined vector ity of generalizing the approach in [6] to our case. fields are required to exist on the manifold to implement the self- Unfortunately, the problem of the formulation of Ma- duality condition everywhere. trix theory on curved manifold is still far from be- 5 This means that the two form potential undergoes a gauge ing understood, but it turns out that these two simi- like transformation from patch to patch and that the patching forms satisfy certain consistency conditions. lar structures likely correspond each other under the 6 Notice the strong resemblance with the no-go theorem in [21] M-theory electric/magnetic duality which exchanges about local interactions between chiral forms. membranes and 5-branes. G. Bonelli / Physics Letters B 521 (2001) 383–390 389

As a consequence of the above observations, it is continue the analysis of the little string theory to try therefore natural to conjecture that the little string to better understand the several unclear points which theory supersymmetric index (and moreover also a are left over. Possible interesting configurations which wider set of correlation functions) can be built from naturally generalize the one we have studied here 3 θ-functions of the period matrix of the covering man- could be M-theory on R × Y , with Y = CY4, K3 × ifolds by combining them both via their characteris- K3 and the 5-branes wrapped on a six holomorphic tics (see the structure of (2.2) where the shifted lattice cycle in Y . corresponds to nonzero characteristic for the relevant The question related to how one should exactly θ-function) and via the covering structure itself as far build (at least a set of) correlation functions in little as the UV completeness with the little string contribu- string theory by using these θ-function building blocks tions is concerned. In particular one expects that the remains open and needs a much more accurate analy- above ratio works for the calculation of correlators of sis. For an example in a close-by perspective see [25]. self-dual strings in the form of surface operators in the In particular one expects that the above ratio works for low energy approximation. the calculation of correlators of self-dual strings in the We expect then that, for any given rank N cover- form of surface operators. ing C of the world-volume manifold on which the N Another important issue which is raised by tak- 5-branes are wrapped, the contribution to the super- ing seriously the little string theory hypothesis is the symmetric index is of the form following. Since it is a superstring theory in 6 di-   mensions, it is noncritical and this means that one α     θ Z0 0 N Z0 , (4.6) should 7 take into account also the Liouville sector β C C which does not decouple from the σ -model. Notwith- 0 where N(ZC) is a modular form built from the periods standing it has been extensively studied, the theory of the covering six manifold which generalizes the of the Liouville (super-)field still lacks a full solution little strings contribution to the generic case while the and this problem, which in critical perturbative super- α string theory was token for avoided, seems to come nonzero characteristic β encodes the twist induced by inequivalent spin structures [25–27] analogous to back into the game again. Let us notice here just the the one which appears in (2.2). These structures and following coincidence. The near horizon geometry of the possible rank N holomorphic coverings has then the M5-brane in eleven-dimensional super-gravity is 4 to be summed up. AdS7 × S . Suppose we study the coupled theory of We expect also that a structure analogous to (4.6) the 5-brane as a membrane theory on this fixed back- applies also to the set of amplitudes relative to a proper ground. Then the longitudinal radial membrane field supersymmetric version of the surface operators con- spans the radial direction in the AdS7. The boundary sidered in [13,24] where a nonzero argument of the value of this field will presumably play a crucial role in θ-function encodes the surface periods. a decoupling procedure and it should be linked to the Liouville field which appears in the six-dimensional string theory in a way similar to the original point of 5. Conclusions and open questions view of [28]. An apparently unrelated question con- cerns the claim that the low energy limit of little string In this Letter we have proposed a six-dimensional theory does not contain gravitational degrees of free- framework for the evaluation of the supersymmetric dom. This issue could also be related to the specific na- index of the M-theory five-brane which correctly ture of the full six-dimensional noncritical string the- reproduces the results in [8] and proved the explicit ory. need of taking into account the full spectrum of the little string theory to reproduce a precise counting in the form of BPS saturated string states. 7 As an alternative, one might try to add to the theory a non- It would be very interesting of course to check geometrical sector to reach the Virasoro central charge saturation, how the methods that we developed in this note but these degrees of freedom should not contribute both to the low extend to other possible M5-brane geometries and to energy effective theory and neither to the supersymmetric index. 390 G. Bonelli / Physics Letters B 521 (2001) 383–390

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