1
Notes on Matrix and Micro Strings∗ Robbert Dijkgraaf a, E. Verlinde b, and H. Verlinde c aDepartment of Mathematics University of Amsterdam, 1018 TV Amsterdam bTH-Division, CERN, CH-1211 Geneva 23, Institute for Theoretical Physics University of Utrecht, 3508 TA Utrecht cInstitute for Theoretical Physics University of Amsterdam, 1018 XE Amsterdam
We review some recent developments in the study of M-theory compactifications via Matrix theory. In particular we highlight the appearance of IIA strings and their interactions, and explain the unifying role of the M-theory five-brane for describing the spectrum of the T 5 compactification and its duality symmetries. The 5+1-dimensional micro-string theory that lives on the fivebrane world-volume takes a central place in this presentation.
1. Introduction with many quantized charges corresponding to the various possible wrapping numbers and in- In spite of considerable recent progress, M- ternal Kaluza-Klein momenta. theory is as yet a theory without a fundamental An alternative but presumably equivalent char- formulation. It is usually specified by means of a acterization of M-theory is that via compactifica- number of fundamental properties that we know tion on a circle S1 it becomes equivalent to ten it should posses. Among the most important of dimensional type IIA string theory [2]. The par- these characteristics is that the degrees of free- ticles with non-zero KK-momentum along the S1 dom and interactions of M-theory should provide are in this correspondence identified with the D- a (mathematically consistent) representation of particles of the IIA model, while the string cou- the maximally extended supersymmetry algebra pling constant gs emerges as the radius R of the of eleven dimensional supergravity [1] S1 via m mn (2) {Qα,Qβ} =(Γ)αβ Pm +(Γ )αβZmn 2 3 `sgs = R, `s = `p/R. (2)
m1...m5 (5) +(Γ )αβZm1...m5 . (1) Here `s denotes the string scale and `p is the 11- dimensional Planck scale. Also upon further com- Here P denotes the eleven dimensional mo- m pactification, all M-theoretic charges now have mentum, and Z(2) and Z(5) represent the two- well understood interpretations in terms of per- and five-index central charges corresponding to turbative and non-perturbative string theoretic the two types of extended objects present in M- configurations, such as wrapped strings, D-branes theory, respectively the membrane and the five- [3] or solitonic five-branes [4]. The precise and brane. Upon compactification, these extended well-defined perturbative rules for determining objects give rise to a rich spectrum of particles, the interactions among strings and D-branes [3] ∗Based on lectures given by H.V. at the APCTP Winter [5] [6] so far provides the best basis for con- School held in Sokcho, Korea, February 1997, and joint crete quantitative studies of M-theory dynamics. lectures at Cargese Summer School June 1997, as well as String theory still forms the solid foundation on on talks given by H.V. at SUSY ’97, Philadelphia, May 27- 31, and by R.D. and E.V. at STRINGS ’97, Amsterdam, which the new structure of M-theory needs to be June 16-21, 1997. built. 2
From both the above starting points, a con- pactified on T 5, based on the 5+1-dimensional vincing amount of evidence has been collected world-volume theory of N NS 5-branes [13]. We to support the conjecture that toroidal M-theory will pay special attention to the appearance of compactifications exhibit a large group of discrete the M5-brane in this proposal, and with the aim duality symmetries, that interchange the various of illuminating the direct correspondence between internal charges [7]. While some of these dual- the framework of [13] and our earlier studies [4]. ities are most manifest from the eleven dimen- In addition we will highlight the main ingredients sional perspective, others become visible only in of Matrix string theory [11], and discuss its appli- the string formulation. In combination, these so- cation to the study of the interacting string-like called U-duality symmetries put very strong re- theory on the 5-brane worldvolume. strictions on the possible non-perturbative prop- erties of M-theory, and in this way provide a wealth of information about its dynamics. 2. U-duality and the M5-brane A very fruitful approach towards unraveling the The degrees of freedom and interactions of M- mysteries of M-theory is provided by the matrix theory compactified on a five-torus T 5 provide a formulation initiated in [8], known as ‘Matrix the- representation of the maximal central extension ory.’ Its basic postulate is that the full eleven- of the 5+1 dimensional N = 4 supersymmetry al- dimensional dynamics of M-theory can be cap- gebra, which in a light-cone frame takes the form2 tured by means of the many-body quantum me- chanics of N D-particles of IIA string theory, in a b ab Qα, Qβ = p+δαβ1 the limit where N is taken to infinity. From the n a b o I ab ab eleven dimensional perspective, this limit can be Qeα, Qeβ = pI γαβ1 + δαβZ (3) interpreted as applying a Lorentz boost, so that nQa ,Qbo = p δ 1ab all degrees of freedom end up with a very large α e β − αβ momentum along the extra eleventh direction. Here the internal labels a, b =1,...,4areSO(5) One can formally combine this limiting pro- spinor indices, and I =1,...,4 label the four cedure with a decompactification limit of the transverse uncompactified space dimensions. The eleventh direction, so that the IIA string and M- central charge matrix Zab comprises a total of theory compactification manifolds become identi- 16 internal charges, which split up in five quan- fied. In this way, a fundamentally new perspec- tized Kaluza-Klein momenta p with i =1,...,5, tive on non-perturbative string theory arises, in i ten charges m = −m that specify the winding which its complete particle spectrum in a given ij ji numbers of the M membranes around the five- dimension becomes identified with a relatively torus, and one single charge q that represents convenient subset of states compactified to one 5 the wrapping number of the M5 brane. To make dimension less. Though in essence a light-cone this interpretation more manifest, it is often con- gauge description, Matrix theory thus manages venient to expand Z in terms of SO(5) gamma to provide a rather detailed theoretical frame- ab matrices as work for exploring the microscopic dynamics of M-theory, while preserving many of the geomet- i ij Zab = q5 1ab + piγab + mij γab. (4) rical foundations of string theory. There is now indeed a rather well-developed understanding of More generally, the decomposition of Zab into how Matrix theory incorporates all perturbative integral charges depends on the moduli of the degrees of freedom and interactions of type II T 5 compactification, which parametrize the coset strings [9] [10] [11]. manifold In these notes we will review some of the re- cent developments in this approach to M-theory. M = SO(5, 5)/SO(5) × SO(5). (5) In particular we will discuss in detail the re- 2Hence α and β are spinor indices of the transversal SO(4) cently proposed formulation of M-theory com- rotation group. 3
The U-duality group is the subgroup of SO(5, 5) charges q5 and pi as fluxes of 5- and 1-form field rotations that map the lattice of integral charges strengths through the 5-cycle and 1-cycles on T 5. on to itself, and is thus identified with Before presenting this construction, let us men- tion that in course we will in fact arrive at a rather U = SO(5, 5, Z). (6) new notion of the M5-brane worldvolume theory, From the eleven dimensional viewpoint, the mod- that is rather different from the usual one in terms uli space (5) is parametrized by the metric G of the physical location of the fivebrane soliton in mn space-time. In particular, the M5-brane world- and three-form potential Cmnk defined on the in- ternal manifold T 5. After replacing the 3-form volume theory we will arrive at lives on an aux- iliary 5+1-dimensional parameter space, denoted C on T 5 by its Hodge-dual 2-form B = ∗C, by T 5. This auxiliary space is only indirectly re- this parametrization of M reduces to the familiar one from the toroidal compactification of type II lated to the embedded M5-brane soliton in phys- string theory on T 5. This correspondence, which icale space-time, via an appropriate geometrical is strengthened by the fact that the U-duality interpretation of the worldvolume fields. group (6) takes the form of the T 5 T-duality In this new set-up, we can introduce a winding group, has become a central theme in recent at- number of the M5-brane via tempts to formulate the T 5 compactification of
M-theory [4,13]. q5 = dV, (8) 5 To study the spectrum of M-theory on T 5,it ZT is natural to start from the maximal dimensional where the 5-form dV is interpreted as the pull- object that can wrap around the internal mani- e back to the auxiliary torus T 5 of the constant fold, which is the M5 brane. From its descrip- volume element on the target T 5. We would like tion as a soliton it is known that its low en- to turn the 4-form potential Ve into an indepen- ergy collective modes form a tensor multiplet of dent field, that is part of the world-brane theory. the =(2,0) world-brane supersymmetry, con- N To this end we imagine that the (2,0) world- sisting of five scalars, an anti-symmetric tensor brane tensor multiplet in fact provides an effec- with self-dual field strength T = dU and 4 chiral tive light-cone description of the M5-brane. Con- fermions. As shown in [4], this worldvolume field cretely this means that of the five scalars, we will theory can be extended to include all the above interpret only four as describing transversal co- 16 charges of M-theory on T 5 via an appropriate ordinates in space-time. This leaves us with one identification of flux configurations. additional scalar field Y . Since the world-brane To make this concrete, consider a M5-brane is 5+1-dimensional, this scalar can be dualized with topology of T 5 R where R represents the × to a 4-form, which we will identify with the 4- world-brane time τ. To begin with, we can in- form V used in (8). Hence we will assume the clude configurations carrying a non-zero flux 5-form field strength W = dV associated with V is normalized such that the associated flux is a mij = dU (7) non-negative integer. T 3 Z ij Quite generally, in a light-cone formalism for 3 through the 10 independent three-cycles Tij in extended objects one typically obtains a residual T 5. An M5-brane for which these charges are gauge symmetry under volume preserving diffeo- non-zero represents a bound state with a corre- morphisms. For the case of the 5-brane these can sponding number of membranes wrapped around be used to eliminate all dependence on the five i the dual 2-cycle. The charges mij are part of compact embedding coordinates X (that map a 16-dimensional spinor representation of the U- the auxiliary T 5 into the target T 5), except for duality group SO(5, 5, Z), which is isomorphic to the volume-form the odd (co)homology of the five-torus T 5.This e i j k l m observation motivates us to try to write the other dV =dX ∧dX ∧dX ∧dX ∧dX ijklm. (9) 4
The five spatial components of ∗V i can roughly known to be advantageous to combine into the be identified with the embedding coordinates worldsheet CFT all superselection sectors corre- Xi. In this correspondence the linearized gauge- sponding to all possible momenta and winding transformation V → V + dΛofthe4-formpo- numbers. T-duality for example appears as a true tential describes a volume preserving diffeomor- symmetry of the worldsheet CFT only if it in- phism, since it leaves dV invariant. cludes all sectors. The notion of the M5 world- We can formalize the duality transformation brane theory used here is indeed the direct ana- between V and Y by taking W to be an inde- logue of this: just as for strings, the various wrap- pendent 5-form and introducing Y as the La- ping numbers and momenta appear as quantized grange multiplier that imposes the Bianchi iden- zero modes of the worldvolume fields. A sec- tity dW =0.ThefactthatWhas integral ond important analogy is that, just like the CFT fluxes implies that Y must be a periodic field, Hilbert space includes unwrapped string states, i.e. Y ≡ Y +2πr with r integer. The 5 remaining we can in the above set-up also allow for un- charges pi can thus be identified with the integer wrapped M5 brane states with q5 =0. winding numbers of Y around the 5 one-cycles This naturally leads to the further sugges- tion that it should be possible to extend the pi = dY. (10) M5 worldbrane theory in such a way that U- T 1 I i duality becomes a true symmetry (just like T- duality in toroidal CFT). In [4] it was shown that Since on-shell ΠV ≡ δS/δV˙ = dY , we deduce that dY is the canonical momentum for V .From this can indeed be achieved by assuming that 5 this it is straightforward to show that these op- the worldvolume theory on T is described by erators are indeed the generators of translations a 5+1-dimensional second quantized string the- along the internal directions on the target space ory, whose low energy effectivee modes describe 5-torus. The formula for the M5-brane wrapping the (2,0) tensor multiplet. Via the SO(5, 5, Z) number now becomes T-duality of this “micro-string theory”, this pro- duces a manifestly U-duality invariant description q5 = ΠY , (11) of the BPS states of all M-theory compactification 5 ZT down to 6 or more dimensions [4]. with ΠY the conjugate momentum to Y . Until quite recently, only limited theoretical In thise way all the 16 charges (q, pi,mij )have tools were available to analyze the detailed prop- been written as fluxes through the odd homol- erties of this conjectured micro-string theory, or ogy cycles on the M5-brane. These 16 bosonic even to confirm its existence. The traditional zero modes form the world-volume superpartners methods usually start from the physical world of the 16 fermionic zero modes, that represent volume of a macroscopic M5-brane soliton, and the space-time supersymmetries that are broken the micro-string theory is in fact not easily visi- by the M5-brane soliton. Based on this result, ble from this perspective. As seen e.g. from the it was shown in [4] that the (2,0) worldvolume identifications (10) and (11), the local physics on theory of the M5-brane could be used to repre- the target space torus T 5 is somewhat indirectly sent the full maximally extended space-time su- related to that on the parameter space T 5 where persymmetry algebra (3). the micro-string lives. The specification of a par-
The distinction between the above and the ticular winding sector q5 , for instance, breakse the standard notion of the 5-brane world-volume is U-duality group to geometric symmetry group perhaps best explained in analogy with the world- SL(5, Z), and thus hides the stringy part of the sheet description of toroidally compactified string micro-string T-duality symmetry. This illustrates theory. According to the strict definition, the that it may be misleading to think about the worldsheet of a string refers to its particular loca- micro-strings directly as physical strings moving tion in space-time, and thus also specifies a given on the space-time location of the M5-brane soli- winding sector. For strings, however, it is well ton. 5
3. BPS Spectrum ing relations of light-cone string theory. In addi- tion, we must impose Further useful information about the relation between the auxiliary and target space can be H = p− (15) obtained by considering the masses of the various which reflects the identification of the world- BPS states. volume time coordinate τ with the space-time The world-brane theory carries a chiral N =2 light-cone coordinate x+. The mass-shell condi- supersymmetry algebra. To write this algebra, 2 tion p+p− = m (we assume for simplicity that it is convenient to introduce chiral SO(4) spinor all transverse space-time momenta pI are zero) indices α andα ˙ for the transverse rotation group relates the space-time mass of M5-brane configu- (which represents an R-symmetry of the world- ration to the world-volume energy H via volume supersymmetry). We then have 2 m = p+ H (16) α β αβ 0 i Qa ,Qb = (γabH + γab(Pi + Wi)), This relation, together with identification of the n o central charge Zab as world-brane fluxes, uniquely α˙ β˙ α˙β˙ 0 i Qa ,Qb = (γabH + γab(Pi − Wi)). (12) specifies the geometric correspondence between the auxiliary torus T 5 and the target space torus n o 5 Here H is the world-volume Hamiltonian, and Pi T . To make this translation explicit, let us de- is the integrated momentum flux through the five- note the lengths ofe the five sides of each torus torus by Li and Li respectively, and let us fix the nor- malization of the fields dU and dY in accordance H = T00,Pi=T0i, (13) withe the flux quantization formulas (7), (10), and 5 5 ZT ZT (11). Since the left-hand sides are all integers, where Tmn denotes the world-volume energy- this in particular fixes the dimensionalities of the momentume tensor. HenceeH and Pi represent two-form U and scalar Y . the generators of time and space translation on To simplify the following formulas, we will fur- the 5+1-dimensional parameter space-time. The ther use natural units both in the M-theory target operators Wi that appear in the algebra (12) rep- space, as well as on the auxiliary space. In the resent a vector-like central charge, and is a re- target space, these natural units are set by the flection of the (possible) presence of string-like eleven dimensional Planck scale `p, while on the objects in the world-volume theory. This vec- world-volume, the natural units are provided by tor charge Wi indeed naturally combines with the the micro-string length scale `s.Apartfromthe 5 momenta Pi into a 10 component vector represen- size of the five torus T˜ , this string length `s is tation of the SO(5, 5, Z) world-volume T-duality indeed the only other length scalee present in the group. 5+1-dimensional world-volume theory. e The world-brane supercharges represent the Now let us consider the physical masses of the unbroken part of space-time supersymmetry al- various 1/2 BPS states of M-theory. These states gebra. The other generators must be identified can carry either pure KK momentum charge, or with fermionic zero-modes, that are the world- pure (non-intersecting) membrane and fivebrane brane superpartners of the bosonic fluxes de- wrapping number. From the target-space per- scribed above. In order to exactly reproduce the spective, we deduce that these states have the 5+1-dimensional space-time supersymmetry (3), following masses (in 11-dimensional Planck units however, we must project onto states for which `p =1) the total momentum fluxes Pi and vector central 2 2 KK momenta: m =(pi/Li) charges Wi all vanish 2 2 Pi =0,Wi=0. (14) membranes: m =(mij LiLj) (17)
2 2 These are the analogues of the usual level match- fivebranes: m =(q5V5) 6
where V5 = L1 ...L5 denotes the volume of the are V5 and p+, and we need to express them in 5 M-theory five torus T in Planckian units. terms of V5 and β in (18). A little algebra show In the world-volume language these 1/2 BPS that states all correspond to the supersymmetric 1 e β V = ,p=. (21) ground states that carry a single flux quantum 5 2/3 +1/3 number. Their mass is therefore determined via V5 V5 the mass-shell relation (16) by the world-volume In particular we find that the wordvolume radii energy carried by the corresponding flux con- are relatede to the M-theorye radii by figuration. To deduce this energy, it is suffi- L L = i . (22) cient to note that at very long distances, the i 1/2 world-volume theory reduces to a non-interacting V5 free field theory of the (2,0) tensor multiplet. Hencee we see in particular that the decompactifi- Hence the flux contribution to the M5-brane cation limit of M-theory corresponds to the zero Hamiltonian can be extracted from the free field volume limit of the auxiliary torus. expression3 Of course, here we have to take into account that T-duality of the micro-string theory can re- 1 5 2 2 2 H = d x |dY | + |dU| + |ΠY | (18) late different limits of the T 5 torus geometry. β T 5 Z This fact plays an important role when we con- where β is an arbitrary parameter (to be related sider partial decompactifications.e For example, later on toe the light-cone momentum p+ in space- if we send one of the radii of the M-theory five- time). Now using (7), (10), and (11), we deduce torus, say L5, to infinity, then (22) tells us that, that the 1/2 BPS states have the following world- after a T-duality on the remaining four compact volume energies (in micro-string units `s =1) directions, the worldvolume of T 5 tends to infin- 2 ity. In this way we deduce that the BPS states of V5 pi KK momenta: e the T 4 compactification of M-theorye is described β Li by the zero slope limit of the micro-string the- e 5 2 ory on a T world-volume. This is of importance (mij LiLj ) membranes: H =e (19) in understanding the SL(5, Z) U-duality group βV5 of M-theory compactifications on T 4 [14,15]. In- e e d q2 deed, all U-duality symmetries for T compact- fivebranes: H = 5 e ifications with d ≤ 5 can be understood in this βV 5 way as geometric symmetries of the micro-string Here V5 = L1 ...L5 denotes the volume of the model. auxiliary space T 5 in string units.e Finally, let us mention that as soon as one con- Frome the conditione e that all these three expres- siders states with more general fluxes, they are sions reproducee the correct mass formulas (17) at most 1/4 BPS and the mass formula becomes via (16), we can now read off the M-theory pa- more involved. One finds rameters in terms of the world-volume parame- m2 = m2 +2V ( /L )2 + 1 ( L )2, (23) 5 BPS 0 5 Ki i V 2 Wi i ters. First of all we see that the two five tori T 5 5 and T in fact have the same shape, i.e. they are with q related by means of an overall rescaling m2 =(qV)2+(p/L )2 +(m L L )2, e 0 5 5 i i ij i j Li = vLi (20) K = q p + 1 m m , (24) with v some dimensionless number. The remain- i 5 i 2 ijklm jk lm ing M-theorye parameters we need to determine Wi = mij pj . 3Here we have used that the self-duality relation Π = dU. U These formulas are completely duality invariant, We further left out the transverse scalars XI since their fluxes are set to zero. and were reproduced via a world-volume analysis 7 in [4]. A crucial new ingredient in the description matrix conjecture of [8] by applying some dual- of BPS states with a non-zero value for Ki is that ity symmetry of the type II string to map the they necessarily contain worldbrane excitations D0-branes to some other type of branes. Com- 5 4 with non-zero momentum on T , while Wi 6=0 pactification of matrix theory indeed generically necessarily describes states with non-zero wind- proceeds by using the maximal T-duality to inter- ing number for the micro-strings.e Hence, with- change the D0 brane charge by that of the maxi- out the micro-string degrees of freedom, the M5- mal dimensional D p-brane that can wrap around brane would have had no BPS-states with non- the internal manifold. In our specific example of 5 zero value for Wi. The geometrical interpretation T this would be the D5-brane. Alternatively, of the bilinear expressions for Ki and Wi will be- as was first suggested in [13] (following up ear- come more clear in the following sections. lier work [14] [15] [4]), one can choose to identify this charge N with the NS five-brane wrapping 4. M-theory from NS 5-branes number around the five-torus T 5. In the following table we have indicated the Theaboveattempttounifythecompletepar- chain of duality mappings that provides the dic- ticle spectrum of a given M-theory compactifica- tionary between the original matrix set-up of [8] tion in terms of one single world-volume theory and the dual language of [13]: has now found a natural place and a far reaching generalization in the form of the matrix formula- N D0 D 5 tion of M-theory initiated in [8]. While the pre- vious sections were in part based on an inspired pi KK NS1 guess, motivated by U-duality and the form of the T5 S = −→ −→ space-time supersymmetry algebra, in the follow- mij D2 D3 ing sections we will describe how the exact same structure can be derived from the recently pro- q NS5 NS5 posed Matrix description [13] of M-theory on T 5. 5 So we will again start from the beginning. MonT5 IIA on T 5 IIB on T 5 According to the Matrix theory conjecture, one can represent all degrees of freedom of M-theory b via an appropriate large N limit of the matrix NS5 NS5 quantum mechanics of the D0-branes [8]. The true extent this proposal becomes most visible D1 D4 when one combines this large N limit with a de- S T5 −→ −→ compactification limit R →∞ of the 11-th M- D3 D2 theory direction. The ratio D5 D0 p+ = N/R (25) represents the light-cone momentum in the new IIB on T 5 IIA on T˜5 decompactified direction. U-duality invariance implies that any charge of Table 1. Interpretationb of the quantum compactified IIA string theory can in principle be numbers and their translation under the du- taken to correspond to the eleventh KK momen- ality chain. tum N. One can thus reformulate the original We notice that all 16 quantized charges, that 4 This is implied by the level-matching condition Pi =0, make up the central charge matrix Zab,havebe- since Ki in fact represents the zero-mode contribution to come D-brane wrapping numbers in the last two P . Notice further that for q 6=0,K contains a contri- i 5 i columns, and furthermore that the SO(5, 5, Z) bution proportional to q5 times the space-time KK mo- mentum pi, which indeed reflects the geometric relation U-duality symmetry is mapped onto the (mani- between world-volume and target space translations. fest) T-duality symmetry of the type II theory. 8
Note further that the D-brane charge vector in- while the string coupling gs of the IIA string after deed transforms in a spinor representation of the the T5ST5 transformation becomes T-duality group. 2 e 2 R The string coupling, string scale and dimen- g˜s = (30) sions of the five torus are related through this V5 duality chain as follows.5 The chain consists of The string scale of the IIA theory is the same as 5 a T5 duality on all five direction of the T ,an in (28). S-duality of the IIB theory, and finally again a In [13] convincing arguments were given for the maximal T-duality on T 5. The end result is a existence of a well-defined limit of type II string mapping from large N D-particle quantum dy- theory, that isolates the world volume dynamics namics to that of N NS five-branes in IIA theory. of N NS five-branes from the 9+1-dimensional On the left, we have started with the conven- bulk degrees of freedom. The key observation tional definition of M-theory compactified on a made in [13] is that, even in the decoupling limit 5 five-torus T , with dimensions Li, i =1,...,5, as g˜s → 0 for the bulk string interactions, one is the strong coupling limit of type IIA string theory still left with a non-trivial interacting 5+1 dimen- 5 on the same torus T . Under the first T5 dual- sional theory that survives on the world volume ity, a D0-brane becomes the D5-brane wrapped of the 5-branes. This theory depends on only two around the dual torus T5 with dimensions dimensionless parameters, namely N and the size ˜5 ˜ 2 of the torus T in units of the string length `s. `s 1 Σi = = . b (26) If one keeps the last ratio finite, the system will Li RLi contain string like degrees of freedom. According Here we used (2), and again work in eleven dimen- to the proposal put forward in [13], this world sional Planck units `p = 1. Following the usual volume theory of N NS 5-branes in the limit of rule of T -duality, the string coupling gs gets a large N may provide a complete matrix theoretic factor of the volume in string units, so the dual description of M-theory compactified on T 5. coupling gs becomes (see again (2) This proposal in fact needs some explanation, since as seen from (30) this decoupling limit g 5 s → gs`s 1 0 in fact contradicts the decompactification limit gs = b= (27) V RV 5 5 R →∞that is needed for obtaining M-theorye on 5 5 T . The resolution of this apparent contradiction whereb V5 is again the five-volume of T (in Planck units). The subsequent S-duality transformation should come from the particular properties of the maps the D5-brane onto the IIB NS5-brane, in- world-volume theory of N 5-branes for large N. verts the string coupling, and changes the string As we will see later, via a mechanism familiar e.g. from studies of black holes in string theory scale by a factor of gs.Sowehave [22] or matrix string theory [9][10][11], the system 0 ˜2 p1 of N wrapped NS 5-branes contains a low energy g˜s=V5R, `s = 2b . (28) R V5 sector with energies of order 1/N times smaller ˜5 The size of the five torus of course remains un- than the inverse size of the five-torus T .These changed under the S-duality. low energy modes are the ones that will become the M-theory degrees of freedom, and the large Finally we perform the T5 duality again, so that N counts IIA fivebrane charge. The dimensions N limit must be taken in such a way that only of the resulting five-torus T˜5 are these states survive. This implies that the typical length scale of the relevant configurations is larger Li by a factor of N than the size of T˜5. L˜i = . (29) RV5 When one analyzes this low lying energy spec- trum for very large N, one discovers that it be- 5For this section we benefited from valuable discussions with G. Moore. A similar presentation can indeed be comes essentially independent of the string cou- found in [25]. pling constant gs,providedgs does not grow too
e e 9 fast with N. It thus seems reasonable to as- volume. sume that in an appropriate large N limit, the world-volume theory of the NS 5-branes itself 5. SYM theory on T 5 becomes independent of gs. This opens up the possibility that the weak coupling limit that was One possible starting point for concretizing this matrix description of M-theory on T 5 is provided used in [13] for proving thee existence of the 5+1- dimensional worldbrane theory, in fact coincides by the large N limit of 5+1-dimensional super with the strong coupling regime that corresponds Yang-Mills theory with the decompactification limit of M-theory. 1 6 1 2 1 I 2 S = d x tr F + (DµX ) Before we continue, let us remark that this Ma- g2 4 µν 2 Z trix theoretic set-up can already be seen to re- 1 produce many qualitative and quantitative fea- +ψΓiD ψ + ψΓI [X I ,ψ]+ [XI,XJ]2 i 4 tures of the framework described in the previous sections. Evidently, the world-volume theory of Here I =1,...,4 labels the 4 Higgs scalar fields. the N NS 5-branes plays a very analogous role to This theory can be thought of as describing the the proposed M5-brane world-volume theory of low energy collective modes of N D5 branes of sections 2 and 3. Most importantly, in both set- IIB string theory (the second column in the above ˆ5 ups the U-duality symmetry of M-theory becomes table), wrapped around the dual five-torus T . identified with a T-duality of a 5+1-dimensional In terms of the parameters above, the Yang-Mills worldvolume string theory. In addition, as we will coupling of the D5-brane world-volume theory is see a little later, all 16 charges of M-theory arise then 1 1 in the present Matrix theory set-up [13] via the = = V R2 (31) g2 g `2 5 exact same identification of fluxes as described in YM s s section 2. Alternatively, we can apply an S-duality and view Finally, by comparing the formulas (28) and the largeb N SYM model as the low energy world (29) with those of the previous section, we see volume theory of N IIB NS 5-branes. Note that that the size and shapes of the respective auxil- the above YM coupling then becomes equal to the 5 ˜5 iary five-tori T and T (notice however the subtle fundamental NS string tension gYM = `s. difference in notation) are expressed via an identi- According to the prescription of Matrix theory, cal set of relationse in terms of those the M-theory finite energy states of M-theory on Te5 are ob- torus T 5. The micro-string torus T 5 of sections tained by restricting to states of the SYM model 2 and 3 is indeed related the Matrix theory torus with energies of order 1/N . Since these states T˜5 used above via a rescaling withe a factor N. are all supported at scales much larger than the Since (as will be elaborated later on) the micro- string scale, it seems an allowed approximation to string scale `s is also N times larger than the discard the stringy corrections to the U(N)SYM fundamental string scale, the two auxiliary five description of the NS 5-brane dynamics. An ob- tori have thee same size relative to the respective vious subtlety with making this low-energy trun- string scales. This fact ensures that the standard cation is that the 5+1 dimensional SYM model T-duality symmetry of the small T˜5 indeed trans- in itself does not represent a well-defined renor- lates into an isomorphic T-duality group that acts malizable field theory. It is conceivable, however, on the large T 5 on which the micro-string lives. that via the combined large N and IR limit that Via this formulation [13], Matrix theory thus is needed for Matrix theory, one can still extract naturally incorporatese our earlier ideas on the well-defined dynamical rules for the subset of the M5-brane. More importantly, however, it also SYM degrees of freedom that are relevant for the provides a concrete prescription for including in- description of M-theory on T 5. teractions, both among the M5-branes as well A second remark is that the Yang-Mills descrip- as for the micro-string theory inside its world- tion in fact seems incomplete. Namely, to de- scribe all possible quantum numbers, one should 10 be able to include all possible bound states of the is the instanton winding number. Hence the SYM NS 5-branes with the various D-branes. Gener- field theory explicitly reveals the presence of the ally these are described by turning on appropri- string like degrees of freedom in the form of the ate fluxes of the SYM theory. For the D1-branes non-abelian instanton configurations. As we will and D3-branes, the corresponding winding num- describe in a later section, the long distance dy- ber are represented via the electric and magnetic namics of these instanton strings may be used as a flux quantum numbers quantitative description of the interacting micro- string theory. pi = tr E, mij = tr F, (32) If ki denotes the SU(N) Yang-Mills instanton T 4 T 2 Z i Z ij number, we evaluate 5 1 through the dual 4 and 2-cycles of the T , respec- NWi = 2(m∧m)i +Nki. (37) tively. The D5 brane wrapping number, however, As we will make more explicit in a moment, the does not seem to appear naturally asb a flux of right-hand side can indeed be seen to correspond the SYM theory. This fact has so far been the to a total string winding number. 6 To see this main obstacle for incorporating the transverse M5 (and to get more insight into how one might re- brane into Matrix theory. cover the M5-brane from the present set-up) it Central in the correspondence between the will be useful to consider the world-volume prop- 5 large N SYM theory and M-theory on T is the erties of the IIA and IIB NS 5-branes from a fact that (apart from this problem of the missing slightly different perspective. q5 -charge) the maximally extended supersymme- try algebra (3) can be represented in terms of 6. Winding Microstrings the SYM degrees of freedom [18]. The construc- tion is quite standard from the general context of After the ground breaking work [20] on the en- string solitons, and relies on the identification of tropy of 5+1-dimensional black holes, it became the world-volume supersymmetry charges of the clear that the internal dynamics of the string SYM model theory 5-branes are most effectively described in terms of an effective world-volume string theory i I Q = tr θ γ Ei + γ ΠI (33) [4,22]. These string-like degrees of freedom, which Z h in the following we will also call micro-strings, arise in part due to one-dimensional intersections 1 ij iJ IJ − 2 (γ Fij + γ DiXJ + γ [XI,XJ]) [20] that occur between D-branes, but also due i to fundamental NS strings trapped inside NS 5- with the generators the unbroken space-time su- branes [19][13]. As indicated in fig. 1. the relation persymmetries, while the generators of the broken between the world-volume string theory and the supertranslations are represented by the fermion low-energy collective modes of the 5-branes in- zero modes deed parallels that between the fundamental type ˜ II strings and the low energy effective field the- Q = tr θ. (34) ory in space-time. This correspondence predicts Z in particular the precise form of the micro-string The dynamical supersymmetry charges (33) gen- ground states. erate an algebra of the exact form (12), where In the past year, this effective string theory has been developed into a remarkably successful P = tr(E F +ΠIDX+θTDθ) (35) i i ij i i 6This is true provided the U(N) symmetry is essentially Z unbroken by the Higgs scalars. More generally, if some is the momentum flux and where of the Higgs scalars have expectation values that are sub- stantially different from zero, Wi will naturally split up W = tr(F F ) (36) into separate contributions corresponding to instantons of i ∧ i the smaller unbroken subgroups. Z 11
N separate strings with reduced string tension Space-time Worldvolume 1/N α0. These fractional strings are also inter- d=10 d=6 section strings, and string duality indeed implies that they are indistinguishable (in the sense of statistics) from the above D-brane intersection Collective Type IIA strings. Denoting the NS string winding charge String Theory String Theory by ki, the total number of intersection strings is given by7 low energy B 1 low energy limit NW = (m m) +q p +Nk (39) limit i 2 ∧ i 5 i i This formula generalizes the formula (37) for the instanton winding charge to the case where Effective Collective q = 0. Notice further the correspondence with FieldTheory Field Theory 5 6 equation (24). Let us now switch to the IIA perspective. In Fig. 1: The effective world-volume dynamics this case D-branes have even dimensions and of the solitonic NS 5-brane defined a d =6 thus D-brane configurations are characterized by “collective string theory”, whose low-energy A 5 degrees of freedom coincide with the usual col- q ∈ Heven(T , Z), and string-like intersections lective fields. occur between 2- and 4-branes. Thus their total winding number is determined via framework for describing the statistical and dy- 1 A A namical properties (such as absorption and emis- 2 (q ∧q )i = mij pj (40) sion processes) of the 5-brane black holes. Until 5 which is the intersection form on Heven(T , Z). recently, however, it was unclear to what extent Hence, via the same mechanism as described for the corresponding 5+1-dimensional string could the IIB case, the total micro-string winding num- really be isolated as a separate theory, that ex- ber (including the intersection strings between ists independently from the 10-dimensional type fundamental strings the N fivebranes) now takes II string. The elegant argumentation of [13] pro- the form vides convincing evidence that this is indeed the A case. We will now collect some useful topological NWi = mij pj + Nki (41) properties of the micro-string theory. where ki denotes the IIA string winding number. A D-brane in type IIB string theory has odd di- As the IIA theory and IIB model are related mension, and thus a general configuration of D- 5 via T-duality, the two micro-string theories are branes wound around the internal manifold T actually the same, except that the total winding will be characterized by a 16 component vector numbers Wi get interchanged with the total mo- B 5 q ∈ Hodd(T , Z) in the odd integral homology mentum fluxes Pi. From the above formulas it is of T 5. Hence the corresponding D-brane intersec- easy to see that the flux contribution to e.g. the A tion strings carry a total winding number given IIA string winding number Wi is indeed identi- 5 by the intersection form on Hodd(T , Z), which cal to the flux contribution to the IIB momentum in terms of our labeling of charges (see Table 1 in B flux Pi given in (35). A similar check can also section 3) takes the form B be done for the IIB winding charge Wi and the A B B dual IIA momentum flux P . 1 (q ∧q ) = q p + 1 (m∧m) (38) i 2 i 5 i 2 i Below we have indicated the Matrix-theory in- Fundamental NS strings can also manifest them- terpretation the total momentum flux Pi and selves as string-like objects within the 5-brane 7Here we assume that all strings and branes are at the world volume. Inside a bound state with N NS 5- same location in the non-compact space-time. See also branes, each NS string can in fact break up into previous footnote. 12
micro-string winding number Wi on the auxiliary different (2,0) tensor multiplets. When they come 5 T , as obtained via the T5 ST5 duality chain: close together, however, new types of interactions occur. A precise low energy description of this in- teracting theory is unfortunately not yet available Pi NS1 KK at this point, although some new insights into its T 5 S 8 = −→ −→ structure have been obtained. Wi D4 D1 Particularly useful information can be ex- tracted by considering the IIA NS 5-branes from 9 MonT5 IIA on T 5 IIB on T 5 the perspective of M-theory .Asfirstpointed out in [24], M2-branes can end on M5-branes b via open boundaries in the form of closed strings attached to the M5-brane worldvolume. These KK NS1 membrane boundaries are charged with respect S T 5 −→ −→ to the self-dual 3-form field strength T = dU NS1 KK on the worldbrane. If we consider an array of N M5-branes, we thus deduce that there are IIB on T 5 IIA on T˜5 (at least) two basic types of string-like objects on the 5+1-dimensional world volume: strings Hence non-zero valuesb of the charges Pi and Wi in with variable tension that come from membranes Matrix theory indicate the presence of longitudi- stretched between two different 5-branes, and sec- nally wrapped M2 and M5 branes. Finite energy ondly, strings with fixed tension that correspond M-theory configurations are therefore selected by to membranes stretched around the 11th direc- imposing level matching constraints Pi =0and tion and returning to the same 5-brane. The Wi = 0, which in the large N NS 5-brane sys- first type of strings are charged with respect tem (the last two columns) project onto the sec- to (the difference of) the 3-forms of the two 5- tor with vanishing KK momenta and NS wind- branes that are connected by the 2-brane. These ing number. The same level matching were in- charged strings are often referred to as tension- troduced already in section 3 from the condition less or self-dual strings. The second type of that the space-time supersymmetry algebra (3) strings, on the other hand, are neutral with re- be reproduced. spect to the self-dual 3-form. They represent the micro-strings, i.e. the fractionated fundamental 7. The Matrix M5-brane IIA strings trapped inside the worldvolume of the N NS fivebranes. The collection of all de- Let us now put the description of sections 2 grees of freedom together look like an (as yet un- and 3 of the M5-brane in a proper Matrix the- known) non-abelian generalization of the abelian oretic perspective. The correspondence will be (2,0) theory. most clear in the formulation of Matrix theory as Charge conservation tells us that the charged the large N limit of N IIA 5-branes. strings with non-trivial winding numbers around The fields parametrizing the low energy col- the T 5 are stable against breaking and unwind- lective modes of a single IIA 5-brane form an ing. For the neutral strings, on the other hand, N =(2,0) tensor multiplet identical to that used there appears to be no similar world-brane argu- in sections 2 and 3. For the compactification on ˜5 T , four of the five scalars represent the trans- 8An (incomplete) list of references include [24][23] [12] [13] verse oscillations of the 5-brane in non-compact [38][39]. directions, while one scalar field Y is extracted 9Here in fact we do not mean the M(atrix) theory per- from one of the RR fields and is indeed periodic. spective related via the T5 ST5 duality chain of section 3, but rather the M-theory whose S1 compactification di- For N IIA 5-branes, as long as they are far apart, rectly gives the IIA theory of the last column of Table 1 we can separate the worldvolume theory into N in section 4. 13 ment available that would demonstrate their dy- the more conventional low-energy formulation ob- namical stability. From the space-time perspec- tained from 11-dimensional supergravity [28]. tive, however, the micro-string winding number is In the following sections we will to obtain some identified with the total NS string winding num- more detailed dynamical information about the ber (times N), and therefore represents a genuine 5+1-dimensional micro-string theory. conserved charge. This conserved quantity is the vector central charge Wi discussed above. 8. SN Orbifolds and 2nd Quantized Strings Among the low energy degrees of freedom of the N IIA 5-brane bound state, we can isolate Before continuing our journey through the ma- an overall (2,0) tensor multiplet, that describes trix world of M-theory, it will be useful to present the collective center of mass motion of the 5- a mathematical identity that will be instrumental branes. With the help of this decoupled tensor in establishing a precise link between the present multiplet, we can include the configurations with formalism for M-theory and the standard geomet- non-vanishing D-brane charges by turning on the ric framework of perturbative string theory. three types of fluxes mij , pi and q5 as described in Consider the supersymmetric sigma model on section 2. The respective IIA and M-theory inter- the orbifold target space, defined by the symmet- pretation of these fluxes are as indicated in table ric product space 1insection4.InthiswayweseethatMatrix N N S M = M /SN . (42) theory indeed exactly reproduces the description in section 2. In addition, we find that the micro- Here M denotes some target manifold, and SN string degrees of freedom, that were first postu- denotes the symmetric group, the permutation lated on the basis of U-duality symmetry, have group of N elements. The two-dimensional world- now found a concrete origin. sheet is taken to be a cylinder parametrized by Based on this new Matrix perspective, we can coordinates (σ, τ )withσbetween 0 and 2π.The now obtain a much more detailed description of correspondence we will show is that this sigma the M5-brane theory, based on the following pro- model, in the limit N →∞,providesanexact posed Matrix definition of its world-volume the- light-cone gauge description of the Hilbert space ory: of second quantized string theory on the target space-time M × R1,1. The world-volume theory of the M5-brane at light- The SN symmetry is a gauge symmetry of the orbifold model, that permutes all N coordinate cone momentum p+ = N/R is identical to that of a bound state of N IIA NS 5-branes, subject to fields XI ∈ M,whereI=1,...,N labeles the N ten level matching constraints (14). The KK mo- various copies of M in the product M .There- menta, M2 and M5-brane wrapping numbers are fore we can define twisted sectors in which, if we 1 in this correspondence identified with respectively go around the space-like S of the world-sheet, the D4, D2 and D0-brane charges. the fields XI (σ) are multivalued, via
XI (σ +2π)=Xg(I)(σ) (43) In essence this identification amounts to a defini- where g denotes an element of the S permu- tion of the M5-brane as the collection of all point- N tation group. A schematic depiction of these like bound states of Matrix M-theory on T 5,in- twisted sectors is given in fig. 2 and correspond to cluding those with zero fivebrane wrapping num- configurations with strings with various lengths. ber. Via this proposed definition, all the prop- e The Hilbert space of this S orbifold field the- erties postulated in [4] have become manifest. N ory is thus decomposed into twisted sectors la- The physical motivation and interpretation of the beled by the conjugacy classes of the orbifold above proposal has been discussed already in sec- group S [29], tion 2. It would clearly be worthwhile to investi- N gate the correspondence between this characteri- N H(S M)= H{Nn}. (44) zation of the M5-brane world volume theory and partitionsM{Nn} 14
where10
M SN SN H = H⊗...⊗H . (49) Ntimes
The spaces| H(n{z) in the} above decomposition de- note the Zn invariant subsector of the space of states of a single string on R8 × S1 with wind- ing number n. We can represent this space via a sigma model of n coordinate fields XI (σ) ∈ M with the cyclic boundary condition
XI (σ +2π)=XI+1(σ), (50) for I ∈ (1,...,n). We can glue the n coordinate fields X(σ) together into one single field X(σ)de- fined on the interval 0 ≤ σ ≤ 2πn. Hence, relative S 1 to the string with winding number one, the oscil- Fig. 2: A twisted sector corresponds to a config- lators of the long string that generate H(n) have a uration with strings of various lengths. 1 fractional n moding. The group Zn is generated by the cyclic permutation Here we used that for the symmetric group, the ω : XI → XI+1 (51) conjugacy classes [g] are characterized by parti- tions {Nn} of N which via (50) corresponds to a translation σ → σ+2π.ThustheZn-invariant subspace consists of nNn = N, (45) those states for which the fractional left-moving n minus right-moving oscillator numbers combined X add up to an integer. where Nn denotes the multiplicity of the cyclic permutation (n)ofnelements in the decomposi- It is instructive to describe the implications of (i) tion of g this structure for the Virasoro generators L0 of the individual strings. The total L0 operator of N1 N2 Ns [g]=(1) (2) ...(s) . (46) the SN orbifold CFT in a twisted sector given by cyclic permutations of length n decomposes as In each twisted sector, one must further keep only i the states invariant under the centralizer sub- L(i) Ltot = 0 . (52) group Cg of g, which takes the form 0 n i i s X Nn (i) Cg = SNn × Zn . (47) Here L0 is the usual canonically normalized op- n=1 erator in terms of the single string coordinates Y X(σ),θ(σ) defined above. The meaning of the Here each factor S permutes the N cycles (n), Nn n above described Z projections is that it requires while each Z acts within one particular cycle (n). n n that the contribution from a single string sec- Corresponding to this factorisation of [g], we tor to the total world-sheet translation generator can decompose each twisted sector into the prod- tot tot L0 − L0 is integer-valued. From the individual uct over the subfactors (n)ofNn-fold symmet- (i) (i) ric tensor products of appropriate smaller Hilbert string perspective, this means that L0 − L0 is a multiple of ni. spaces H(n) 10Here the symmetrization is assumed to be compatible Nn H{Nn} = S H(n) (48) with the grading of H. In particular for pure odd states N N n>0 S corresponds to the exterior product ∧ . O 15
To recover the Fock space of the second- quantized type IIA string we now consider the following large N limit. We send N →∞and consider twisted sectors that typically consist of a finite number of cycles, with individual lengths ni that also tend to infinity in proportion with N. The finite ratio ni/N then represents the fraction Fig. 3: The splitting and joining of strings of the total p+ momentum (which we will nor- via a simple permutation. tot malize to p+ = 1) carried by the corresponding string11 supercharges are given by (in the normalization tot p+ =1) (i) ni p+ = . (53) N 1 N Qa = √ dσ θa, So, in the above terminology, only long strings N I I I=1 survive. The usual oscillation states of these X strings are generated in the orbifold CFT by cre- √ N Qa˙ = N dσ Ga˙ , ation modes α−k/N with k finite. Therefore, in I I=1 the large N limit only the very low-energy IR ex- I X citations of the SN -orbifold CFT correspond to with string states at finite mass levels. Ga˙ (z)=γi θa∂xi (56) The Zn projection discussed above in this limit I aa˙ I I effectively amounts to the usual uncompactified In the SN orbifold theory, the twisted sectors of (i) (i) level-matching conditions L0 − L0 =0forthe the orbifold corresponding to the possible multi- individual strings, since all single string states for string states are all superselection sectors. We which L0 − L0 6= 0 become infinitely massive at can introduce string interactions by means of large N. The total mass-shell condition reads a local perturbation of the orbifold CFT, that (here we put the string tension equal to α0 =1) generates the elementary joining and splitting of
tot tot strings. p− = NL0 (54) The mass-shell conditions of the individual 9. String Interactions via Twist Fields strings is recovered by defining the individual light-cone momenta via the usual relation Consider the operation that connects two dif- ferent sectors labeled by SN group elements that (i) (i) (i) are related by multiplication by a simple trans- p+ p− = L0 . (55) position. It is relatively easy to see that this sim- All strings therefore indeed have the same string ple transposition connects a state with, say, one tension. string represented by a cycle (n) decays into a In the case that M = R8,wecanchoosethe state with two strings represented by a permu- i a a˙ fields X , θ , θ to transform respectively in the tation that is a product of two cycles (n1)(n2) 8v vector, and 8s and 8c spinor representations with n1 + n2 = n,orviceversa.Sothenum- of the SO(8) R-symmetry group of transversal bers of incoming and outgoing strings differ by rotations. The orbifold model then becomes ex- one. Pictorially, what takes place is that the two actly equivalent to the free string limit of the un- coinciding coordinates, say X1 and X2 ∈ M,con- compactified type IIA string theory in the Green- nect or disconnect at the intersection point, and Schwarz light-cone formulation. The spacetime as illustrated in fig. 3. this indeed represents an
11 elementary string interaction. The fact that the length ni of the individual strings spec- ifies its light-cone momentum is familiar from the usual In the CFT this interaction is represented by a light-cone formulation of string theory. local operator, and thus according to this phys- 16 ical picture one may view the interacting string the left-moving and right-moving twist fields and theory as obtained via a perturbation of the SN - to sum over the pairs of I,J labeling the two pos- orbifold conformal field theory. In first order, this sible eigenvalues that can be permuted by the Z2 perturbation is described via a modification of the twist CFT action i i j j Vint = τ Σ ⊗ τ Σ . (62) IJ 2 I
d 8−d N sical SYM action. In other words, the classical (T × R ) /SN . (68) moduli space of zero action configurations of the When we combine this with the results described two-dimensional SYM model is an infinite dimen- in the previous two sections, we deduce that sional space, that looks exactly like the universal the super-Yang-Mills model in the strict IR limit moduli space of all two-dimensional Riemann sur- gives the same Hilbert space as the free light-cone faces! The suppression of string interactions in quantization of second quantized type IIA string the IR field theory therefore entirely arises due theory. to the quantum fluctuations around these classi- The conservation of string number and indi- cal field configurations. vidual string momenta will be violated if we turn From this Yang-Mills set-up, it is straightfor- on the interactions of the 1+1-dimensional SYM ward to derive the correct scaling dimension of theory. When we relax the strict IR limit, one the twist-fields (that create the branch cuts) by gradually needs to include configurations in which computing the one-loop fluctuation determinant the non-abelian symmetry gets restored in some around such a non-trivial zero action configura- small space-time region: if at some point in the tion. To start with, this computation only in- volves the eigenvalue fields xi , and thereby re- (σ, τ ) plane two eigenvalues xI and xJ coincide, I we enter a phase where an unbroken U(2) sym- duces to the standard CFT analysis. In the metry is restored. We should thus expect that present context, the orbifold CFT is automati- for non-zero λ, there will be a non-zero transition cally provided with a ultra-violet cut-off by means amplitude between states that are related by a the non-abelian SYM dynamics that takes over at simple transposition of these two eigenvalues. a short distance scale of the order 1/gYM .Inprin- We now claim that this joining and splitting ciple it should be possible to determine in this process is indeed first order in the coupling con- way the exact numerical coefficient that relates stant λ as defined in the SYM Lagrangian. In- the CFT perturbation parameter λ and the YM stead of deriving this directly in the strongly coupling. coupled SYM theory, we can analyze the effec- The result (69) for the conformal dimension of tive operator that produces such an interaction the interaction vertex Vint of course crucially de- pends on the number of Higgs scalar fields being in the IR conformal√ field theory. The identifica- 0 equal to 8. The fact that for this case V is tion λ ∼ gs α requires that this local interaction int indeed irrelevant in the IR limit provides the jus- vertex Vint must have scale dimension tification afterwards of the above naive derivation 3 3 (h, h)=( , ) (69) of the orbifold sigma model as describing the IR 2 2 phase of the U(N) SYM model. Another way of 8 under left- and right-scale transformations on the stating the same result is that the R /Z2 cannot two-dimensional world-sheet. The analysis of the be resolved to give a smooth Calabi-Yau space. previous section confirms that this is indeed the It is indeed interesting to try to repeat the case. same study for SYM models with less super- As a further confirmation of this physical pic- charges. Concretely, we could consider the di- ture, it is relevant to point out that there is no mensional reduction of the N =1 SYMmodel classical barrier against the occurrence of string in 5+1-dimensions, which will give us a matrix 19 string theory with d = 4 transverse Higgs fields. becomes a free field theory but instead reduces to 4 Via the above reasoning we then deduce that in a sigma model on the moduli space MN,k(Tˆ )of the strict IR limit in the S1 direction, the string kU(N) instantons on the four-torus Tˆ4.Since interactions are not turned off, since the interac- this instanton moduli space is hyperk¨ahler, the tion vertex Vint is now exactly marginal. We will corresponding supersymmetric sigma model is in- return to this case in the next section. deed conformally invariant. Finally we remark that, for SYM theories with In [12] it was proposed that this sigma-model even less supersymmetries, the naive reasoning could be used as a concrete way of representing appliedinthissectioninfactbreaksdown,since the interacting micro-string theory on the world- in this case the interaction vertex Vint becomes volume of the k 5-branes. Crucial for this in- a relevant perturbation. Hence in this case the terpretation is the fact that the moduli space of correct IR dynamics is no longer accurately rep- instanton configurations on T 4 with rank N,in- resented by a (small) deformation of the naive stanton number k, and fluxes m can be thought orbifold IR field theory. Instead one expects that of as a (hyperk¨ahler) deformation of the symmet- the twist field V obtains a non-zero expecta- n 4 1 int ric product space S T with n = Nk + 2m∧m. tion value, and the IR theory presumably takes Hence via the results of the previous sections, the 4 the form of some Ising type QFT. sigma model on MN,k(Tˆ ) is indeed expected to behave in many respects as a light-cone descrip- 11. Microstring Interactions tion of a 5+1-dimensional interacting string the- ory. A relatively concrete model of the micro-string The interpretation of this sigma model as de- theory can be obtained by applying the reason- scribing the micro-string dynamics on k five- ing of [13] to the 5+1 SYM formulation of Matrix branes becomes a bit more manifest via the so- theory. Namely, in this formulation we can de- called Mukai-Nahm duality transformation. This scribe longitudinal NS 5-branes (that extend in transformation provides an isometry the light-cone direction) by means of YM instan- M (Tˆ4)=M (T 4) (71) ton configurations [35]. Here the 5-brane num- N,k k,N ber simply coincides with the instanton number between the moduli space of kU(N)instantons k.13 Similar as in [13], we can then consider the on Tˆ4 to the moduli space of N instantons on a world-volume theory of these k longitudinal NS U(k) Yang-Mills theory on the dual four-torus T 4. 5-branes in the limit of zero string coupling. As This dual U(k) SYM theory can be thought of as also explained above, this corresponds to taking the world-volume theory of k fivebranes wrapped the limit where one of the compactification cir- around T 4,andtheNinstantons describe the cles becomes very large. The theory thus again micro-string degrees of freedom on this world- reduces to the IR limit a two-dimensional matrix volume. This dual language is most suitable for string theory of the form (66), but now with ma- discussing the decompactification limit of the M- trices XI that carry a non-zero topological charge theory torus, which leads to the description of k [35][18] uncompactified fivebranes as a non-linear sigma 4 model on Mk,N (R ). k =trX[iXjXkXl] (70) The fact that there exists a superconformal de- n 4 representing the instanton number on the trans- formation of the orbifold sigma model on S T is verse T 4. It can be seen that in this case, the IR in direct accordance with the results of the previ- limit of the matrix string theory (66) no longer ous section, since it is reasonable to suspect that the marginal operator that represents this defor- 13 This representation follows from the table in section 7, mation can be identified with an appropriate Z2 where the longitudinal M5 branes (via their IIA identifi- twist operator that generates the splitting and cation with D4-branes) get mapped by the T 5 duality to D1 strings, which manifests themselves as instantons in joining interactions among the microstrings. As the large N D5 SYM theory. the string in this case has 4 transverse dimen- 20 sions, the twist-field indeed has total dimension part that describes the constant zero-modes in 2. Below we would like to make this description the instanton background, satisfying more explicit. Z Y =0,γiZ ηβ =0. (74) To this end, it will be useful to consider the [i j] αβ i situation in Matrix theory, which describes the k Since Zi corresponds to a k-instanton configura- 5-branes in combination with a single IIA matrix tion we can use the index theorem to determine string. In fact, to simplify the following discus- that there are k such zero-modes, both for the sion, we will give this string the minimal p+ mo- Y and η field. In the Coulomb phase they de- mentum, that is we will represent it simply by scribe the low energy degrees of freedom that me- adding one row to the matrix coordinates Xi,so diate the interaction between the string and the that we are dealing with a U(N +1)matrixthe- 5-brane. In this phase, the Y -modes are mas- ory. Next we write sive and thus are naturally integrated out in tak- ing the IR limit on the S1. In the Higgs phase, Zi Yi Xi → (72) on the other hand, they become massless two- Y i xi dimensional fields, and represent the additional micro-string degrees of freedom that arise from Here the matrix entry x represents the coordi- i the IIA string trapped inside the k 5-branes. No- nate of a propagating type IIA string (with in- tice that in this transition from the Coulomb finitesimal light-cone momentum p ), and the + phase to the Higgs phase the single IIA string U(N)partZ describes the k 5-branes. Similarly, i indeed “fractionates” into k individual micro- we can write strings.
ζa ηa The quadratic part of the action for the Y de- θa → (73) η θa grees of freedom takes the form a 2 2 2 2 The idea is now to fix Zi to be some given in- S = d σ (DYi) + |xI −zI| |Yi| 4 stanton configuration on T plus a diagonal part Z zI 1N×N that describes the location of the 5-brane 2 α α α˙ α˙ in the uncompactified space, and to consider the + d σ ηa D+ηa + ηa D−ηa (75) propagation of the IIA string in its background. Z This set-up will be useful in fact for studying α I β˙ +ηa γ ˙ (x−z)I ) ηa two separate regimes, each of which are quite in- αβ teresting [39] [40]. First there is the micro-string Here we suppressed the summation over the index regime, which corresponds to the case where the labeling the k zero-modes. The discrete symme- extra string x is captured inside the k NS 5- tries of the instanton moduli space imply that the branes. In this case, the Y variable need to be model (75) possesses a Sk gauge symmetry that thought of as the extra degrees of freedom that acts by permuting these k coordinates. In [37] the extend the moduli space of kU(N)instantons presence of the fivebrane background was incor- to that of kU(N+ 1) instantons. Secondly, we porated in matrix theory by adding an appropri- can also consider the regime where the IIA string ate hypermultiplet to the Matrix action. In our described by the x-coordinate is far enough re- set-up, the role of these additional hypermulti- moved from the NS 5-branes so that it propa- plets is taken over by the off-diagonal fields Y,η. gates just freely on the background geometry of Let us explain the fermion chiralities as they the 5-brane solitons. This outside regime cor- appear in the Lagrangian (75). In the original responds to the Coulomb branch of the matrix matrix string action the SO(8) space-time chiral- string gauge-theory, while the micro-string regime ities are directly correlated with the world-sheet represents the Higgs branch. chirality, reflecting the type IIA string setup, For both regimes, it will in fact be useful to giving left-moving fermions θa and right-moving isolate from the Y degrees of freedom in (72) the fermions θa˙ .TheT4compactification breaks the 21
SO(8) into an internal SO(4)thatactsinside fifth scalar Y , and the 3 helicity states of the self- the four-torus times the transversal SO(4) rota- dual tensor field. These are indeed the fields that tions in space-time. The spinors decompose cor- parametrize the collective excitations of the NS- respondingly as 8s → (2+, 2+) ⊕ (2−, 2−)and brane soliton. 8c → (2+,2−)⊕(2−,2+). The instanton on This 5+1-dimensional string thus T 4 with positive charge k>0 allows only zero forms a rather direct generalization of the 9+1- modes of positive chirality, selecting the spinors dimensional Green-Schwarz string. Clearly, how- that transform as 2+ of the first SO(4) factor. ever, the linearized theory can not be the whole So, through this process the space-time chirality story, because it would not result in a Lorentz co- of the fermion zero modes gets correlated with variant description of the M5 world-brane dynam- the world-sheet handedness, giving k left-moving ics. Micro string theory is indeed essentially an α α˙ fields ηa and k right-moving fields ηa ,where interacting theory, without a weak coupling limit. α, α˙ =1,2 denote chiral space-time spinor indices Its interacting dynamics is obtained by including and a =1,2 label chiral world-volume spinors. the non-linear corrections to (75), obtained by The Higgs phase of the matrix string gauge the- properly taking into account the non-linear ge- 4 ory corresponds to the regime where xI − zI ' 0. ometry of the moduli space MN,k(T )ofU(N) The above lagrangian of the 4k string-coordinates k-instantons on the four-torus Tˆ4 . (Y,η) then represents a linearized description of As argued above, the sigmab model on 4 the micro-string motion. inside the fivebrane MN,k(T ) should be representable in terms of a worldvolume. The linearized world-sheet the- finite marginal deformation of the above model by ory has 4 left-moving and 4 right-moving super- means ofb a twist field. Concretely, the linearized charges orbifold model has four different twist fields
aα˙ ab˙ α a˙ α˙ ab˙ α˙ F = ∂Y ηb , F = ∂Y ηb (76) Vab = Va ⊗ V b (79) I I which correspond to the unbroken part of the where the left-moving factor Va is world-brane supersymmetry and satisfy i b Va = γabτiΣ (80) ˙ ˙ {Faα˙ , Fbβ} =2a˙bαβ L , (77) 0 and a similar expression holds for the right- b Here L0 is the left-moving world-sheet Hamilto- moving V a.HereΣ denotes the fermionic spin nian. field, defined via the operator product relation
The ground states of the micro-strings con- 1 α − 2 ˜ α tain the low-energy collective modes of the five- ηa (z)Σb(0)∼z abΣ (0) (81) brane. To make this explicit, we note that the with Σ˜ α(0) the dual spin field. Both spin fields left-moving ground state must form a multiplet have conformal dimension 1/4, while τ has di- of the left-moving fermionic zero-mode algebra i mension 3/4. α β αβ {ηa ,ηb }=ab (78) So, all the above operators describe possible marginal deformations, that at least in first or- which gives 2 left-moving bosonic ground states der conformal perturbation theory, preserve the |αi and 2 fermionic states |ai. By taking the ten- conformal invariance of the sigma model QFT. sor product with the right-moving vacua one ob- Theses 4 physical operators V are in one-to- tains in total 16 ground states, which represent ab one correspondence with the RR-like micro-string the massless tensor multiplet on the five-brane. ground states |ai⊗|bi,thatrepresenttheexcita- Specifically, we have states |αi⊗|β˙i that describe tions of T = dU and the scalar field Y . Rotation αβ˙ αβ˙ i four scalars X = σi X transforming a vector symmetry on the world-volume, however, selects of the SO(4) R-symmetry, while |αi⊗|ai and out the scalar combination α α˙ |ai⊗|α˙i describe the fermions ψa and ψa .Fi- ab nally, the RR-like states |abi decompose into the VY = Va ⊗ V b (82) 22
4 that describes the emission of a Y mode, as the space Mk,N (R ) allows for a number of defor- only good candidate for representing the string mation parameters in the form of the Higgs ex- interactions. pectation values of the U(k) SYM model. These Geometrically, the twist fields Vab represent the Higgs parameters modify the geometry of the in- 4 four independent blow-up modes of a R /Z2 orb- stanton moduli space, and indeed have the effect ifold singularity. The three symmetric combi- of softening most of its singularities. This is in nations deform the orbifold metric to a smooth direct accordance with the micro-string perspec- ALE space and the three parameters correspond tive, since for finite Higgs expectation values the to the different complex structures that one can charged self-dual strings always have a finite ten- pick to specify the blow-up. These deformations sion. lead to regular supersymmetric sigma models, but are not rotational invariant. The antisymmetric 12. Future Directions combination VY can be interpreted as turning on a B-field θ-angle dual to the vanishing two-cycle To end with, we would like to mention some future directions, where useful new insights may around the Z2 orbifold point. h The value of this θ-angle away from the pure orbifold CFT, which be obtained. determines the strength of the micro-string inter- Space-time Geometry. actions, is in principle fixed by the description We have seen that the matrix setup is quite of the matrix micro-string as the sigma-model on well adapted to extract all the effective degrees 4 MN,k(Tˆ ). of freedom that describe the type IIA fivebrane, The value of the θ-angle modulus can be de- and one might hope that the present context of termined by considering the instanton moduli matrix string theory opens up new ways of ex- space around the locus of point-like instantons. amining the interaction with their environment. This problem was considered in [39], where it In particular, at weak coupling one would like to is proposed that this naturally fixes its value at be able to recover the classical fivebrane geome- θ = π.14 A puzzling aspect of this proposal is try including its anti-symmetric tensor field. This that this special value of θ is actually known to geometry should arise in the CFT description of lead to a singular CFT [41]. This is problematic, the Coulomb phase of the matrix string theory, since it would make the sigma model description where the string x(σ) propagates at a finite dis- of the micro-string dynamics inconsistent, or at tance |xI − zI | > 0 from the fivebrane soliton. least incomplete. There could however be a good This description of the string and fivebrane is physical reason for the occurrence of this singu- closely related to the recently studied case of a D- lar CFT. In the linear sigma model description string probe in the background of a D-fivebrane of the Higgs branch, the singularity seems closely [36]. One can derive an effective action for the related to the presence of a Coulomb branch on x-part of the U(N)matrixbyintegratingoutthe the moduli space that describes the emission of off-diagonal components Y and η. In [36] this was strings from the fivebrane [39]. An alternative shown that the one-loop result for the transversal physical interpretation, however, is that the sin- background metric gularity of the CFT reflects the possible presence 2 k 2 of the tensionless charged strings. ds = 1+ (dx) . (83) |x−z|2 The latter interpretation in fact suggests that the singularity may possibly be resolved by in- is indeed the well-known geometry of the NS five- troducing a finite separation between the k five- brane [42]. branes. In the M-theory decompactification limit Most characteristic of the solitonic fivebrane is in particular, we know that the relevant moduli its magnetic charge with respect to the NS anti- symmetric tensor field B
14 Here we defined θ = 0 as the value at the free orbifold H = k (84) CFT. ZS3 23 where H = dB for a three sphere enclosing Besides the fact that it acquires a corresponding the brane. This result can be recovered anal- conformal scale dimension, this also means that ogously, via a direct adaptation of the analysis it provides the natural cut-off scale at which the of [37]. The transversal rotation group SO(4) ' short-distance singularities need to be regulated. SU(2)L × SU(2)R acts on the relative coordinate In this light, it would be very interesting to field (x − z)I via left and right multiplication. study in more detail the transition region be- This group action can be used to decompose the tween the Coulomb and Higgs phase of this two- αβ˙ I dimensional gauge theory, describing the captur- 2×2matrixγI (x−z) in terms of a radial scalar ϕ field r = e and group variables gL,gR as ing process of the micro-string inside the five- brane. The relevant space-time geometry in this I ϕ −1 γI (x − z) = e gLgR . (85) case should contain all the essential features of a −1 black hole horizon, which initially is located at The combination g = gLgR labels the SO(3) an- r = eϕ = 0. We expect that via the above CFT gular coordinate in the transversal 4-plane. In- I correspondence, and in particular by studying the serting the above expression for x into (75), the limits on its validity, useful new insights into the angles gL and gR can be absorbed into η via a reliability of semi-classical studies of black hole chiral rotation, producing a model of kSU(2) −1 physics can be obtained. fermions coupled to a gauge field A+ = gL ∂+gL, A = g−1∂ g . Via the standard chiral anomaly − R − R Micro-strings on K3 × T 2. argument, one derives that the one-loop effective Interesting new lessons can be learned from the action includes an SU(2) WZW-model for the beautiful geometric realization of heterotic string field g with level given by the five-brane number theory via M-theory compactified on K3. In this k. The background three-form field H in (84) is correspondence the perturbative heterotic string reproduced as the Wess-Zumino term of this ac- becomes a natural part of the M5-brane spec- tion. trum wrapped around the K3, which thereby pro- We thus recover the CFT description of the vides a far reaching generalisation of its world- type IIA string moving near the fivebrane [42]. sheet dynamics in terms of the above 5+1- This action consists of a level kSU(2) WZW dimensional micro-string theory. A matrix for- model combined with a Liouville scalar field ϕ mulation of the interactions between different I wrapped M5-branes may ultimately provide a de- S(x )=SWZW(g)+SL(ϕ). (86) scription of heterotic string dynamics, in which with non-perturbative duality is realized as a geomet- ric symmetry. 2 2 (2) SL(ϕ)= dσ |∂ϕ| + γR ϕ (87) As a concrete illustration, we consider the gen- Z eralization of the one-loop heterotic string par- The background charge γ of the scalar ϕ repre- tition function. In perturbative string theory, sents a space-time dilaton field that grows linearly this one-loop calculation proceeds by considering for φ →−∞, and is normalized such that the to- the CFT free energy on a world-sheet with the tal central charge is c =6. topology of T 2, i.e. the closed world-trajectory It is quite meaningful that the radial coordi- of a closed string. In the M-theory realization, nate r = eϕ is described in terms of a Liou- the string world-sheet expands to become the ville type field theory (87). The role of r is in- world-volume of the M5-brane, while the CFT deed very similar to that of the world-sheet met- get replaced by the micro-string theory. The one- ric in two-dimensional quantum gravity: since loop diagram thus takes the form of a M5-brane r represents the mass of the lightest fields that world-trajectory with the topology of K3 × T 2. have been integrated out, it also parametrizes the The direct analogue of the perturbative string short-distance world-sheet-length scale at which calculation thus involves the partition function the CFT description is expected to break down. second quantized micro-string theory on this 6- 24 dimensional manifold. The above partition function can indeed be For the restricted case where one considers only thought as the direct generalization of the genus the BPS contributions, this calculation can be one partition function of perturbative heterotic explicitly done. The BPS restriction essentially string BPS-states. The fact that Z now depends amounts to a projection on only the left-moving on three moduli instead of one reflects the fact micro-string modes, and thereby eliminates the that the worldsheet CFT is now replace by a effect of its interactions. The M5-brane BPS par- world-volume micro-string theory, which in par- tition sum can thus be computed by taking the ticular also has winding sectors that are sensitive exponent of the relevant one-loop free energy of to the size of the T 2. As a consequence, the parti- the micro-string tion function Z has the intriguing property that it is symmetric under duality transformations, that =exp micro−string (88) ZM5−brane F1−loop in particular interchange the shape and size pa- The above one-loop string partition function has rameters T and V. in fact been computed already in [44]. Let us The general group of dualities is best exhibited briefly describe the result, because it in fact de- by writing the three moduli into a 2 by 2 matrix splays some rather suggestive symmetries. TV This partition sum Z will depend on the mod- Ω= (92) uli that parametrize the shape and size of world- VU! 2 volume manifold K3 × T . It turns out that which takes the suggestive form of a period ma- there are three relevant moduli T, U and V,that trix of a genus 2 curve. The above partition func- 2 parametrize the shape and size of the T and a tion Z indeed turns out to be a modular form un- Wilson line expectation value. The expression for der the corresponding SO(3, 2; Z) modular group, the partition function (88) now takes the follow- and is in fact invariant under a SL(2, Z) sub- 15 ing form group. It is quite tempting to identify this 2πi(T+U+V) SL(2, Z) symmetry with the electro-magnetic du- Z(T,U,V)= e− × (89) ality symmetry that arises upon further compact- −c(4kl−m2) 2πi(kT+lU+mV) ification of this theory to four dimensions. A con- 1 − e crete suggestion that was made in [43] is that the (k,l,mY)>0 coefficients in the expansion (91) represent the where the coefficients c(n) are defined by the ex- degeneracy of (bosonic minus fermionic) dyonic 22,4 pansion of the K3 elliptic genus as BPS states with a given electric charge qe ∈ Γ 22,4 and magnetic charge qm ∈ Γ , via the identifi- 2 2πi(hτ+mz) χτ,z(K3) = c(4h − m )e (90) cation h 0,m Z ≥X∈ 1 2 1 2 d(qm,qe)=D(2qm,2qe,qe·qm) (93) and are non-zero for n ≥−1. The expression (89) In [43] it was shown that this formula survives can indeed be recognized as a second quantized a number of non-trivial tests, such as correspon- string partition function [29]. If we expand Z as dence with the perturbative heterotic string as a formal Fourier series expansion well as with the Bekenstein-Hawking black hole 2πi(kT+lU+mV) Z(T,U,V)= D(k, l, m)e− (91) entropy formula. If true, this result suggests that k,l,m the presented formulation of M-theory via the mi- X crostring theory and the M5-brane may in fact be the integer coefficients D(k, l, m)representtheto- extended to compactifications all the way down to tal (bosonic minus fermionic) number of states four dimensions. Clearly, further study of matrix with given total momentum, winding and U(1) theory compactifications is needed to confirm or charge, as specified by the three integers k, l, m. disprove this conjecture. 15Here (k, l, m) > 0meansthatk, l ≥ 0andm∈Z,m<0 for k = l =0. Acknowledgements 25
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