We consider states with large angular momentum to facilitate the study of the M-theory regime of the AdS/CFT correspondence. More precisely, we study the duality between M-theory on AdS and supersymmetric Chern–Simons-matter (ABJM) theory with gauge group U()U() and level , in the regime where is of order one and is large. In this regime the study of both sides of the duality is challenging: the lack of an explicit formulation of M-theory in AdS makes the gravity side difficult, while the CFT side is strongly coupled and the planar approximation is not applicable. In order to overcome these difficulties, we focus on states on the gravity side with large orbital angular momentum associated with a single plane of rotation in . We then identify the corresponding operators in the CFT, thereby establishing the AdS/CFT dictionary in this large angular momentum sector. We show that there are natural approximation schemes on both sides of the correspondence as a consequence of the presence of the small parameter . On the AdS side, the sector we focus on is well-approximated by the matrix model of M-theory – with matrices of size – defined on the maximally supersymmetric eleven-dimensional pp-wave background. The pp-wave approximation to M-theory in AdS is justified for , while loop corrections in the matrix model are suppressed compared to tree-level contributions for . On the CFT side, we study the ABJM theory defined on with large magnetic flux . Using a carefully chosen gauge, we find an expansion of the Born–Oppenheimer type which arises naturally for large in spite of the theory being strongly coupled. The energy spectra computed on the two sides agree at leading order. This provides a highly non-trivial test of the AdS/CFT correspondence including near-BPS observables associated with membrane degrees of freedom, therefore extending the validity of the AdS/CFT duality beyond the previously studied sectors corresponding to either BPS supergravity observables or the type IIA string regime.
DIAS-STP-13-09, KEK-TH-1640, OIQP-13-09
Membranes from monopole operators in ABJM theory:
large angular momentum and M-theoretic AdS/CFT
Stefano Kovacs aaa E-mail address: firstname.lastname@example.org , Yuki Sato bbb E-mail address: email@example.com and Hidehiko Shimada ccc E-mail address: firstname.lastname@example.org
Dublin Institute for Advanced Studies, Dublin, Ireland
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
Department of Particle and Nuclear Physics, Graduate University for Advanced Studies (SOKENDAI), Tsukuba, Ibaraki 305-0801, Japan
Okayama Institute for Quantum Physics, Okayama, Japan
Our understanding of non-perturbative aspects of string theory is still quite limited, although important progress has been made in recent years, thanks, in particular, to work on string dualities and D-branes. It is very important to consolidate and further this progress. M-theory [1, 2], a conjectured eleven-dimensional theory which arises as strong coupling limit of type IIA string theory, plays a crucial role in this area. Various general features of M-theory are understood – it does not contain a dimensionless coupling constant (the only parameter in the theory is the eleven-dimensional Planck length), it reduces to eleven-dimensional supergravity in the low-energy limit and it contains among its excitations M2- and M5-branes, for which a classical action is known. These classical properties have many non-trivial consequences and implications for non-perturbative string theory. However, a well established formulation of M-theory in terms of its fundamental degrees of freedom is still lacking. In order to fully exploit the power of M-theory and elucidate its role in establishing a truly non-perturbative picture of string theory, it is crucial to develop a better understanding of the microscopic formulation of the theory including a consistent framework for its quantisation. The best candidate for such a formulation is currently the matrix model of M-theory.
In this paper we present a proposal for the study of a sector of M-theory combining the matrix model approach with the AdS/CFT correspondence. We show how the AdS/CFT duality can be studied in a genuinely M-theoretic regime by focussing on a particular set of states characterised by a large orbital angular momentum. Taking advantage of the dual description of these states in terms of a CFT allows us to independently confirm the results of the matrix model analysis. In this way, we simultaneously check the validity of both the matrix model proposal and the AdS/CFT correspondence.
The matrix model of M-theory can be considered as a regularised version of the theory describing (super)membrane degrees of freedom [3, 4] 111More precise statements are the following: (i) for a given regularisation parameter (the size of the matrices), a sufficiently smooth configuration in the membrane theory, which in general describes multiple membranes, has a corresponding configuration in the matrix model; (ii) the classical action functionals for the configuration in the continuum membrane theory and that for the corresponding configuration in the discrete matrix model approximately match; (iii) the approximation becomes better, for a fixed configuration in the continuum theory, when the size of the matrices becomes larger, provided that the parameters of the discrete theory have the appropriate dependence on the regularisation parameter; this dependence defines the classical continuum limit. The above properties imply that the semi-classical approximation to the path integral of the matrix model includes contributions which are governed by a Boltzmann factor associated approximately with the action functional of the membrane theory. In this sense the matrix model contains (multi-)membranes. In order to have a better understanding of the relation between matrix model and membrane theory it is necessary to address questions such as “Does the matrix model contain other degrees of freedom such as M5-branes?” and “What should the quantum continuum limit be?” . In this approach the embedding coordinates of the membrane and their fermionic superpartners are replaced by matrices 222In the literature the size of the matrices in the matrix model is usually denoted by . Here we use the letter to avoid confusion with the parameter used in the context of the AdS/CFT correspondence. and the resulting theory describes a quantum mechanical system with a finite number of degrees of freedom. The size of the matrices plays the role of a regulator and the quantum theory of the (super)membrane is expected to arise in the limit. The same matrix model is found in type IIA string theory as describing the low-energy dynamics of a system of D-particles (D0-branes) [5, 6]. In this context the size of the matrices is associated with the number of D0-branes. In  it was conjectured that the limit of this supersymmetric matrix model capture the entire dynamics of M-theory.
A complete and satisfactory understanding of the large limit of the matrix model is still lacking and this represents a major obstacle in establishing it as a viable description of M-theory. Another unresolved issue concerns the emergence of the eleven-dimensional Lorentz symmetry [7, 8, 9, 10, 11]. No complete proof that a Lorentz invariant quantum theory arise in the large limit is known. In particular the construction of the matrix model is closely tied to the use of light-front quantisation and no manifestly Lorentz-invariant formulation is available.
In order to substantiate the matrix model proposal it is necessary to address the fundamental issue of identifying proper observables in M-theory and then understanding how to realise them in the matrix model itself. Moreover a concrete scheme for the calculation of such observables should be identified and this is rendered challenging in particular by the absence of a dimensionless coupling constant. The majority of the tests of the matrix model approach to M-theory involve either the low-energy supergravity approximation or compactification to type IIA string theory in ten dimensions. A comprehensive review can be found in . Without considering such limits it is difficult to decide whether any results obtained from the matrix model are correct, although strong constraints should come from consistency requirements associated with unitarity and Lorentz invariance.
In this paper we propose an approach which brings the AdS/CFT correspondence into the picture in order to overcome some of these limitations and make progress on these issues. More specifically the use of the AdS/CFT dictionary allows us to identify quantities which are dual to CFT observables as “good” observables in the matrix model. Moreover, being able to independently compute such observables on the two sides of the duality, we are able to justify the results of the M-theory calculations. We will carry out this programme in a sector containing M2-brane states in M-theory, without resorting to a limit in which eleven dimensional supergravity or type IIA string theory can be used.
The specific AdS/CFT duality that we focus on in this paper, which we refer to as the AdS/CFT correspondence hereafter, was proposed in [13, 14]. It relates M-theory in an AdS background to a Chern-Simons-matter gauge theory with supersymmetry. This theory, which we will refer to as the ABJM theory, has U()U() gauge group – with level and for the two factors – and was first constructed in , following previous work [15, 16, 17, 18, 19, 20, 21, 22, 23]. It describes the low-energy limit of the dynamics of coincident membranes in .
The AdS background arises as near-horizon geometry of such a stack of M2-branes. The action is generated by rotations acting simultaneously in the , and planes of in which the is embedded. We denote the angular momentum generators associated with rotations in these four planes – which can be chosen as basis for the Cartan subalgebra of the SO(8) isometry group of – by and respectively. The can be described as an fibration over , where the has constant radius and is generated at each point by . This is the which is identified as the M-theory circle [24, 14] and the quotient has the effect of dividing the circumference of this circle by . For with fixed the theory is compactified to ten dimensions and reduces to type IIA string theory in AdS . This limit has been extensively studied after the original proposal  and corresponds to the ’t Hooft limit in the CFT, where is large with fixed.
We are instead interested in studying a genuinely eleven-dimensional, M-theoretic, regime where is of order and is large.
One reason to study the M-theory regime of the AdS/CFT duality is that one hopes to learn about M-theory in this way, as already discussed above. In particular, since the ABJM theory is conjectured to describe the low-energy dynamics of M2-branes, it is natural to ask whether there is a direct connection between this theory and the matrix model. One of the main results in this paper is to establish a natural and very direct connection between a certain sector of the ABJM theory and the pp-wave matrix model first formulated in .
Another motivation for our work comes from the possibility of gaining new insights into fundamental aspects of the AdS/CFT correspondence by studying it in a regime which is essentially different from what has been considered before. Although the AdS/CFT duality has been extensively studied, especially in its canonical version relating the supersymmetric Yang–Mills (SYM) theory in four dimensions to type IIB string theory in an AdS background, important open questions remain concerning its foundations. In particular the fundamental mechanism underlying the correspondence is not fully understood. Analysing a non-stringy AdS/CFT, of which the M-theoretic regime of the AdS/CFT duality is a prime example, should help to shed light on this aspect, as certainly in this case the correspondence cannot be explained in terms of open/closed string duality. Another important feature of the regime we focus on is that it is not compatible with the use of the planar approximation, since it requires large but , so that cannot be fixed. This is natural as the ’t Hooft expansion suggests that the gauge theory should have a description in terms of string-like degrees of freedom, which is not the case in the M-theory regime. Therefore the sector we consider allows us to analyse the gauge/gravity duality independently of the special role played by the planar approximation.
The duality in the M-theoretic regime is considered to be rather non-tractable. On the CFT side, the theory is strongly coupled as . Furthermore, one cannot focus on the planar diagrams and all non-planar contributions are in principle relevant. On the AdS side, one has to face the problem of formulating M-theory in AdS, in particular when trying to calculate observables including quantum corrections.
In this paper, we present evidence that when one introduces a large orbital angular momentum, , the presence of the small parameter makes it possible to identify good approximation schemes on both the CFT and the AdS sides. We discuss the relevant observables on both sides and establish a dictionary between them. The spectra computed on the two sides match, verifying the AdS/CFT conjecture in an M-theoretic regime.
The idea of using a large angular momentum to obtain a workable approximation is natural as the WKB approach is usually applicable in cases where one has large quantum numbers (in our case ). In the AdS/CFT context this idea has been put forward in [25, 26, 27]. As first shown by Berenstein, Maldacena and Nastase (BMN) in , focussing on a large angular momentum sector leads to a situation in which both sides of the duality are weakly coupled and the AdS/CFT correspondence is directly testable. Our work is in many ways analogous to the BMN analysis, although with some important differences. We construct operators in the ABJM theory, which play a role analogous to the BMN operators. The construction of such operators is, however, totally different and this reflects the fact that they correspond to excited states of membranes rather than strings.
On the gravity side of the correspondence we describe the physics of states in AdS which belong to a sector characterised by large angular momentum. M-theory states are classified by the eigenvalues of the Cartan generators and . We focus on states which have large and the other components of the angular momentum of order one. The dynamics of such states can be described using the maximally supersymmetric eleven dimensional pp-wave geometry to approximate the AdS background. Following the proposal to use the matrix model as a microscopic formulation of M-theory, it is then natural to adopt as framework for our calculations the pp-wave matrix model . An important aspect of our proposal is that the size of the matrices in this matrix model should be identified with .
The possible vacuum states in the large angular momentum sector are the BPS states of the pp-wave matrix model, which were studied in [25, 28, 29]. The simplest such state is a fuzzy sphere configuration corresponding to a spherical membrane which extends in the AdS directions and is point-like in , where it moves along a great circle with large angular momentum . In general the BPS states correspond to a collection of concentric fuzzy spheres, labelled by a set of integers corresponding to the portion of the total angular momentum carried by the individual membranes. The radii of the fuzzy spheres are proportional to their angular momentum. The use of the pp-wave approximation is justified if these radii are much smaller than the radius of curvature of the AdS and factors in the original background. This leads to the condition
for the applicability of the pp-wave approximation.
After describing the ground state in the large sector, we discuss the spectrum of fluctuations around the classical vacuum configurations following [28, 29, 30]. We present the tree level spectrum, which is determined by the pp-wave matrix model Hamiltonian at quadratic order in the fluctuations. We then discuss the behaviour of quantum corrections associated with cubic and quartic terms in the fluctuations 333 The spherical configurations discussed above can also be obtained as solutions to the equations of motion derived from the classical membrane action. However, we emphasise that the matrix model is a formulation at the quantum level and this is a definite advantage because it provides a framework to compute quantum corrections to the spectrum and to compare them with the dual CFT.. The condition that one-loop effects produce small corrections to the tree level result turns out to be
Having discussed the large angular momentum sector on the gravity side using the pp-wave matrix model, we then describe the dual large observables in the CFT. These are gauge-invariant operators in the ABJM theory with quantum numbers matching those of the membrane states we discussed. The requirement of gauge invariance leads to identify monopole operators as dual to membrane states in the large sector. Monopole operators , which play a crucial role in the ABJM theory and also in three dimensional gauge theory in general [32, 33, 14], are classified by a set of integers, the so-called GNO charges , which satisfy a Dirac quantisation condition [35, 36]. The BPS operators we consider in this paper are special cases, characterised by a large R-charge, of those already considered in  and further studied in [37, 38, 39, 40, 41]. We show, by focussing on BPS or ground states, that it is possible to identify the GNO charges of the relevant CFT operators with the angular momenta of the dual membrane states associated with motion along the great circle in . This correspondence was also observed in .
Monopole operators are associated with a Dirac monopole singularity at the insertion point. As such they do not have a simple manifestly local description in terms of the elementary fields in the theory. In order to deal with this complication it is convenient to use radial quantisation and study the ABJM theory on in Hamiltonian formulation in the presence of magnetic flux through the [32, 33]. Using the state-operator map we identify the states in the radially quantised ABJM theory – in a sector characterised by large magnetic flux, – which are dual to membrane excitations in the bulk. An important ingredient in this construction is the identification of a suitable gauge.
In this framework the dictionary relating the gravity and gauge sides arises in a natural way, leading to a very direct correspondence. Bulk states corresponding to spherical membranes and their excitations have a dual description in terms of states of the ABJM theory on . Therefore states on the two sides of the duality are described in terms of the same spherical harmonics. The energy spectrum of the membrane excitations, which are in general non-BPS, corresponds to the energy spectrum of the ABJM theory in radial quantisation.
In the case where the ground state on the gravity side is a single membrane, we verify that the tree-level spectrum obtained from the matrix model calculation agrees with the leading order result on the CFT side for all types of bosonic and fermionic excitations.
Despite the fact that the ABJM theory is strongly coupled for , we argue that a perturbative expansion is possible using a Born-Oppenheimer type approximation. The presence of a large magnetic flux, , induces a separation of energy scales which leads to a natural identification of slow (or low-energy) modes and fast (or high-energy) modes. Integrating out the fast modes leads to an effective low-energy Hamiltonian for the slow modes which is weakly coupled for large . We propose that this approach provides a framework for the systematic study of quantum corrections in the ABJM theory in the large sector that we defined.
In our construction leading to the formulation of the Born-Oppenheimer approximation for the large sector of the ABJM theory we will assume that it is possible to use the classical action as a starting point to identify the BPS states even for small . This assumption is partially justified by supersymmetry and by the consistency of the results of related work which uses localisation techniques  in combination with a similar premise. A full justification of this assumption will be provided a posteriori by the emergence of an expansion in which the effective coupling constant controlling quantum corrections is not the bare , but a combination involving inverse powers of . A more detailed discussion of these issues is presented in the sections devoted to the analysis of the CFT side.
We also discuss the generalisation to the case in which the ground state contains multiple membranes. The dual CFT sector involves monopole operators characterised by multiple non-zero GNO charges, corresponding to the angular momenta of the individual membranes. In this case the pp-wave matrix model vacuum consists of block-diagonal matrices [25, 28]. The excited states built on such vacua involve fluctuations in off-diagonal blocks, which do not correspond to degrees of freedom associated with individual membranes in the continuum. We will identify the dual states in the ABJM theory and show that in some cases – specifically when there are two membranes of approximately the same size and hence close to each other – these extra degrees of freedom on the two sides of the correspondence can be compared reliably and quantitatively within the limits of validity of our approximations. The agreement between the corresponding spectra is a strong indication that these states describe true degrees of freedom of M-theory, which are captured by the matrix model, but are not present in the conventional continuum membrane theory.
The AdS/CFT duality proposed in  has been extensively studied in the type IIA regime. Many of the techniques originally developed for the AdS/CFT correspondence have been adapted to this case. In particular integrability has been exploited in the ABJM theory following the early results in [42, 43, 44]. For a review see chapter IV.3  of . Also, large angular momentum operators (with vanishing total monopole charge) in the type IIA limit were first studied in [47, 43]. In the small regime, on the other hand, localisation techniques were successfully applied to the calculation of the superconformal index in . Similar methods have been used to obtain exact results for other BPS observables such as the free energy starting with the work of [48, 49, 50, 51]. Our analysis is also devoted to the small (M-theoretic) regime, however, we focus on non-BPS quantities. In the large sector described above, we develop an approach which makes it possible to systematically study quantum corrections to certain non-BPS observables on both sides of the correspondence.
This paper is organised as follows. In section 2 we describe the AdS side of the correspondence. We discuss the pp-wave approximation for membranes in AdS and present the associated matrix model and its energy spectrum. In section 3 we describe the CFT side. We first discuss the Hamiltonian formulation of the ABJM theory in . We then explain the separation between fast and slow modes in the framework of the Born-Oppenheimer approximation and present the energy spectrum in the large sector. Particular attention is devoted to the discussion of gauge-fixing which plays an essential role in our analysis. In the discussion of both sides of the duality we first consider BPS states (ground states) and then near-BPS states (fluctuations around the ground state), which are not protected and receive quantum corrections. In section 4 we discuss the case of multi-membrane vacua. We conclude in section 5 with a discussion of our results and an outline of possible extensions and generalisations.
2 AdS side
In this section, we describe the AdS side of the correspondence. We begin by recalling some basic formulae in M-theory and the AdS/CFT duality.
M-theory has only one length scale and the membrane tension, , is directly related to the eleven dimensional Planck length. We use the conventions of [13, 14] in which the Planck length is defined so that the Einstein-Hilbert part of the supergravity action reads
The relation between the membrane tension and the Planck length is then 
The AdS/CFT correspondence proposed in [13, 14] was constructed considering the near horizon limit of a stack of M2-branes in (which may be understood as a certain projection of M2-branes in flat space). The resulting geometry is AdS, where the radius of the , , in terms of the eleven-dimensional Planck length satisfies
while the radius of curvature of the AdS factor is
We shall now specify the kinematical regime we study in this paper. Corresponding to rotations in the and planes of in which the is embedded, there are four angular momentum quantum numbers, and . The states we focus on are those for which one of them, which conventionally we take to be , is large and the other angular momentum quantum numbers are of order .
Another important quantum number is . This is related to the momentum along the M-theory circle, which in the AdS background is identified with the great circle (or rather the family of great circles) corresponding to the orbit of the generator 444For points in the plane, the M-theory circle coincides with the equator generated by rotations. [14, 24]. The states we are interested in have , and . In most instances we will simply write to refer to either or . Since we focus on the leading order terms in a expansion, the difference will often be irrelevant. When the distinction between the two is relevant, we will explicitly specify whether we are referring to or .
In the following we first consider and then generalise to the case of , which is obtained via a certain projection. Since the quotient acts on the M-theory circle, the projection requires the quantum number of any individual object to be a multiple of (while of course can take any integer value).
The dynamics of objects (both point-like and extended, such as strings or membranes) propagating in a curved geometry with large spatial momentum can be described using an approximation scheme referred to as the pp-wave approximation [25, 26, 27]. This can be understood as an extension of the familiar infinite momentum frame argument (or the ultra-relativistic limit) in flat space to the case of a curved background. As is well known, the dynamics of an object having very large spatial momentum in flat space-time is approximately governed by a free non-relativistic Hamiltonian. If the background space-time is curved, the dynamics of objects with very large spatial momentum, proportional to a parameter , is instead approximately controlled by a non-relativistic Hamiltonian containing an external harmonic oscillator potential, whose strength is determined by the curvature radius and by .
The same Hamiltonian with a suitable identification of parameters also describes the dynamics of objects in a so-called pp-wave geometry. There is a limiting process, referred to as Penrose limit [53, 54, 55], which produces the pp-wave geometry starting from the original background. However, we stress that the point of view that we take in this paper is to treat the procedure as an approximation scheme to describe the dynamics of special states in the AdS/CFT correspondence. Rather than viewing the pp-wave background as arising from a formal limit between two geometries, we consider it as an approximation which allows to capture the dynamics of states with large spatial momentum propagating in the original space-time [26, 27].
Let us recall the essential points of the pp-wave approximation by using a simple example, a massless particle in the space-time with metric
where is the line element on the -dimensional unit sphere with . The dynamics of the particle is governed by the mass shell condition
We temporarily use the indices to label the coordinates of the space-time (2.5). We focus on a great circle in . We then assume that the particle has large momentum along this fixed circle and does not deviate far from it. Let the spatial coordinate be defined as the angle around the fixed large circle multiplied by the radius . The longitudinal momentum conjugate to is by assumption large. We choose the transverse coordinates , , in the directions orthogonal to the great circle. In terms of these coordinates the metric is approximately
Large longitudinal momentum therefore implies large energy . The finite difference, which plays a role analogous to the light-cone gauge Hamiltonian, is given by
where we used . Equation (2.9) shows that, for fixed (large) longitudinal momentum, , the dynamics of the particle in curved space is approximately that of a non-relativistic harmonic oscillator. Notice that the longitudinal momentum is actually quantised – , where is an integer – because its conjugate coordinate is periodic with period . Therefore in this approximation
Equation (2.7) is valid if . Classically one can assume a particle to remain arbitrarily close to the fixed great circle. However, in the quantum theory the wave function of the particle has finite extension. For the -th excited state, the extension can be estimated using (2.9) and ,
Hence we see that the condition , which validates the use of the pp-wave approximation, gives an upper bound on the excitation number,
and also implies
Another way of understanding the above formulae is in terms of a centrifugal potential. Because of the large angular momentum, the particle experiences a strong centrifugal force confining it around the equator (where the radius of the trajectory is the largest – the centrifugal force pushes objects in the direction where the radius becomes larger). The pp-wave approximation keeps the leading order term in this centrifugal potential, which as expected has the harmonic oscillator form. The strength of the potential is determined by the curvature radius of the background and the (angular) momentum.
The use of the pp-wave approximation in the context of the AdS/CFT correspondence involves an additional subtlety. In order to have a consistent dictionary between the gravity and CFT sides, it is necessary to change the space-time picture on the AdS side to the one given in [56, 57] which is particularly suited for studying holographic aspects (i.e. the computation of correlation functions following the prescription in [58, 59]). More specifically, one should not consider objects (particles, strings or membranes) propagating in the AdS space (with oscillating wave functions), but rather one should consider objects undergoing a tunnelling process (with exponentially decreasing or increasing wave functions). In practice this is achieved by a certain double Wick rotation. This prescription was proposed in  for the pp-wave approximation to string theory in AdS. It solves various puzzles regarding the signature of the bulk/boundary, including the identification of energy and conformal dimension and the signature of vector type fluctuations. Although the new interpretation is different leading to a better, consistent correspondence, the end result of the pp-wave approximation is mathematically equivalent . Both of the interpretations, with or without the double Wick rotation, lead to the same effective Hamiltonian in the pp-wave approximation. This is the case even for more general backgrounds corresponding to near horizon limits of D-brane configurations . The same interpretation has also been applied to the computation of correlation functions using methods derived from the study of integrable systems in [61, 62]. We shall not elaborate on this issue any further and we refer the reader to [56, 57] for additional details. In the following we assume that the identification of observables between the gravity and CFT sides of the correspondence is made adopting the prescription discussed in these papers.
Applying the above considerations to the study of M-theory in AdS, we conclude that the dynamics of states with large angular momentum in this background can be described using a suitable pp-wave approximation. Combining this idea with the matrix model proposal leads us naturally to use a matrix model which has the same form as the one arising in the maximally supersymmetric eleven-dimensional pp-wave geometry [63, 55]. This matrix model was first proposed in  and it was later derived in [28, 64] from the regularisation of the supermembrane theory in the pp-wave background. This matrix model is the main ingredient in our analysis of the gravity side of the AdS/CFT duality.
Our discussion in this section is based on a reinterpretation of previous results on the pp-wave matrix model [28, 29, 30]. In the spirit of using the pp-wave background as an approximation scheme to study a large angular momentum sector of M-theory in AdS, we will write the matrix model in terms of parameters characterising the original geometry, i.e. the radii and , and the angular momentum parameter 555In particular we do not introduce a mass parameter, , as commonly done in the literature. This introduction of is not necessary for the comparison between observables on the AdS side and the CFT side and moreover it makes the analysis of the limits of validity of the pp-wave approximation less transparent.. We first consider the membrane theory in AdS in the pp-wave approximation and then regularise it to obtain the matrix model. Rather than providing a detailed derivation of the membrane Hamiltonian starting from the supermembrane theory in AdS (analogous to that in [27, 56] for the type IIB string in the AdS background), we will justify its form based on the same arguments that led to (2.9). The physics of membranes in AdS can be captured by simply restricting the attention to those special states of the supermembrane theory in AdS  for which all individual membranes have a quantum number which is a multiple of .
The bosonic part of the membrane Hamiltonian 666We will often refer to the combination as the Hamiltonian on the AdS side of the correspondence. in the pp-wave approximation is
where is the membrane tension (2.2) and the nine transverse coordinates have been denoted by and , with , , indicating three scalars originating from AdS directions and , , referring to six scalars originating from directions. We also use to refer to the set of all nine transverse directions. In the following we will use the notation to collectively denote all the membrane coordinates when we do not need to distinguish between AdS and directions. The Lie bracket, , in (2.14) is defined as
for any functions, and , on the membrane world-volume. The constant is the total area of the base space,
It should not of course appear in observable quantities and we will see later that does not appear after the regularisation. is the momentum along the equator of the . It is related to the (integer-valued) quantum number by
where to be precise in the numerator should be understood as the value of . is a similar quantity associated with a “time-like” direction in AdS, which, by the conventional dictionary of the AdS/CFT duality, is related to the conformal dimension of the dual CFT operators by 777The identification becomes quite direct and transparent in the interpretation discussed in [56, 57].
The various terms in (2.14) can be understood as follows. The quadratic terms in the and coordinates come from the harmonic oscillator potential arising in the pp-wave approximation, analogous to the quadratic term appearing in (2.9). The cubic term for the ’s is induced by the coupling of the membrane to the three-form potential, which has non-zero background value in the AdS space. The remaining terms are those appearing in the membrane Hamiltonian in flat space in the ultra-relativistic limit 888Those familiar with the light-cone gauge formulation of the membrane theory might wonder whether we are working in the light-cone gauge or using the ultra-relativistic limit (also called the infinite momentum frame in the case of flat space-time). Arguably, it makes sense to distinguish the two points of view in flat space since the light-cone gauge gives exact results and it is applicable to generic states, whereas the ultra-relativistic limit is an approximation valid only for special states. However, this distinction is meaningless in the present case of a curved space-time in which we have to make an approximation – the pp-wave approximation – and consider special states with large angular momenta.. We have partially fixed the reparametrisation invariance of the membrane in a way analogous to that used in the light-cone gauge for membranes in flat space-time [4, 3]. The Hamiltonian (2.14) can be rewritten in the form of a sum of squares, which simplifies the study of the minima of the potential,
There is also the phase space constraint
associated with the residual gauge symmetry corresponding to the area preserving diffeomorphisms. The membrane Hamiltonian (2.14), (2.19) and the constraint (2.20) are mathematically equivalent to those of the membrane theory on the pp-wave background [28, 64] by appropriate rewriting of the parameters.
The matrix model which we will use in the following was obtained by regularising the Hamiltonian described in the previous paragraphs. An essential element of our proposal is that the proper matrix model regularisation, suitable to describe the large sector of the AdS/CFT duality, should use matrices of size .
One way to understand this identification is to notice that the D0-brane charge, which should be the matrix size , is equal to . This follows from the identification of the M-theory circle in the AdS/CFT correspondence with the orbits of the generator acting on the AdS space-time.
Another way to understand the identification of the matrix size with the angular momentum, which is based on the interpretation of the matrix model as regularised membrane theory, is the following. In our gauge fixing of the membrane theory, we choose the space-like coordinates on the world-volume so that the longitudinal momentum density is constant on a time-slice of the world-volume. This implies that the (base-space) area of a certain portion of the time-slice of the world-volume is proportional to the longitudinal momentum contained in that portion. The longitudinal momentum is approximately equal to the momentum along the M-theory circle to leading order in our approximation. Because of the periodicity of the angle along the M-theory circle, the associated momentum has a minimum, . This minimum of the momentum implies a minimum for the area in the time-slice of the world-volume of the membrane. The total area is proportional to the total momentum, , and the minimum of the area is proportional to with the same coefficient of proportionality. Hence the time-slice of the world-volume is divided into pieces. This is achieved by regularising the membrane world-space by matrices, as the matrix regularisation corresponds to dividing the membrane world-space into equal area pieces. The number of these pieces is equal to the matrix size . This can be understood using an analogy with the quantisation of a system with a single degree of freedom: the Bohr-Sommerfeld quantisation says that the minimum area of the phase space (the membrane world-space) is quantised in units of which is equal to the total area divided by the matrix size in the membrane context. Therefore the size of the matrices should be 999 A similar interpretation can be applied to the case of the BMN analysis of the AdS/CFT duality. In that case fixed-time slices of the string world-sheet are discretised to a lattice with sites, conforming with the construction of BMN operators on the CFT side.. The use of finite dimensional matrices in the presence of a compact longitudinal direction is reminiscent of the DLCQ argument presented in .
Let us recall some basic relations used in the matrix regularisation. A comprehensive review can be found in . In this paper we follow the conventions of . Functions on the membrane world-volume at fixed time, , , are replaced by matrices, , , where the map is linear. These matrices provide a discrete approximation to the corresponding functions. The basic operations on functions have counterparts on the associated matrices. This correspondence can be summarised as follows,
The symbol indicates that the two sides of these relations are equal up to higher order corrections in . The first equation simply states that the product of two functions becomes the multiplication (or more precisely one half the anti-commutator) of the corresponding matrices. The second equation relates the Lie bracket of two functions to the commutator of the associated matrices multiplied by a factor proportional to .
Following this procedure, one introduces the matrix version of the membrane coordinates,
The canonical conjugates of these matrices, , are related to the matrix version of the continuum momentum, , by
Using , the complete matrix model Hamiltonian takes the form
where we have also included the fermionic terms which were omitted in the membrane theory. Here are SO(9) gamma-matrices and . As in the case of the membrane Hamiltonian, the bosonic part of (2.26) can be rewritten as a sum of squares,
where we used .
The use of implies that the M-theory charge should be a multiple of for any state in the matrix model. Based on this observation, we propose that the matrix model describes physics in AdS, rather than AdS, in the pp-wave approximation.
The canonical (anti-)commutation relations are
where , , collectively denotes the matrices associated with all nine membrane coordinates, are SO(9) Majorana spinor indices and are matrix indices.
The phase space constraints are
where again the sum over runs from 1 to 9.
2.1 BPS states
The classical stable solutions of the pp-wave matrix model with zero energy are known [25, 28]. They are the BPS states (or the ground states) in the sector we are studying in this paper. They are given by a collection of so-called fuzzy spheres extending in the 3 transverse directions originating from AdS and are point-like in the directions.
From the form of the matrix model Hamiltonian (2.27), which is written as a sum of squares, it is clear that minimum energy configurations have for , so that the only non-vanishing fields in the classical solution are , . They should satisfy
This equation is solved by taking the ’s to be proportional to generators, , of a representation of SU(2). The explicit form of the solution is
where the ’s obey
The simplest solution corresponds to choosing the ’s to be the generators of the irreducible dimensional SU(2) representation. Taking the proportionality constant so that is written as
one finds for the parameter
where we used and . Equations (2.34) and (2.35) have then a simple geometric interpretation. A spherical membrane of unit radius, described by coordinates , , with , is approximated in the matrix model (with matrices of size ) by the configuration , referred to as a fuzzy sphere of unit radius [3, 4]. Therefore the solution (2.34) corresponds to a fuzzy sphere of radius given by (2.35).
where the -th block on the diagonal, , , is of size . It is given by
and are the generators of the irreducible SU(2) representation of dimension .
Block-diagonal configurations in the matrix model (for which the equations of motion factorize into those for the individual blocks) are interpreted as describing collections of classically independent objects. In the present case, (2.36)-(2.37) represent distinct fuzzy spheres. More precisely the block diagonal matrices (2.36) describe a collection of concentric fuzzy spheres of radii given in (2.38). They extend in the AdS directions and carry momentum along a great circle in .
The general solution minimising the matrix model Hamiltonian is therefore characterised by a set of integers, , , satisfying
The ’s must be multiples of , because the size of the -th block in (2.36), , , is necessarily an integer. This gives further support to our proposal that the matrix size should be , since the projection associated with the quotient implies that the angular momentum of each membrane in a multi-membrane configuration should be a multiple of .
In  it was shown that the states in the pp-wave matrix model can be organised into multiplets of the SU(24) supergroup and special states belonging to BPS multiplets were identified. The vacua we described in this section were shown to belong to multiplets termed doubly-atypical in . These multiplets have energies which are non-perturbatively protected. Therefore the degeneracy of the vacua corresponding to different numbers of spherical membranes is not lifted in the full quantum theory.
The theory contains distinct sectors associated with the vacuum configurations (2.36)-(2.38) and the fluctuations around them. It would be interesting to study the possibility of tunnelling connecting these sectors corresponding to different perturbative vacua 101010 The fact that the energies of the ground states are non-perturbatively protected  suggests that the tunnelling processes may be allowed only between excited states and not between pairs of ground states. Some properties of instanton solutions associated with tunnelling processes were studied in [28, 68, 69]. . Such an effect should be understood as corresponding to the interaction of membranes. For example, in a two-membrane vacuum, interactions can lead to a transfer of longitudinal momentum between the two membranes. This corresponds to a transition between an initial state characterised by two angular momenta, and , and a final state in which the angular momenta are and , with . Similarly it is possible to have tunnelling processes corresponding to the splitting or joining of membranes. For example a single membrane with angular momentum could split into two membranes with angular momenta and , with . Since the angular momenta are quantised (being integers and multiples of ), these transitions are not allowed in perturbation theory. We expect the effect of these tunnelling processes to be negligible compared to the leading order perturbative corrections to the spectrum which will be discussed in section 2.3.
Let us more closely examine the formula (2.38) for the radii of the minimal energy fuzzy spheres,
where in a vacuum with membranes. This shows that the size of the spherical membranes grows with their angular momentum, . However, for our analysis to be valid we should require that the membranes do not extend beyond the region in which the AdS background is well approximated by the pp-wave geometry. More precisely for the pp-wave approximation to be applicable we should require that the radii satisfy . Using (2.3) this amounts to . Combining this result with the requirement (2.13) that the ’s be large we obtain the condition
for the pp-wave approximation to be valid. In section 2.3 we will discuss how a stricter lower bound on arises if one requires that quantum corrections in the matrix model be small.
The pp-wave approximation we have discussed so far can be considered as keeping the leading order terms in an expansion in powers of
It should be possible to compute corrections to the pp-wave approximation and incorporate higher orders in this expansion into the matrix model.
As observed above, the various perturbative vacua are expected to be non-perturbatively connected through tunnelling processes. Therefore it may be more natural to require that the pp-wave approximation be applicable to all possible vacua and not just to a particular one corresponding to a given set of ’s. If we take this point of view, considering the perturbative vacuum consisting of a single membrane, it follows that the total should satisfy
This condition in turn implies a bound on the number, , of membranes. Since the individual ’s are integers and multiples of , the vacuum with the largest number of membranes with a given total corresponds to the case in which for all . Combining (2.43) and (2.39) for this vacuum we get
This condition is consistent with the fact that we are describing configurations of membranes in a fixed background, obtained as near-horizon limit of a black brane solution corresponding to coincident membranes, without including any back-reaction.
At first sight, requiring the validity of the pp-wave approximation for all possible perturbative vacua may appear to be incompatible with the lower bound in (2.41). Considering for simplicity , in the extreme case in which for all , the condition is not satisfied, implying that the vacuum fluctuations of the centre of mass of the membranes will invalidate the use of the pp-wave approximation, as explained in the general discussion around (2.13). However, this problem may be resolved if we use a dual description of this membrane configuration in terms of M5-branes, using the proposal in [25, 70]. According to these papers the vacuum corresponding to the partition , should be identified with a configuration of a single M5-brane. Since the angular momentum of the M5-brane is , (2.13) is satisfied from the M5-brane point of view. The size of this M5-brane is given by [25, 70]
which is satisfied automatically in our regime.
Similar considerations can be applied to other states containing multiple membranes with small angular momentum, which can be identified with configurations of M5-branes carrying large angular momentum, satisfying the conditions of applicability of the pp-wave approximation. For intermediate values of the angular momenta, more complicated configurations, such as the five-branes discussed in , may be relevant.
2.2 Near BPS fluctuations
We next consider the fluctuations around the ground states discussed in the previous section. The spectrum of such fluctuations for the pp-wave matrix model has been studied in detail in [28, 30]. We will present the results in terms of parameters, , and , which are suitable for the comparison with the ABJM theory to be discussed in section 3.
We focus on the single membrane vacuum, i.e. the case in which the minimal energy configuration corresponds to matrices of the form (2.32), where the ’s are the generators of the dimensional irreducible representation of SU(2). The case of multi-membrane vacua will be discussed in section 4.
In order to study the spectrum of excited states in the single membrane sector all the fields are expanded in terms of fluctuations around the classical solution . For the scalars, which are the only variables with a non-trivial background value, we denote the fluctuation by and write
Substituting into the matrix model Hamiltonian one obtains quadratic, cubic and quartic terms in the fluctuations,
The tree-level spectrum is determined by computing the eigensystem of the quadratic Hamiltonian, , which takes the form
This Hamiltonian is diagonalised by expanding the fluctuations, , and , in a basis of matrices, which consists of discretised versions of the spherical harmonics [3, 4]. This should be expected, since the matrix model is the regularised version of the continuum membrane theory. In the continuum the vacuum solution is a spherical membrane and the spherical harmonics are the natural basis to use to expand its fluctuations. The discretised versions of the spherical harmonics are referred to as matrix spherical harmonics. They are classified by a pair of quantum numbers, , where , and . The excited states in the matrix model spectrum are correspondingly labelled by integers .
For the scalars associated with directions, , , there are polarisations and the spectrum is
The upper bound on the quantum number reflects the effect of discretisation introduced by the matrix regularisation: matrix spherical harmonics constructed from the generators of the dimensional irreducible representation of SU(2) exist only with . Each level in (2.51) has a degeneracy , corresponding to the allowed values of the quantum number .
We note that the mass term and the contribution from the Laplacian ( and respectively under the square root in (2.51)) combine in such a way as to result in a rational energy spectrum. The same is true for the spectrum of the fluctuations and the fermions that we present below. This fact does not seem to have a simple explanation in the matrix model. However, we will see in section 3.2 that it has a simple interpretation on the CFT side.
The three scalars coming from AdS directions, , contain only two physical transverse polarisations. This is because of the presence of the constraint (2.30) associated with the residual gauge symmetry corresponding to area preserving diffeomorphisms 111111 Since the membranes are point-like in the directions, fluctuations of all ’s are transverse and there is no similar reduction of degrees of freedom. . Diagonalising the quadratic Hamiltonian in this sector yields energies
In order to study the fermionic fluctuations one first decomposes the SO(9) Majorana spinors according to the SO(3)SO(6) isometries of the pp-wave matrix model. Diagonalising the quadratic Hamiltonian yields two sets of states with energies respectively
The multiplicity of the corresponding states is for both sets, with the factor of coming from the fact that the fermions are spinors of the SO(6) isometry group associated with rotations in the transverse directions in .
The spectrum of the pp-wave matrix model is summarised in table 1.
2.3 Perturbation theory
Quantum corrections to the energy spectrum reviewed in the previous subsection are computed using standard quantum mechanics perturbation theory [28, 29, 30]. The majority of the fluctuations are non-BPS and therefore their spectrum will be corrected, but there are some BPS fluctuations whose spectrum is protected . The situation is reminiscent of the open string spectrum around giant gravitons in the pp-wave approximation . Leading order corrections for some of the states in the spectrum were computed in [28, 30].
The perturbation part of the Hamiltonian consists of cubic and quartic terms in the fluctuations around the classical solution. Expanding the Hamiltonian (2.27) one gets
The leading order correction to the energy of a generic state, , is computed using the familiar formula
Note in particular that the degeneracy of the un-perturbed states due to the SO(3) symmetry does not require the use of degenerate perturbation theory as the perturbed Hamiltonian still possesses the SO(3) symmetry. We will not present explicit perturbative calculations in the pp-wave matrix model. We will limit ourselves to recalling the relative weight of the perturbative corrections compared to the tree level result. This was studied in [28, 30] and we present here the result in terms of parameters which are more suitable in the AdS/CFT context for the comparison with the ABJM theory. The tree level energies summarised in table 1 are of order . The ratio of the one loop corrections (2.59) to the tree level result is of order [28, 29, 30]
The first term in (2.59) involves a sum over intermediate states and both terms contain sums over the and quantum numbers arising from the expansion in matrix spherical harmonics. Each summand is the matrix element between individual states. The matrix elements, estimated using the Hamiltonian (2.26), are of order , which is the same as (2.60). The fact that the matrix elements are small for large is expected since the strong centrifugal force for large suppresses the fluctuations (see (2.11)) making the interaction terms smaller than the quadratic terms. However, the sums in (2.59) can potentially produce factors of and alter (2.60). Hence the dependence of the loop corrections on is the result of the competition of two effects: as grows, each matrix element is suppressed, but at the same time the number of degrees of freedom increases. The explicit calculations in [28, 30] show that at leading order no extra factors of arise from the summations, thanks to non-trivial cancellations due to supersymmetry. This was proven in  for all states in the single membrane vacuum and it is natural to expect (2.60) to hold for the leading order corrections in all vacua. The absence of extra factors of in the perturbative expansion at leading order is related to the one loop finiteness of the membrane world-volume theory in the matrix regularisation, where the size of the matrices, , plays the role of UV cut-off. Further work is needed to establish whether similar cancellations persist at higher orders, so that (2.60) can be considered a genuine coupling constant for the pp-wave matrix model.
The ratio (2.60) can also be related to the ratio of the eleven dimensional Planck length to the size of the spherical membranes,
where is given in (2.40). This is natural, since in a theory of quantum gravity, such as M-theory, loop corrections should be suppressed when the relevant length scale is much larger than the Planck scale. Only when is sufficiently large such that , it is possible to distinguish the extended spherical membranes we are discussing from point-like gravitons.
We can also estimate the size of the fluctuations of the membrane coordinates around the stable fuzzy sphere. These fluctuations should be small compared to the radius of the sphere. The magnitude of the fluctuations of the membrane coordinates is of order , as can be deduced from the simple particle picture of the pp-wave approximation presented at the beginning of section 2. The ratio of this to the size of the spherical membrane, , is therefore