Split noncommutativity and compactified brane solutions [1ex] in matrix models
UWThPh201120
Harold Steinacker^{1}^{1}1
Faculty of Physics, University of Vienna
Boltzmanngasse 5, A1090 Vienna (Austria)
Abstract
Solutions of the undeformed IKKT matrix model with structure are presented, where the noncommutativity relates the compact with the noncompact space. The extra dimensions are stabilized by angular momentum, and the scales of are generic moduli of the solutions. Explicit solutions are given for and . Infinite towers of KaluzaKlein modes may arise in some directions, along with an effective UV cutoff on the noncompact space. Deformations of these solutions carry NC gauge theory coupled to (emergent) gravity. Analogous solutions of the BFSS model are also given.
Contents
 1 Introduction
 2 Matrix models and their geometry
 3 Extra dimensions and split noncommutativity
 4 Higherdimensional compactification
 5 Effective gauge theory and KaluzaKlein modes
 6 Conclusion
1 Introduction
Matrix models such as the IKKT respectively IIB model [2] provide fascinating candidates for a quantum theory of fundamental interactions. Part of the appeal stems from the fact that geometry is not an input, but emerges on suitable solutions. For example, it is easy to see that flat noncommutative (NC) planes are solutions. More generally, one can consider geometric deformations of such brane solutions which correspond to embedded NC branes . Their effective geometry can be described easily in the semiclassical limit [3, 4], resulting in a dynamical effective metric which is strongly reminiscent of the open string metric [5]. Since the maximally supersymmetric matrix model is expected to provide a good quantum theory on 4dimensional branes, a physically interesting quantum theory of gravity should arise on such branes.
However, a single 4dimensional brane is clearly too simple to reproduce the rich spectrum of phenomena in nature. In order to recover e.g. the standard model, additional structure is needed. One possible origin of such additional structure are compactified extra dimensions, as considered in string theory. Another very interesting possibility are intersecting branes, which play an essential role in recent attempts to recover the standard model from string theory. At present, it appears that intersections of branes with compactified extra dimensions provide the most promising avenue towards realistic physics, cf. [6].
It is wellknown that compactified extra dimensions such as fuzzy spheres arise as solutions of matrix models with additional terms, such as quadratic of cubic terms [7, 8, 9, 11, 10]. This should allow in principle to obtain sufficiently rich solutions in order to recover the standard model. However, the addition of such extra terms spoils much of the appeal of the matrix model: the geometry of the spacetime branes is then strongly constrained, and much of the essential (super)symmetries is lost.
In the present paper, we show that there are indeed solutions of the undeformed IKKT model with geometry , where can be , , or and . For all these solutions, the Minkowski signature of the model is essential. Analogous solutions for the BFSS model are quite obvious and known to some extent; however, these solutions of the IKKT model appear to be new. Our constructions are inspired by the wellknown “supertube” or fuzzy cylinder solution [12, 13, 14, 15, 16] of the BFSS model [17, 18], carried over by a twisting procedure to the IKKT case. These solutions should provide sufficient structure towards physically realistic solutions of the matrix model, along the lines of [20].
All the solutions under consideration here have an interesting common feature: the noncommutative structure – which underlies all interesting matrix model solutions – does not respect the compact resp. noncompact spaces in , but connects them in an essential way. The structure is reminiscent of the canonical symplectic structure of cotangent bundles, where compact and noncompact coordinates are canonically conjugated. This is indicated by the name “split noncommutativity”.
There are many reasons why split noncommutativity is interesting. First, the solutions presented here only exist in the case of Minkowski signature. This is of course welcome from a physical point of view^{2}^{2}2This also suggests that the Euclidean model may not be suitable to understand the vacuum structure.. The underlying mechanism is that the compact extra dimensions are stabilized by (internal) angular momentum. Moreover, there are even solutions whose noncompact sector is in fact commutative and hence isotropic, along with an intrinsic UV cutoff. This would seem to resolve many of the problems associated with noncommutative field theory  the violation of Lorentz invariance, causality, etc.  which should be hidden in the compact sector.
However, things are not that simple. In the Minkowski case, the effective (open string) metric has a different causality structure^{3}^{3}3As discussed in [4], this change of causality structure might be avoided using complexified . A similar issue would also arise in string theory with timelike fluxes. The appropriate treatment of this issue is unclear. than the naive embedding (closed string) metric . Thus the completely isotropic solutions turn out to be nonpropagating with compact timelike curves, which is clearly undesirable. On the other hand, we do obtain physically meaningful solutions with standard Minkowski metric on the noncompact spacetime, at the expense of admitting some spacetime noncommutativity in the noncompact sector.
The geometrical degrees of freedom provided by the extra dimensions are also welcome for the effective (emergent) gravity on such branes. The higherdimensional compactification provides additional degrees of freedom associated to gravity due to the Poisson structure [21, 22, 23], and fewer embedding degrees of freedom. Moreover, the scale and to some extent the shape of the compact space are free moduli of the solutions, and not fixed by the model. We will discuss these gravitational aspects only briefly in this paper, and postpone a systematic analysis to future work.
2 Matrix models and their geometry
We briefly collect the essential ingredients of the matrix model framework and its effective geometry, referring to the recent review [4] for more details.
2.1 The IKKT matrix model
The starting point is given by a matrix model of YangMills type,
(2.1) 
where the indices run from to , and is the invariant tensor of . This is the bosonic sector of the the 10dimensional maximally supersymmetric IKKT or IIB model [2]. The “covariant coordinates” are Hermitian matrices, i.e. operators acting on a separable Hilbert space . The equations of motion take the following simple form
(2.2) 
for all . Indices of matrices will be raised or lowered with . We denote the commutator of two matrices as
(2.3) 
We focus on matrix configurations which describe embedded noncommutative (NC) branes. This means that the are quantized embedding functions
(2.4) 
of a dimensional submanifold, and
(2.5) 
is interpreted as quantized a Poisson structure on . Here denotes the semiclassical limit where commutators are replaced by Poisson brackets, and are locally independent coordinate functions chosen among the . Such a collection of matrices defines a quantized embedded Poisson manifolds , denoted as “matrix geometry”. We will assume that is nondegenerate, so that its inverse matrix defines a symplectic form on . The submanifold is equipped with a nontrivial induced metric
(2.6) 
via the pullback of . Finally, we define the following (effective) metric
(2.7) 
on , dropping possible conformal factors [3] which are not of interest here. It is not hard to see that the kinetic term for scalar fields on is governed by the effective metric . The same metric also governs nonAbelian gauge fields and fermions on (up to possible conformal factors) [3, 24], so that must be interpreted as gravitational metric. Since the embedding is dynamical, the model describes a dynamical theory of gravity, realized on dynamically determined submanifolds of .
2.2 The BFSS matrix model
Although our focus is on the IKKT model, it is very instructive to recall also the BFSS model, which was proposed as a nonperturbative definition of M(atrix) theory [17], cf. [18, 19]. It is a timedependent matrix model with 9 bosonic matrices , and appropriate fermions. Rather than discussing the action, we only write down here the bosonic equations of motion:
(2.8) 
dropping all dimensionful parameters.
3 Extra dimensions and split noncommutativity
The basic idea of this paper is to study NC brane configurations in the matrix model with geometry , where the noncommutative structure mixes the spacetime with the compact space . This means that the (nondegenerate) Poisson structure on satisfies , so that it contains terms of the form
(3.1) 
where are coordinates on and are coordinates on . This will be indicated by the name “split noncommutativity”. If and have the same dimension, then we may even impose , i.e. is isotropic. A standard example is the canonical symplectic structure on the cotangent bundle .
There are several reasons why split noncommutativity is interesting. First, if is an isotropic submanifold, then  as the name indicates  it does not carry any Poisson tensor field which could break Lorentz invariance. Indeed there are very strong bounds on Lorentz violation, and thus on possible Poisson background fields. Moreover, we will see that (maximally) split noncommutativity implies an effective UV cutoff on , due to the NC structure on ; this will be discussed in section 5. Another motivation is that such a structure will allow us to find solutions of YangMills matrix models with compact extra dimensions, without any additional terms in the action that would break some symmetry or introduce scale parameters^{4}^{4}4There are solutions of the IKKT model which can be interpreted as compactification on a torus [26]. However these are very different types of “stringy” 10dimensional solution involving infinite (winding) sectors, which do not fit into the framework of embedded branes under consideration here. The IIB model is divergent on such solutions at one loop, but it is expected to be finite on the present (lowerdimensional) solutions.. In particular the shape and scale parameters are free moduli, which means that these solutions should admit deformations with nontrivial effective 4D geometry. This should be important for (emergent) gravity, where some of the metric degrees of freedom come from the brane embedding. These aspects will be discussed briefly in sections 4.7 and 5.2.
3.1 Basic example: the fuzzy cylinder
A simple prototype of a space with split NC is given by the fuzzy cylinder [13, 25, 12] :
(3.2) 
Defining and , this can be stated more transparently as
(3.3) 
This algebra has the following irreducible representation^{5}^{5}5More general irreducible representations are obtained from this basic representation by a (trivial) constant shift .
(3.4) 
on a Hilbert space , where form an orthonormal basis. We take , since the are hermitian. Then the matrices can be interpreted geometrically as quantized embedding functions
(3.5) 
This defines the fuzzy cylinder . It is the quantization of with canonical Poisson bracket , i.e. locally.
Wavefunctions.
A basis of functions on is given by
(3.6) 
so that the most general function on i.e. matrix can be expanded as
(3.7) 
Note that the set of linear momenta is in fact compactified on a circle. This follows from , so that
(3.8) 
as operators on . This observation is very important: it means that there is an effective UV cutoff in the momentum space for . This is a consequence of the uncertainty relations combined with split noncommutativity: since the compact space has an IR cutoff , the noncompact space has a UV cutoff . This is physically very welcome, and in sharp contrast to noncompact noncommutative spaces such as the MoyalWeyl which has no intrinsic UV cutoff in spite of the uncertainty relations. On the other hand, there is no cutoff in the winding modes . A related observation has been made in [25]. In particular, the space of all functions in (3.7) (and the spectrum of the Laplacian (5.4)) has the characteristics of a onedimensional space, as in a 1dimensional QFT. This is also consistent with the relation , which has a 1dimensional volume divergence. The relevance of these observations to the noncommutative gauge theory on will be discussed in section 5.
Matrix Laplacian.
The matrix equations of motion (2.2), (2.8) are governed by the following matrix Laplace operator
(3.9) 
We note the following useful identity
(3.10)  
where . For the fuzzy cylinder algebra (3.3), this implies
(3.11) 
i.e.
(3.12) 
Thus is “harmonic” while are in some sense “massive”. Therefore the fuzzy cylinder is not a solution of either the IKKT or the BFSS matrix model. However, it is quite obvious how to build a corresponding solution for the BFSS model: the cylinder should be rotating.
3.2 Rotating cylinder solutions
BFSS solution.
Starting with a fuzzy cylinder as above with NC modulus and radius , define the following 3 timedependent matrices
(3.13) 
It is obvious using (3.12) that this gives a solution of the BFSS matrix model, which is wellknown [12]. In the semiclassical limit, this matrix geometry describes . This is a D2brane solution which is stabilized because the is rotating, extended along an arbitrary direction.
IKKT solution.
Next we want to find a corresponding solution of the more geometric IKKT model. One cannot apply the same trick directly, since there is no commutative time. However based on general arguments [2], there should be a corresponding solution.
Consider a fuzzy cylinder as above with NC modulus and radius . We can embed the noncompact direction along a lightlike direction . For example, consider the matrices defined as
(3.14) 
where is the timelike direction. This is indeed a solution of the IKKT model
(3.15) 
for any and . Note that this works only in the Minkowski case. The semiclassical limit is given by the geometry with Poisson structure . This defines the propagating fuzzy cylinder, which is propagating in a lightlike direction. The induced metric in the coordinates is , and the effective metric (2.7) is .
Several remarks are in order. First, note that the induced metric is degenerate. This means that vanishes identically, i.e. there is no “cosmological constant”. However this applies only to the 2dimensional case, and the higherdimensional generalizations below will have nondegenerate metrics.
Furthermore, note that we obtained a compactification without adding any other terms (such as cubic terms) and scales to the matrix model. Such additional terms would necessarily break the symmetry of the model, and strongly constrain the geometry of the noncompact space^{6}^{6}6unless the fluxes arise purely dynamically, perhaps through fermion condensation; however such a mechanism has not been established.. This would be in conflict with gravity. Accordingly, the radius of as well as are free moduli, and not determined by some explicit scale or potential in the action. This aspect will be discussed further in section 4.7.
This cylindrical solution of the IKKT model can be interpreted as a closed (D) string. However, it is quite different from the BFSS solution (3.13): the cylinder is propagating along a lightlike direction while the is essentially constant. We will obtain different types of solutions below.
3.3 Propagating plane wave solution
The following simple solution of the IKKT model describes a 2dimensional plane wave which propagates along a noncompact time direction. We first make a trivial but useful observation: If generate the quantum plane, then the two matrices (for fixed ) satisfy the relations of a fuzzy cylinder (3.3), with and NC modulus .
Now let generate the quantum plane , and define 4 hermitian matrices as follows:
(3.16) 
where is the timelike direction and . This is reminiscent of (3.14) except that and no longer commute. It is again easy to see that
(3.17) 
thus we obtained a solution of the IKKT model. This defines the propagating plane wave. The semiclassical limit is given by the geometry with a planewavelike embedding, and Poisson structure . The induced metric is given by
(3.18) 
which in lightcone coordinates is
(3.19) 
This is flat with Minkowski signature. The effective metric is then also flat and Minkowski, where the role of time and space is switched.
Although this solution has no compactified extra dimensions, we can use a similar construction to generate such solutions. We will start with some solution of the spacelike matrix equation , and turn it into a rotating solution of the IKKT model. This will be discussed next, focusing on the case of 4 noncompact directions.
4 Higherdimensional compactification
4.1 Stabilization by angular momentum
Assume we have a matrix geometry in the dimensional Euclidean matrix model which satisfies
(4.1) 
There are many explicit examples corresponding to quantized compact space , such as the fuzzy sphere (6.3), the fuzzy torus (6.1), fuzzy [27, 28, 29], and others. We would like to obtain a corresponding solution of the IKKT or BFSS matrix model, by giving angular momentum to .
In the timedependent BFSS model, this can be achieved simply by assembling the hermitian matrices into complex ones
(4.2) 
and giving them a timedependence as follows
(4.3) 
It is obvious that this solves the matrix equations of motion . Note that the rotation may or may not be a symmetry of . Since in the BFSS model, only can be rotated in this way. If we want to have solutions with the topology in the IKKT model, then only is admissible.
Now we describe two simple constructions which provide similar solutions of the IKKT model.
4.2 Twisting via a fuzzy cylinder
Lemma 1
Suppose , are hermitian matrices which satisfy
(4.4) 
Let be a fuzzy cylinder (3.3) with radius and NC modulus , which commutes with the above matrices. Collect the into complex matrices as
(4.5) 
and assume for . Then the 6 hermitian matrices defined via
(4.6) 
satisfy
(4.7)  
(4.8)  
(4.9)  
(4.10)  
(4.11) 

(4.7) and (4.8) are immediate using the variables. (4.9) can be seen using the identity (3.10), e.g.
(4.12) because commutes with and . The heuristic reason is that the twisting amounts to an orthogonal transformation, which leaves the matrix Laplacian invariant. The same computation with instead of , e.g.
(4.13) gives (4.11). Similarly, (4.10) follows from e.g.
(4.14) using (3.10) and (3.3), because commutes with the fuzzy cylinder and .
To understand the geometrical significance, assume that the original matrices describe a quantized embedding of a symplectic manifold . Let be the fuzzy cylinder. Then the above construction in the semiclassical limit amounts to a map
(4.15) 
where stands for the action on corresponding to (4.6), and
(4.16) 
is a quantized embedding map for . The corresponding Poisson structure on is the pushforward of the Poisson structures on and on via (4.15). One must distinguish two cases. First, if the action defines a flow on , then is odddimensional. The Poisson structure is thus degenerate, with symplectic leaves labeled by the eigenvalues of some central function. We will give an example below. Second, if the map (4.15) is free i.e. (at least locally) a diffeomorphism (hence does not preserve ), then the image becomes a symplectic manifold, with quantized embedding map given by (4.16).
Note that one can apply a transformation on the before defining the complex combinations (4.5); this will be exploited below. Similarly, the noncompact direction of the added cylinder may be oriented along an arbitrary direction, which maybe spacelike, timelike, or lightlike. Finally, some of the matrices are allowed to vanish . This will be useful below.
4.3 Twisting via a plane wave
Recall that if generate the quantum plane , then the two matrices (for fixed ) satisfy the relations of a fuzzy cylinder (3.3), with and NC modulus . We thus obtain the following analog of lemma 1:
Lemma 2

This follows easily from lemma 1 applied to the fuzzy cylinder algebras .
The geometrical significance of this construction is clear: describes a quantized embedding with nontrivial Poisson structure along some noncompact direction , and the modified embedding induces a rotation along this .
With these constructions at hand, we can obtain new solutions of the IKKT model with compact extra dimensions stabilized by angular momentum, analogous to (4.3). While adding a cylinder will lead to undesired closed timelike circles, adding the plane waves will give the desired compactifications which propagate along a noncompact direction.
4.4 Higherdimensional cylindrical solutions
BFSS solutions for .
As an example, we can take 3 mutually commuting copies of the fuzzy cylinder realized by for as defined above, and give the a timedependent factor with . Thus define 9 hermitian timedependent matrices as follows
(4.23) 
where etc. This clearly gives a solution of the BFSS matrix model of type , interpreted as 3 rotating cylinders. Note that the noncommutative structure is indeed split as discussed previously, and the subspace is commutative.
There are obviously many variations of this solution, such as , or with different winding numbers by replacing compensated by . It is also possible to replace one by a noncommutative torus (6.1), and obtain e.g. .
Nonpropagating IKKT solution for .
Now we want to construct similar solutions of the IKKT model, based on lemma 1. Consider again 3 mutually commuting fuzzy cylinders with NC modulus and radius , realized by for , embedded along spacelike directions:
(4.24) 
where etc. Now we add a “timelike” fuzzy cylinder with NC modulus , which commutes with the remaining generators. Thus define 10 new hermitian matrices as follows
(4.25) 
Now lemma 1 can be applied along with (3.11), which implies that
(4.26) 
Therefore we obtain a solution of the IKKT model for . Solutions with different winding numbers can be obtained by adjusting the accordingly. However, according to the discussion below lemma 1 there is a constraint. The central generator is easily identified as
(4.27) 
Therefore the symplectic leaves define D5branes^{7}^{7}7In accord with the string literature we denote  dimensional submanifolds with Minkowski signature as branes. with the structure , compactified along . In particular, the noncompact space is completely isotropic. However, such irreducible solutions can be obtained more directly:
Nonpropagating IKKT solution .
We modify the above construction in order to avoid the degenerate Poisson structure. Starting again with (4.24) and coinciding , we define
(4.28) 
(note that and have been interchanged). This amounts to an orthogonal transformation among the . Then clearly the new still define and satisfy and . Now let them rotate again by adding a timelike cylinder with NC modulus as follows
(4.29) 
As above, this is a solution provided . The point is that now the timelike rotation does not preserve , but sweeps out . Therefore the above matrices define a quantized , with classical coordinates and Poisson structure . In particular, the noncompact space is completely isotropic.
Now consider the induced metric, which in the coordinates is given by
(4.30) 
Therefore the effective metric (2.7) is
(4.31) 
This has indeed Minkowski signature, however the timelike direction is now in the compact space. This change of causality structure is a typical phenomenon in the context of emergent gravity (which should also occur for the open string metric in similar contexts). One possibility to avoid this is to consider complexified Poisson structures corresponding to complexified matrices, as discussed in [4]. However in the present paper, we insist that all are hermitian matrices, so that (4.31) must be taken serious. In that case, the timelike directions are compactified, and there is no propagation along the noncompact space . This will apply in particular for the lowest Kaluza–Klein modes. Therefore these solutions are interesting but unphysical, and we must look for solutions with Minkowski signature on the noncompact space. Such solutions will be found below, using a twist along a quantum plane.
There are obviously many variations of this solution. By letting some cylinders degenerate we obtain solutions for . In particular, the reduced form of (4.24) can be recovered in this way. We can also introduce different winding numbers provided the are adjusted accordingly. Finally, it is instructive to note that one can also use the semiclassical result [4] to see that (4.29) is a (semiclassical) solution of the model.
4.5 Propagating cylindrical IKKT solutions.
In order to obtain an effective metric which has Minkowski signature in the noncompact directions, we will use the construction in lemma 2 with Minkowski signature on .
Propagating and .
As a first example, we start with a fuzzy cylinder with NC parameter and radius , and twist it with the noncommutative plane wave (which commutes with the cylinder) as follows
(4.32) 
where . Then lemma 2 gives