###### Abstract

We analyze the possibility of having a constant spatial NS-NS field, . Cosmologically, it will act as stiff matter, and there will be very tight constraints on the possible value of today. However, it will give a noncommutative structure with an associative star product of the type . This will be a fuzzy space with constant radius slices being fuzzy spheres. We find that gauge theory on such a space admits a noncommutative soliton with galilean dispersion relation, thus having speeds arbitrarily higher than c. This is the analogue of the Hashimoto-Itzhaki construction at constant , except that one has fluxless solutions of arbitrary mass. A holographic description supports this finding. We speculate thus that the presence of constant (yet very small) , even though otherwise virtually undetectable could still imply the existence of faster than light solitons of arbitrary mass (although possibly quantum-mechanically unstable). The spontaneous Lorentz violation given by is exactly the same one already implied by the FRW metric ansatz.

hep-th/0601182

BROWN-HET-1461

Constant H field, cosmology and faster than light solitons

Horatiu Nastase

Brown University

Providence, RI, 02912, USA

## 1 Introduction

A constant NS-NS B field has zero field strength, thus could maybe have been considered just a choice of gauge, except it was shown in detail in [1] that in a certain limit it makes the space noncommutative. Noncommutative geometry would imply certain spontaneous violation of Lorentz invariance, given that, for instance, is a constant tensor (making the direction 3 preferred). As such, there were limits put on the size of . A constraint based on a noncommutative version of QCD drives down to about [2], whereas a more conservative constraint based on just noncommutative QED gives it the maximal possible value of [3] (see also [4, 5] for related bounds). More interestingly, noncommutativity also predicts the existence of gauge theory solitons [6, 7, 8, 9, 10, 11], which were shown in a very interesting paper [12] to have galilean dispersion relations, thus moving at speeds arbitrarily higher than the speed of light, but only in the noncommutative directions (in fact, it was even shown in [13], that low momentum wavepackets can move faster than c due to UV/IR mixing). It is not clear what the phenomenological implications of this fact are, yet there is something disconcerting about such a strong breaking of rotational invariance (clearly choosing, be it even locally, a preferred direction in space). It certainly seems to imply that the B field cannot be constant over the whole Universe.

One can however also ask what happens if one chooses a constant H=dB field. This is a solution of the equations of motion in Freedman-Robertson-Walker (FRW) geometry, as we will see, but also has a more satisfying feature compared to a constant B field: one can choose a constant field with only 3+1d spatial indices, i.e. , which doesn’t break any more Lorentz invariance than the one already assumed in the FRW metric ansatz, i.e. a choice of time slicing, physically corresponding to the cosmic microwave background radiation (CMBR) reference frame. Indeed, there are only two types of Lorentz violations that are consistent with the FRW ansatz: choosing a nonzero constant antisymmetric tensor , or its 4d dual, a nonzero component of a vector. One can take two points of view about that: either to say that and are consistent with the FRW ansatz, or maybe even to take it, in the case of , as a seed for the FRW cosmology, thus as a possible explanation for our expanding 3+1 dimensional space. We will not explore this possibility here, but it is conceivable that if one has a constant tensor at the Big Bang, it could provide the initial condition for our FRW space. Among the fields of string theory, the NS-NS fields are universal, so they are model independent, and the only field that satisfies the previous requirement is . In this paper we will try to explore the consequences of a constant .

Constant H field will imply a varying B field, and one can ask whether one has also a noncommutative structure. A space-time varying noncommutativity was analyzed before, first for a time-dependent one in [14], shown to be consistent in [15], then for a space-dependent one in [16], and further studied in [17, 18] (see also [19]). We will use cosmology to impose constraints on the constant and find that it is small enough that the noncommutative structure will be of the type . The faster than light solitons on the fuzzy sphere were treated in [20], but that case, even if similar to ours, is different in that it deals still with 2d solitons, whereas we deal with 3d solitons. It would still be interesting to see if there are any connections though. After this paper was finished, we became aware of [21], where the same fuzzy space we considered was analyzed, and called .

We will find that behaves like stiff matter of negative energy density, and thus falls off on cosmological time scales as , thus it cannot have much impact on recent cosmology, but at most on the initial conditions. Yet given the example of [12], there could be still measurable phenomenological implications in the form of faster than light solitons. We will look for such solitons in the gauge theory on the space with and see if we get the same result from a holographic description.

The paper is organized as follows. In section 2 we analyze the possible noncommutative structure, using cosmology to set bounds on H and see the implications. In section 3 we ask whether can have any important effects on cosmology. In section 4 we build noncommutative solitons in the gauge theory on the space, paralleling the analysis in [12]. In section 5 we derive a holographic description of the gauge theory and check that our solitons can be described in it. In section 6 we conclude. The Appendix contains a review of the analysis in [12], for use in the paper.

## 2 Constant H field and FRW cosmology

In FRW cosmology, one always breaks spontaneously Lorentz invariance, i.e. there is a preferred cosmological time, thus a preferred cosmological frame (the rest frame of the CMBR). Thus one can obtain some superluminal motion just from the cosmological evolution (see for instance [22]). Of course, rotational and translational invariance is preserved (the FRW solution is in fact the unique homogeneous and isotropic cosmology, i.e. the unique cosmological solution that preserves rotational and translational invariance). But that means that in FRW cosmology one can have a constant field (zero component of a vector) or (123 component of an antisymmetric field) without further spontaneous breaking of Lorentz invariance (i.e. by the vacuum solution). But we know we generically have such a field.

Energy-momentum tensor

In string theory we always have a dilaton and a NS-NS B field, together with gravity, as the general bosonic closed string sector (NS-NS). It has the action

(2.1) |

which has as B equation of motion , and as Bianchi identity, which means that H=constant (and =constant) is a solution. In particular, we will be interested in the case =constant. Then the H field energy momentum tensor will be (we put )

(2.2) |

and with () we have

(2.3) |

where . If we consider instead the (4 dimensional) dual field () to be the relevant dynamical field, and =constant the relevant solution, with , then from

(2.4) |

we get now

(2.5) |

except now . Thus in both cases , giving positive energy density and pressure in the form of stiff matter (). We can try to construct a with only energy density or only pressure.

If we add a negative cosmological constant , i.e.

(2.6) |

to the constant H field energy-momentum tensor, we get no energy and only pressure,

(2.7) |

while if we add a positive cosmological constant we get energy density with no pressure (dust matter):

(2.8) |

However, in both cases the equality cannot persist in time, as a cosmological constant stays constant in time, whereas we will see that drops quickly with time.

Before we go on, let us pause and try to understand the constant H-field
in string theory. At this point, is just an arbitrary
constant, and we didn’t make any assumptions about how we will get this.
We took a model independent approach, and assumed there will be such an
H-field, as this was allowed by the classical supergravity equations of
motion, and one can understand the as a limit of an H field on
a spatial 3-sphere, , when the
radius of the sphere becomes infinite. As such, this could provide the
initial conditions for our FRW cosmology, by having 3 spatial dimensions,
curled up in a sphere, expand due to the H flux. However, such a model
would depend on the existence of a consistent string theory model at
the Planck era, and there are possible problems with that^{*}^{*}*I
thank Aki Hashimoto for pointing these out to me. We will deal with
open strings in the following, which assumes the existence of Dp branes
with , such that our 3+1 dimensional world is embedded in them.
However, for such branes (in particular, for D3 branes) in the presence of
an H flux, there exists a baryon anomaly, which requires the addition of
stretched branes a la Hanany-Witten. So a particular consistent picture
involving our constant H field and Dp branes (such that our 3+1 dimensions
are embedded in them) will be difficult to find.
However, we will go on in this model independent way, assuming that
at least nowadays this will not be an important problem, since as we will
obtain, needs to be extremely small.

Open string variables and noncommutative geometry

Let us now see what kind of cosmology will open strings (on an assumed Dp brane that contains our 3+1 dimensions) feel in the H field backgroud. Consider a constant H field with spatial components only, i.e. = constant, living in flat Minkowski space. Then we can find a gauge where (rescaling constants to 1) .

One can define open string variables and in the usual way (following Seiberg and Witten [1], these will be the variables felt by open strings moving in the above background) by (but we put for the moment ), giving

(2.9) |

Inverting to one finds that the open string variables are (the closed string variables were )

(2.10) |

Of course, as shown in [1], these variables are the ones appearing in the open string n-point functions, but in order to have just a noncommutative geometry structure, one needs to decouple the corrections from the effective action. This was done in [1] (for the constant B field case) by taking the limit , (closed string metric in the noncommutative directions), and keeping (open string metric) and fixed. Then one obtains, for instance that in the limit, (matrix inverse).

But first of all, notice that a 3 dimensional antisymmetric matrix has no inverse, as one can explicitly check (unlike, say, a 2 dimensional or 4 dimensional antisymmetric matrix). So clearly something special happens when 3 coordinates are involved. We can gain a better understanding of this fact by looking at the case studied in [16], where only was nonzero. Specifically, there one had the metric and B field

(2.11) |

and after going to open string variables by one obtains

(2.12) |

One can check that the limit is exactly the Seiberg-Witten limit for decoupling of corrections (the closed string metric in the 2,3 directions goes like , whereas B, G, are fixed), however the open string variables are actually independent of ! So we don’t need to take to obtain the correct G and !

Moreover, we immediately recognize that the rotationally invariant form of this case is just the case we were analyzing in (2.9), except taken in reverse (exchange for ). Specifically, the closed string metric and B field

(2.13) |

gives the open string variables

(2.14) |

again independent of the value of ! Thus we can say that gives noncommutative geometry with these variables, however we don’t need to take such a limit to obtain the same open string variables. In particular, we can take the case of for the whole Universe, in which case both the open and the closed string variables have approximately flat metric and constant and . That means that we choose to neglect the difference between the open and closed string metrics (or rather, we will analyze the case when this is true).

The independence of the open string variables we take it to mean that corrections will still respect the noncommutative structure. In other words, if we work at energies below the string scale, the corrections should decouple even though we don’t have the Seiberg-Witten scaling of the variables. This assumption of course deserves further tests, but we will consider it plausible at this point and consider that at energies much smaller than we will have noncommutative geometry.

But what kind of noncommutativity is the one we found? In [23] it was argued that constant H field means nonassociativity.

However, one can check that in (2.14) (as well as the one in (2.9)) satisfies the associativity condition found in [23],

(2.15) |

meaning that the star product defined with it is still associative! How can this be? After all, the argument in [23] is very simple: the left hand side of the associativity condition (2.15) can be rewritten as

(2.16) |

and in the Seiberg-Witten limit , and . Of course, as [23] noticed, this is only valid if is invertible, but that is not true by definition for a 3 dimensional antisymmetric matrix, as we noticed. Moreover, we chose a spherically symmetric gauge for B, such that . But we can easily check that for any gauge we choose (for instance the one in [16], with only nonzero, if is constant, the associativity condition will be satisfied!

However, is still nonzero, meaning that the various star products can be different. Indeed, for constant , one trades the noncommutativity for the usual star product of functions,

(2.17) |

and if one extends this to the varying case, it matters whether we take or in the exponent (or a different combination). A well defined prescription for a star product was given by Kontsevich [24] and up to second order in it is

(2.18) | |||||

and in fact the different between all possible star products is given by terms.

Still, when acting on spherically symmetric functions , also gives zero, meaning that the star product on spherically symmetric functions is still unique!

Open string cosmology

For completeness, let’s further consider what happens when we go to open string metrics.

The metric in (2.10) can be rewritten as

(2.19) |

that is, conformal to the open FRW model. Indeed, the FRW metric is

(2.20) |

where is closed 3d Universe, is critical (flat), and is open. Other forms of for the open Universe are

(2.21) |

none of which are manifestly homogenous. A space conformal to the homogenous space like the one we have is however not homogenous (translationally invariant)! The transformation of coordinates is and it gives ()

(2.22) |

One could also analyze the constant spatial H field in the general homogenous Universe (FRW) in variables, i.e. for closed string variables

(2.23) |

Then we get the open string variables

(2.24) |

In the calculation we used since we could generalize it to any a(r).

So why is the open string variables space not homogenous? In both cases, we start with a homogenous (translationally invariant) metric and ! But is not homogenous: a translation is equivalent to adding a constant B field! In any case, we were interested in having minimal variation between the open string and closed string metrics, so approximate homogeneity should still hold. But is this automatic?

Experimental constraints

Before deciding whether this fact contradicts experimental observations, we have to see what are the constraints that observational cosmology puts on the H field.

First, let us see what is the maximal possibility. If the energy density of a constant H field is of the order of the cosmological constant today (which is the maximum allowed value), then , thus .

But then the noncommutativity in (2.10) (in open string variables) is (with and dropping the indices, reintroducing dimensions)

(2.25) |

and if , then , and thus on the scale of the Universe, i.e. for , we have (the size of the Universe is about ; ). Then just on the scale of the Universe we get deviations from homogeneity (translation invariance), since the factor , that quantifies the deviations from homogeneity in both the metric and becomes equal to 1 just at the size of the Universe. That might still be important for cosmology, but we will examine it in detail in the next section, and the result will be that H is much smaller than the maximal value, thus the deviations from homogeneity are negligible, and for all intents and purposes one can take to be the only effect of the constant H on the open string variables, with the noncommutativity length scale.

## 3 Cosmological relevance of H field and experimental data

In this section we will study the cosmological implications of adding an energy-momentum tensor for the H field ( of the previous section) to the one for matter, radiation and cosmological constant.

We have seen in the previous section that the constant (or the constant 4 dimensional dual ) energy momentum tensor gives equal positive energy and pressure (stiff matter).

FRW models

The general FRW equations are

(3.26) | |||

(3.27) |

The two are related by the conservation equation (except for the integration constant )

(3.28) |

For k=0 (flat Universe), we have ()

(3.29) |

If the H field would dominate, we would have in the previous. A closed Universe () with H field dominating is not very plausible cosmologically (as it would contradict observation), so let us instead examine a closed Universe, with . For we have (assuming as usual an equation of state )

(3.30) |

As an example of open Universe, implies , thus

(3.31) |

By redefining t we can put C=0, and , thus

(3.32) |

H field as stiff matter

In reality, for constant we have , and thus in an H-field dominated Universe we would have , which seems hard to solve. But we approximate at late times

(3.33) |

and matching coefficients we get , thus

(3.34) |

meaning that .

At early times, we get

(3.35) |

For comparison, let us look at another case of open Universe, AdS space (), which has constant acceleration. We have

(3.36) |

thus in that case we have a maximum size of the Universe. For , at large times we get a rapidly dropping acceleration instead (like ):

(3.37) |

But let us now understand what happens when the H field is added to other types of matter (dust, radiation, cosmological constant). From the conservation equation it should be clear that we have for individual components

(3.38) |

and thus H behaves like stiff matter, with . These behaviours are independent of what type of matter dominates, i.e. of what actually is. Remember that if the component with dominates, then

(3.39) |

thus for matter domination we have , for radiation domination we have , for domination we have , for curvature domination we have (open Universe at late times).

Observe that

(3.40) |

Thus indeed constant is compatible with the equations of motion, as it generates stiff matter, that in turn implies constant.

Also note that for a constant H field in a matter dominated flat Universe (at this moment, we go from matter dominated to acceleration - ?- dominated cosmology), we can solve the FRW equations and find that

(3.41) |

as expected.

Experimental constraints

Is it possible to have a time-dependent ? The equation of motion would be and the Bianchi identity . Clearly constant satisfies both. But if we put and the only one nonzero, the equation of motion is still satisfied (it reduces to , which is true), but the Bianchi identity is not true anymore: it would imply . One could try and see if that works, but it doesn’t. Indeed, that would imply

(3.42) |

which is not true!

In conclusion, the only thing that works is that the H field is constant, and then it behaves like stiff matter , thus if it is to be comparable to the cosmological energy density now, it should have been dominant before (as it decreases as , much faster than matter and radiation). That cannot be true, as it would violate all the established cosmology.

Instead, one has to have of the order of the overall at the Planck scale or the string scale ( not to disturb usual cosmology and from naturalness). Then, at the current time, the energy density of the H field will be completely negligible, thus will most likely be irrelevant for cosmology.

But that is not entirely excluded from interesting experimental consequences. It cannot drive the current acceleration, for that we actually need , since drops quickly, and it cannot be a significant contribution to , since it would have dominated in the past. But it could give noncommutativity at a cosmological scale, as we saw, without affecting cosmology:

If the string scale is the lowest possible, i.e. if , in order to have at the size of the Universe, , one needs , thus now. As it drops as , it would have been of order at , i.e. 81 e-foldings ago. More relevant maybe, it would have been equal to at , i.e. 58 e-foldings ago, thus it could be of just about the right order of magnitude to be present, but not have contributed to cosmology, as about 50-60 e-foldings are necessary for inflation anyway.

Also, then the only scale relevant for noncommutativity (the length scale in is

(3.43) |

## 4 Noncommutative faster than light soliton solutions

In [12] it was shown that in the case of constant noncommutativity (coming from constant B field), there exist noncommutative gauge field string-like solitons (defined in 2 spatial dimensions, with trivial extension in the third), which can have arbitrarily high velocities (larger than c), because of the spontaneous breaking of Lorentz invariance by the choice of , say. The arbitrary velocity is then in the (2,3) plane, and this seems to provide an explicit breaking of rotational invariance (not just Lorentz invariance), which seems hard to understand given the experimental observation of perfect rotational invariance of cosmology. The details of the construction are given in the Appendix. We try to parallel that construction here, in order to find faster than light solitons in the rotationally symmetric case, that doesn’t break further Lorentz invariance othen than the one broken by the FRW cosmology.

Representing the algebra and finding the action

We want to work on the 3d space with noncommutativity , (we put for the moment the scale to 1) i.e.

(4.44) |

in other words, on the SU(2) space, or space of quantum angular momenta. As is well known, with one can rewrite the algebra as

(4.45) |

and then one can define states that are eigenvalues of and , which commute. Then , thus by definition can be thought of as the radius in spherical coordinates. We see then that we must think of acting on the space of all possible representations of SU(2), i.e. arbitrary j. Thus the space is like the fuzzy sphere, except the radius (representation of SU(2)) is a radial coordinate of space.

It is straightforward to show that the algebra can be represented in terms of commuting coordinates and their derivatives as

(4.46) |

for instance

(4.47) |

Now we would like to define also derivatives on the noncommutative space, by putting (no summation). But that is not enough: We need to satisfy consistency conditions, derived from applying derivatives to the algebra and using the Jacobi identities

(4.48) |

Then we obtain, for instance,

(4.49) |

We first want to take care of the definition of derivatives, i.e. . We note the action of on a few combinations:

(4.50) |

and find that

(4.51) |

and then we can represent the derivative as

(4.52) |

We can check that this satisfies not only , but also the consistency conditions. This leaves however open the value of , which as we can see in the Appendix is very important for the construction of solitonic solutions. We will thus decide on the correct representation of derivatives (and thus on ) when we build solutions.

Note that at both the coordinates and the derivatives are represented on a unit radius commuting 2-sphere (in terms of the phase space, i.e. coordinates and derivatives). We observe this by noting that scaling doesn’t affect and .

We further observe that in this representation, is actually , where for instance is the angle in the plane formed by the commuting cartesian coordinates on , , etc. Of course, there are only two independent angles, thus the three angles are related (hence the noncommutation relations). Also, at , the derivatives are just , thus .

The introduction of derivatives implies however also the need to extend the usual understanding of the representation. Let’s see this (in the case ) for the simplest operator, , and its conjugate . We have two representations, the representation, and the (“Fourier transformed”) unit radius commuting representation, in terms of angles. In terms of we have as usual

(4.53) |

which translates into the (spherical harmonic) representation of as

(4.54) |

and relies upon our abstract expression (4.46), written as

(4.55) |

But now going backwards, we defined on the derivatives as

(4.56) |

which means on spherical harmonics

(4.57) |

which can be obtained from

(4.58) |

which however is not a usual operator on states (which would give a complex number instead of the function ), but rather is of the type obtained for usual conjugate operators () in the x-space basis:

(4.59) |

Thus the introduction of derivatives implies an extension of the usual understanding of the basis into a -type basis. It becomes clear then why we need to specify the derivatives as well in order to define the representation space. As mentioned, we will do that when we build solutions.

Now we can define covariant derivatives and the YM action for the gauge field A in a similar manner to the constant noncommutativity case. Define first

(4.60) |

and then the field stregth of A is

(4.61) |

Integration is defined over a sphere by comparing the integral of , thus

(4.62) |

and thus over the volume of a sphere as

(4.63) |

Then the action in the gauge is (putting also the noncommutativity length scale, )

(4.64) |

Its static equations of motion are

(4.65) |

and the (Gauss law) constraint (coming from varying with respect to and then imposing the gauge condition) is

(4.66) |

(We can think of it roughly as ).

Building solutions

We will now try to systematically analyze possible solutions analogous to the ones in [12].

If we have an operator S satisfying as in the constant noncommutativity case (see the Appendix)

(4.67) |

then we can look for a solution of the type

(4.68) |

where are numbers, that can be identified with the positions of the solutions if behaves as (by an argument analogous to the one given in the Appendix for the constant noncommutativity case), and the equations of motion then reduce to (if )

(4.69) |

and we want to look for a solution with nonzero field strength

(4.70) |

such that one has nonzero magnetic flux. For a static solution, the Gauss constraint is automatically satisfied.

In a spherically symmetric situation, one would define the magnetic flux in the commutative case as

(4.71) |

where is the area element on the sphere, and the magnetic flux a doesn’t depend on the radius of the sphere chosen. We can check that this formula reproduces the charge of a monopole (). Before we generalize this to our noncommutative case, we have to understand the basis of states, . The identity operator is