###### Abstract

We give matrix and supergravity descriptions of type IIA F-strings polarizing into cylindrical D2 branes. When a RR four-form field strength is turned on in a supersymmetric fashion (with 4 supercharges) , a complete analysis of the solutions reveals the existence of a moduli space of F1 D2 polarizations (Caesar) for some fractional strengths of the perturbation, and of no polarization whatsoever (nihil) for all other strengths of the perturbation. This is a very intriguing phenomenon, whose physical implications we can only speculate about. In the matrix description of the polarization we use the Non-Abelian Born-Infeld action in an extreme regime, where the commutators of the fields are much larger than 1. The validity of the results we obtain, provides a direct confirmation of this action, although is does not confirm or disprove the symmetrized trace prescription.

hep-th/0111156

The polarization of F1 strings into D2 branes:

‘‘Aut Caesar aut nihil ^{1}^{1}1Latin for “either Caesar or nothing””

Iosif Bena

Department of Physics

University of California

Santa Barbara, CA 93106

## 1 Introduction

Recently, several papers have appeared describing states where strings and branes are polarized into higher dimensional branes, following the ideas in [1]. Usually the higher brane is supported against collapse by the presence of the field with which it couples. Nevertheless, only rarely is the back-reaction of this field on the other supergravity fields taken into account. This is because taking back-reaction into account does not considerably change the physics in the case of polarization into objects two or more dimensions higher. Nevertheless, when one studies the polarization of a string or brane into a brane one dimension higher (like F1 D2 or D4 NS5) the effect of this back-reaction on the physics is crucial.

The polarization of long fundamental strings into a tubular D2 brane in the presence of a four-form background was first studied in [2]. Ignoring back-reaction, the potential was found to be of the form

(1) |

where is the radius of the tube. This naive potential seems to imply that for any the potential is unbounded below at large , and the strings have a finite life time as they can tunnel into a D2 brane. Furthermore, if one expands the square root for ,

(2) |

one can also see that for this potential implies that the strings are not stable classically. Thus, a large number of coincident F1 strings could spontaneously disappear from the spectrum when a tiny is present. We got this puzzling result because the back-reaction of on the dilaton and graviton was not taken into account. This back-reaction gives a term in (2) of the form

(3) |

which usually prevents the decay of the strings. Finding the exact coefficient of this term is crucial if one wants to understand the physics of this system. A very large coefficient may prohibit polarization entirely, while a very small one leaves the puzzle about the strings disappearing at infinity unresolved.

There are several way to find this coefficient. The hard and straightforward way is to compute the back-reaction explicitly using the supergravity equations of motion. This has been done for other polarization cases in [3, 4]. A slicker approach is to use supersymmetry [5, 6, 7] and to find this coefficient by completing the square in the less naive version of (2). One can ascertain how much supersymmetry is preserved by this background either by doing a supergravity computation [8, 9] or by relating this background to a boundary field theory via an AdS-CFT type correspondence [10, 11]. Since the latter method only involves examining the supersymmetry of a two dimensional field theory, we will be using it here. Chapter 3 will be devoted to this subject.

Another aspect which makes the study of F1 D2 polarizations transparent is the existence of a matrix description for the resulting state. This is the subject of chapter 2. As shown in [1] and expanded in [5], Dp D(p+2) polarizations can be understood in terms of the matrices describing the position of the Dp branes becoming noncommutative. This description lacks however in other polarization cases.

At first glance, to give a matrix description to F1 D2 polarizations involves using matrix string
theory [12]. ^{2}^{2}2In a recent preprint a matrix string theory description of the D1/F1 string
polarizing into a D5 brane was atempted [13]
Nevertheless, a much simpler way to describe these polarizations is to add a small
number of D0 branes to the configuration, and to use the D0 brane noncommutative coordinates to describe it.
Our system can be recovered by taking the D0 brane density to zero ^{3}^{3}3We thank W. Taylor for this
suggestion..

This configuration of strings blowing up into a tubular D2 brane in the presence of D0 brane charge was considered recently by Mateos and Townsend [14]. In a following paper [15], a Matrix Model description for the supertubes was found. Nevertheless, this Matrix Model description does not work in the limit we are interested in (when the D0 contribution to the energy vanishes). To describe this limit one must use the full Non Abelian Born Infeld action of the D0 branes [16, 1, 17, 18]. We should note that in this limit the field strength is much larger than one.

It is interesting to ask whether the validity of the result we get (which agrees exactly with the supergravity result and the one in [14]) confirms the validity of the symmetrized trace prescription. The answer is probably negative, because our ’s are very simple, and their commutators vanish. Thus the computation is probably insensitive to the inability of the symmetrized trace prescription to produce the right terms when commutators of F’s are involved [19].

Anticipating, we will find that the polarization of F1 strings into a tubular brane is impossible except for values of proportional to fractions whose denominator is smaller than . For these values a number of strings polarizes into one or more tubular D2 branes. The radius of these polarization configuration is a modulus.

## 2 Matrix description

It is possible to describe F strings with D0 charge blown up into a tubular D2 brane using the D0 brane matrix degrees of freedom. This description is valid both with and without an flux turned on [20]. In order to give an ensemble of D0 branes a D2 charge, one has to take an ansatz where three of the D0 brane coordinates are not commuting. F1 charge can be given as well, by having the noncommuting coordinates be time dependent. This is expected both from the form of the nonabelian coupling with , which contains a time derivative of the coordinates, and also by noticing [14] that a tubular D2 brane with string and D0 charge has a nonzero angular momentum.

Let us assume to have D0 branes uniformly distributed along the direction. An ansatz for the matrices which gives the right charges is [20]

(4) |

In the infinite volume limit, the matrices and are infinite, and obey several useful identities:

(5) |

Note that we work in the normalization where the matrices are dimensionless. To obtain the physical quantities corresponding to dimensionless symbols (such as and ) one has to multiply by the appropriate powers of . Since is the separation between the individual D0 branes, is the size of the direction, which we take to be very large. The local D0 charge density is . For weak fields the F string charge density along the direction is given by [17]

(6) |

and is quantized. As we will see shortly, this quantization proceeds differently for large string charge.

The D2 dipole charge is proportional to

(7) |

The first two terms in the left hand side are nonzero, while the third is zero. This means there is local D2 charge in the and planes, but not in the plane, exactly as it should be for a cylindrical D2 brane.

As the matrices , and are infinite in the large volume limit, the ciclicity of the trace is not respected. Without any turned on, the action is given [16] by

(8) |

One can plug in the defined in (5), and evaluate the determinant above. By using the fact that the determinant of a matrix with noncommuting entries ^{4}^{4}4The determinant is defined as takes all orderings into account, or alternatively the fact that and commute, we notice that the terms proportional to and their permutations cancel. The only combination of matrices which appears under the square root is . Since this combination is proportional to , the Lagrangian simply becomes

(9) |

In order to obtain a Hamiltonian from this Lagrangian, one has to find the conjugate momentum corresponding to the string charge:

(10) |

where the normalization of the string charge with respect to was found by comparing the small string charge limit of (10) with (6). The Hamiltonian is therefore

(11) |

the same as the one found in [14]. This formula is valid for all values of the D0, F1 and D2 charges. We can take the limit when the D0 charge vanishes ( keeping the length of the tube fixed) and the string charge dominates to obtain

(12) |

The first term is the energy of coincident static strings, and the second one reproduces exactly the Born-Infeld contribution to the action (27) when one identifies the parameters in the matrix and supergravity descriptions:

(13) |

This exact agreement provides a confirmation of the nonabelian Born-Infeld action, as well as of the nonabelian coupling with the field found in [17]. Nevertheless, it does not imply anything about the validity of the symmetrized trace prescription because the only matrix whose trace we had to take was .

## 3 The supergravity analysis

Let us for the beginning explore the supergravity background created by a large number of coincident F strings. By a generalization of the AdS-CFT correspondence in the spirit of [11], the two-dimensional field theory living on the worldvolume of these coincident F strings is dual to string theory living in the near-horizon background of these strings. As explained in [21], turning on an operator in the boundary theory Hamiltonian corresponds to turning on a nonnormalizable mode in the bulk. One can study the effect of the operator on the boundary theory by examining the supergravity dual which the corresponding nonnormalizable mode creates. Nevertheless the study of this theory, although straightforward once all the computations are done, is not our primary interest. Rather, we are looking for a supergravity background with a nonzero which preserves 4 of the original 16 supercharges, and we are using this correspondence only as a tool to keep control on the supersymmetry.

As the fermions of the IIA F string in static gauge transform in the 8s of the SO(8) R symmetry group, a fermion bilinear transforms in the 35 or the 28 of this group. The bilinear in the 35 corresponds to a self dual RR 4-form field strength on the space transverse to the strings. This 4-form polarizes the strings into D4 branes, creating the dimensional reduction of the M2M5 setup in [6]. The fermion bilinear in the 28 corresponds to a bulk RR 2 form and 6 form field strengths on the transverse space. The 6 form on the transverse space is equivalent by Hodge duality to a 4 form with 2 legs along the strings, which is what we need for F1 D2 polarization.

The background

The string frame supergravity background of fundamental strings is

(15) |

where is a harmonic function on the transverse space which becomes

(16) |

when the strings are coincident. If , the supergravity solution is everywhere valid and it is not dual
to anything. However, for the beginning we are interested in a regime where we could use the AdS-CFT type of duality discussed above, and thus we will focus for now on ^{5}^{5}5Once we establish the form of the supersymmetric supergravity perturbation, we can also use it for (as long as the 1 in the harmonic function can be ignored) since the supersymmetry of the background does not depend on ..
For the dilaton becomes large, and physics starts being described by the supergravity solution of N coincident M2 branes. For supergravity in the background (15) gives a weakly coupled description of the physics, and the 1 in the harmonic function can be ignored. We will work in this regime. For even smaller radii, the theory has a weakly coupled description in terms of an orbifold CFT, analyzed in [12].

The perturbation

We are interested in perturbing the above background with the RR fields which could cause
polarization. As explained above, these fields are and on the space transverse to the strings.
If one defines ^{6}^{6}6This is not the canonical definition of
but it is the most convenient to work with. , the IIA equations of motion become:

(17) |

Combining (15) with (17) and expressing the ten dimensional Hodge dual in terms of the 8 dimensional transverse space Hodge dual , the equations satisfied by the first order perturbations become:

(18) |

We should note that acting on even forms , while . To find the perturbations with the same R-symmetry transformation properties as the fermion bilinear it is convenient to group the 8 transverse coordinates and the fermions into 4 complex pairs:

(19) |

Under the rotations the fermions transform as :

(20) |

Thus a fermion bilinear of the form has the same SO(8) transformation
properties as
, (defined in the Appendix) and their Hodge duals and ^{7}^{7}7The standard properties of these tensors are reviewed in the Appendix..

The most straightforward way to examine the supersymmetry preserved by a specific fermion bilinear is to examine the free 2-dimensional boundary theory which has 8 bosons in of SO(8) and both the left and right moving fermions and in . The original supersymmetry transformations are

(21) |

where are matrices given in [24]. It is a straightforward
exercise to show that adding a fermion bilinear, together with its
bosonic partner preserves 4 of the original 16 supercharges and .
^{8}^{8}8On can also examine the supersymmetry preserved by a certain fermion bilinear by doing a 9-11
flip. A bilinear of the form becomes a vector bilinear on the M2 brane when lifting to
M-theory. Reducing along a different direction, this becomes a vector bilinear on the D2 brane, of the
form which upon T duality along becomes again a scalar fermion
bilinear on the D1 brane: . Reexpressing the complex fermions in real ones, this bilinear is , which can be expressed as coming from a superpotential of the form . Thus after changing the sign of a field this superpotential becomes the dimensional reduction of a hyper bilinear in the 2D SYM on the D2 branes. Since that superpotential preserves 4 supercharges ( in 2+1 dimensions), this is the supersymmetry preserved by our operator.

Thus, a 2 form and a dual 6 form corresponding to the fermion bilinear can be written as combinations of and , and respectively and multiplied by powers of . Plugging this ansatz in (18), we find 4 solutions. Two of them correspond to a mode which is not a chiral primary, and the other two, one normalizable and the other non normalizable correspond to the fermion bilinear we are interested in. Adding the fermion bilinear to the boundary theory Hamiltonian corresponds [21] to adding to the background (15) the non normalizable solution:

(22) |

where is proportional to the coefficient of the fermion bilinear added to the boundary Lagrangian.

A tubular D2 brane in this geometry couples with . In our background, this combination can be found using

(23) |

Since is a harmonic form, it only depends on its value at infinity, which is given by the boundary theory. Therefore the expression above remains true when the F1 strings are coincident or when they are distributed. This implies that the WZ term in the action of a cylindrical D2 brane is independent on where the F1 strings are.

Probing the perturbed solution

In order to find the self interaction potential of a cylindrical D2 brane with large F1 charge , one has to find first the potential felt by a D2 brane probe with smaller F1 charge , and then to build the D2 brane by bringing test D2 branes from . Anticipating, we will find that similarly to other Polchinski-Strassler type problems [5, 6, 7, 22, 23], the probe potential will be independent of the distribution of the strings; thus the self interaction potential of a tubular D2 brane containing all the strings will be the same as the probe brane potential with replaced by .

A cylindrical D2 brane can be given F1 charge by turning on a nontrivial on its worldvolume. The quantized F1 string charge in a nontrivial , and is

(24) |

If we call

(25) |

the Hamiltonian density per unit string length will be

(26) |

We are interested in probes with dominant string charge, , which is easily realized for in the regime where supergravity is valid. In this regime the square root can be expanded, and the main contributions to the Hamiltonian cancel, as they represent the interaction energy between parallel F1 strings. After this cancellation, the subleading term of the Born-Infeld action is

(27) |

As we mentioned in the previous chapter, this supergravity result reproduces exactly the one obtained using the nonabelian Born Infeld action in the limit of vanishing D0 charge. For a D2 brane of geometry, this contribution to the action does neither depend on the orientation of the , nor on . It is straightforward to generalize this action for the circle replaced by an ellipse. This can be easily done in a similar way to [5, 6, 7, 22, 23]. If one turns on the bilinear which corresponds to , it is natural to expect the probe to have a minimum if the two semiaxes of the ellipse lie in the two planes spanned by components of and . If the lengths and orientations of the semiaxes are given by the modulus and phase of and , the result is obtained by just replacing by in (27).

One can also find the interaction potential coming from the Wess Zumino action. Using (23) and integrating over the ellipsoid one finds after a few straightforward steps

(28) |

which is also independent of the harmonic function . Thus, the first two terms of the action of the probe are

(29) |

As we said in the Introduction, besides these two terms there is another one coming from the backreaction of the and perturbations on the metric, dilaton and 2-form. One can see from the equations of motion that this term should be proportional to
^{9}^{9}9The factor comes from the square of the first order perturbation, comes from the dominant F-string charge of the probe, and the proportionality with is not hard to derive..
Supersymmetry allows us to find this term exactly by simply completing the square in (29). Thus

(30) | |||||

The full potential

The last term of (30) is also independent of . Therefore, the potential of the probe tube does not depend on the position of the F1 strings sourcing the geometry. One can now find the self interaction potential of a D2 brane tube containing all F1 strings, in its own geometry, by bringing from infinity small probes and using them to construct the final configuration. Since the potential of each of these small probes is independent of where the rest of the probes are, the final potential will be the sum of the probe potentials, which gives the same result as (30) with replaced by . The most general configuration will consist of several D2 tubes with F1 charges . The potential will be the sum of terms, each of them equal to (30) with replaced by :

(31) |

where parametrize the semiaxes of the ’th D2 tube, which contains F1 strings. This potential is very interesting. It implies that for a generic strength of the perturbation, there is only one minimum at . Contrary to what one might expect, increasing the strength of the four-form does not make the polarization to a D2 brane more likely. This is because a large causes a large backreaction on the graviton, dilaton and 2 form, which do not favor the polarization.

Even more remarkable, for the particular values , the zero energy solution is given by for the ’th tube, and for the others. Thus, for these values of the perturbation a moduli space of possible polarization vacua opens up; of the original strings can be polarized. The lowest absolute value of which allows the polarization to occur is . In this case all strings polarize. For , of the strings polarize into one D2 brane. For it suddenly becomes possible for half the strings to polarize into one D2 brane, and the other half to polarize into another D2 brane. Both D2 tubes can be at any radius. In general, for concentric D2 branes at different radii, with F1 charge each. , the ground state will consist of

Consistency checks

Before we continue, we should address two small but potentially dangerous issues. First, we have to make sure that the supergravity perturbation fields are weaker than the background fields. The energy of the background and perturbation fields are respectively

(32) |

The condition for validity of the perturbation expansion is thus . For the smallest value of which allows for a moduli space this is equivalent to which is trivially satisfied. However, for larger values of which allow strings to polarize into a D2 brane, this condition is satisfied only for .

The second issue has to do with the attraction between the opposite sides of a tube, or between concentric tubes. Because the large F1-string tension and the D2 brane tension add in quadratures, the interaction coming from the gravitational D2-D2 attraction has a negligible effect on the energy. What could spoil the interaction is the electric attraction between the opposite sides of the tube.

The issue of this possible interaction is usually not addressed in most of the polarization papers. In principle this can give an energy contribution which may affect the polarization ground state. In our context, since the balance of the energies is so delicate, such a contribution could in principle easily lift the moduli space. One could probably argue that supersymmetry prevents this contribution from affecting the potential, but nevertheless it is instructive to see how it happens.

As we can see from (23), our D2 brane tube couples with a field combination which is proportional to the harmonic form . The presence of extra D2 brane charge affects the Bianchi identity for only. Nevertheless, remains harmonic. Therefore the presence of D2 charge (from the opposite side of the tube) does not affect the Wess-Zumino term of the potential. Thus, as we expected, the attraction between the opposite sides of the tube or between different tubes does not have any effect on the energy of the system. The lack of interaction between tubes placed at different radii is characteristic to Polchinski - Strassler type setups, and is also present for the supertubes with D0 charge [25].

Usually the appearance of a moduli space of vacua signals some symmetry enhancement. It is interesting to ask what this symmetry might be. One certain thing is that supersymmetry is not enhanced, since both the field strengths which allow polarization and those who do not allow it preserve 4 supercharges. Thus, this is probably a symmetry which appears in the UV of the mysterious orbifold CFT living on the strings.

## 4 Conclusions and future directions

We have given matrix and supergravity descriptions to the polarization of type IIA fundamental strings in tubular D2 branes. For the matrix description we have used the matrix degrees of freedom of D0 branes and their nonabelian Born-Infeld action to describe an F1 D2 tubular configuration in a nontrivial background.

In order to give a supergravity description while maintaining control over the supersymmetry we have turned on a fermion bilinear in the theory living on coincident fundamental strings. We have then related this bilinear to a supergravity nonnormalizable mode via an AdS-CFT like correspondence, and showed that the nonnormalizable mode induces F1 D2 polarization. We have found that polarization occurs only when the value of the bulk fields is proportional to fractions whose denominator is smaller than ; morover, in these cases the polarization radius is a modulus. For any other values of the bulk fields there is no polarization.

For the supergravity description we have used the setup and technology developed by Polchinski and Strassler [5]. Although this technology is sometimes heavy, it has certain advantages (such as the control on the supersymmetry and on the backreaction of the fields, or the lack of attraction between the opposite sides of the polarized object) which in our opinion make it the only correct setup for the study of polarization into objects one dimension higher.

One interesting question which one might ask is what happens if one turns on a nonsuper-symmetric traceless scalar bilinear in the potential, which might create an instablity along some direction. Understanding the effect of this other piece in the potential can be easily done in our framework (see [7] or [26] for an example). Such a piece makes the strings cease to be a classical ground state, and seems to leave the puzzle in [2] unsolved. The only consolation is that only a very special combination of 4-form and graviton will create this instability. In general however, turning on an will not afect the stability of the strings.

It is intriguing to see what is the fate of this configurations when one lifts them to M theory. A D2 brane with large F string charge lifts to an M2 brane helix which winds around the eleventh dimension many times while circulating once around an . The radius of the large circle is a modulus, so there is no potential for the helix to be stretched in a radially symmetric fashion. Even more puzzling to our intuition is the fact that a state of many such helices, non concentric and possibly intertwined preserves 4 supercharges. When one takes the size of the 11’th dimension to infinity these helices should naturally dissolve into the “Coulomb branch” of the theory on parallel M2 branes.

The F1 D2 configuration falls in the general category of objects polarizing into objects one dimension higher. Similar situations can occur in the case of D4 branes polarizing to NS5 branes. We expect a similar potential, which allows a moduli space to open up for a very specific value of the perturbation, and does not allow polarization for any other value. On general grounds one expects the D4 NS5 polarization to happen for supersymmetry in 4+1 dimensions, and therefore the moduli space not to be lifted up by quantum corrections. Again, the lift of the configuration in M-theory would give a spiral-shaped M5 brane.

We have studied a very intriguing system, and found many phenomena which challenge our intuition, such as the presence of a moduli space of polarization vacua or the fact that M2 brane spherical helices in M theory at finite can be supersymmetric. We cannot but hope that in time a more fundamental explanation of these phenomena and other less understood aspects of M theory will be given.

Acknowledgments: I would like to thank Wati Taylor for many useful comments and help in understanding the matrix description of this polarization. I would like to thank Joe Polchinski, K. S. Narain, S. Sinha, B. de Wit, R. Roiban and A. Buchel for useful discussions. This work was supported by a University of California Special Regents Fellowship, and by NSF grant PHY97-22022.

## 5 Appendix

We list several properties of the transverse space antisymmetric 2- and 6-tensors which form a basis for the forms with the R-symmetry transformation properties of a fermion bilinear.

(33) | |||||

(34) | |||||

(35) | |||||

(36) | |||||

(37) | |||||

(38) |

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