Contents

On the gravity dual of Chern-Simons-matter theories with unquenched flavor


Eduardo Conde 111eduardo@fpaxp1.usc.es and Alfonso V. Ramallo222alfonso@fpaxp1.usc.es

Departamento de Física de Partículas, Universidade de Santiago de Compostela

and

Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782, Santiago de Compostela, Spain

Abstract

We find solutions of type IIA supergravity which are dual to three-dimensional Chern-Simons-matter theories with unquenched fields in the fundamental representation of the gauge group (flavors). In the holographic dual the addition of flavor is performed by means of D6-branes that are extended along the Minkowski gauge theory directions and are delocalized in the internal space in such a way that the system is supersymmetric and the flavor group is abelian. For massless flavors the corresponding geometry has the form of a product space , where is a six-dimensional compact manifold whose metric is obtained by squashing the Fubini-Study metric of with suitable constant factors which depend on the number of flavors. We compute the effect of dynamical quarks in several observables and, in some cases, we compare our results with the ones corresponding to the supergravity solutions generated by localized flavor branes. We also show how to generalize our results to include massive flavors.

1 Introduction

The recent results on the correspondence constitute a rich framework in which fundamental questions about the holographic correspondence [1] can be posed and, in some cases, highly non-trivial answers can be obtained. The crucial breakthrough in this subject took place with the results of Bagger and Lambert [2] and Gustavsson [3], who proposed that the low energy theory on multiple M2-branes is given by a new class of maximally supersymmetric Chern-Simons-matter theories. Inspired by these results, Aharony et al. (ABJM) [4] constructed an supersymmetric Chern-Simons-matter theory which is now believed to describe the dynamics of multiple M2-branes at a singularity.

The ABJM theory is a Chern-Simons gauge theory with levels and bifundamental matter fields. In the large limit this theory admits a supergravity description in M-theory in terms of the geometry. If we represent as a Hopf bundle over , the orbifold acts by quotienting the fiber. When the Chern-Simons level is large the size of the fiber is small and the system is better described in terms of type IIA supergravity by performing a dimensional reduction to ten dimensions along the Hopf fiber of . In this ten-dimensional description the geometry is of the form with fluxes and preserves 24 supersymmetries.

The precise knowledge of the field theory dual to a system of multiple M2-branes on has allowed to test some of the non-trivial predictions of the AdS/CFT correspondence. In particular, in [5] it was checked by means of a purely field theoretic calculation, using matrix model techniques and localization, that the number of low energy degrees of freedom of coincident M2-branes scales as for large , as predicted by the gravity dual (see also [6]). Moreover, the ABJM model has been generalized in several directions. By adding fractional M2-branes, the gravity dual of Chern-Simons-matter theories with was constructed in [7] . If the sum of Chern-Simons levels for the two gauge groups is non-zero, the corresponding gravity dual can be found in massive type IIA supergravity by considering solutions in which the Romans mass parameter is non-vanishing [8] (see also [9]). The ABJM construction has been extended to include different quivers and gauge groups with several amount of supersymmetry in refs. [10]-[15].

In this paper we will study the generalization of the ABJM model which is obtained by adding flavors, i.e. fields transforming in the fundamental representations and of the gauge group. The holographic dual of such a system was proposed in refs. [16, 17]. In the type IIA description the addition of massless flavor is achieved by considering D6-branes that fill the space and wrap an submanifold inside the , while preserving supersymmetry (see also [18]-[21] for different setups with D6-branes in Chern-Simons-matter theories). When the number of flavors is small one can study the system in the quenched approximation, in which the D6-branes are considered as probes in the background. This quenched approach has been adopted in refs. [22]-[24], where different observables of the Chern-Simons-matter theory with flavor have been analyzed.

In the present article we will study the holographic dual of the ABJM theory with unquenched flavor. For a system of localized and coincident D6-branes, the corresponding gravity dual in M-theory is a purely geometric background which, in the near horizon limit, is a space of the type , where is a tri-Sasakian seven-dimensional manifold whose cone is an eight-dimensional hyperkähler manifold [25]. Notice that the backreacted metric always has an Anti-de-Sitter factor. This is related to the fact that the dual Chern-Simons theory remains conformally invariant after the addition of flavor (see ref. [26] for a perturbative calculation of the beta functions and a study of the corresponding fixed points). However, the tri-Sasakian metric of has, in general, a complicated structure which makes difficult to use it for many purposes. For this reason, in this paper we study the backreaction induced by a smeared continuous distribution of a large number of flavor branes. This approach was initiated in [27] in the context of non-critical strings and has been successfully applied to study unquenched flavor in several other setups (see [28] for an extensive review and references to the original works).

In order to obtain the gravity dual of a field theory with unquenched flavor one has to solve the equations of motion of supergravity with brane sources. These sources typically modify the Bianchi identities of the forms and, as they contribute to the energy-momentum tensor of the system, they also modify the Einstein equations. If the flavor branes are localized the sources contain Dirac -functions and, as a consequence, solving the equations of motion is, in general, a difficult task. On the contrary, if the sources are delocalized there are no -function terms in the equations of motion and finding explicit analytic solutions is much simpler. Notice that when the branes are smeared, they are not coincident anymore and, therefore, the flavor symmetry for flavors is rather than . Moreover, the solutions with smeared unquenched flavor are generically less supersymmetric than the ones with localized flavor, due to the fact that we are superposing branes with different orientations in the internal space. Indeed, in our flavored ABJM case the solutions will be SUSY instead of being .

The backreaction of the flavor branes induces a deformation of the unquenched solution which, in particular, results in a suitable squashing of the metric. In order to determine precisely this flavor deformation one has to write the metric in a way such that it can be squashed without breaking all supersymmetry. We will argue below that for the ABJM case in the type IIA description the convenient way of writing the metric is as -bundle over . After representing the metric in this way, the flavor deformation just amounts to squashing the fiber with respect to the base, as well as to changing the radii of both and factors of the metric (similar squashed deformations of were considered recently in [29, 30, 15]). In our solutions, the squashing factors are constants which have a precise dependence on the number of flavors and encode the effects due to loops of fundamentals. Indeed, we will be able to determine the effects due to dynamical flavors in several observables such as, among others, the free energy on the sphere, the quark-antiquark potential energy or the dimension of meson operators.

The organization of the rest of this paper is the following. In section 2 we review the ABJM solution and we rewrite it in terms of the fibration over . In section 3 we study the deformations of the ABJM unflavored background with supersymmetry which preserve the -bundle structure. The corresponding supergravity solutions are obtained by solving a system of first-order BPS equations which we partially integrate in general. In particular we find two Anti-de-Sitter solutions which correspond to the original unflavored ABJM model and to the squashed model of reference [15]. In section 3 we also find running solutions which approach the ABJM background in the IR.

In section 4 we study the supersymmetric embeddings of flavor D6-brane probes in the deformed geometries. The backreaction for branes corresponding to massless quarks is analyzed in section 5, where an ansatz for the background deformed by the massless fundamental fields is proposed and a set of first-order BPS equations is obtained. Section 6 is devoted to studying the solutions of the flavored BPS system that are Anti-de-Sitter. In section 7 we analyze the effect of the unquenched flavor in several observables. In section 8 we study the backreaction of unquenched massive flavors. Finally, in section 9 we summarize our results and discuss some directions for future work. The article is completed with several appendices, where some detailed calculations not included in the main text are performed and some other aspects of our work are explored.

2 The ABJM solution

The metric of the ABJM background is given by:

(2.1)

where and are respectively the and metrics. The former, in Poincare coordinates, is given by:

(2.2)

with being the Minkowski metric in 2+1 dimensions. This solution depends on two integers and which represent, in the gauge theory dual, the rank of the gauge groups and the Chern-Simons level, respectively. In string units, the radius can be written in terms of and as:

(2.3)

Moreover, for this background the dilaton is constant and given by:

(2.4)

This solution of type IIA supergravity is also endowed with a RR two-form and a RR four-form whose expression can be written as:

(2.5)

with being the Kähler form of and is the volume element of the metric (2.2). This solution is a good gravity dual of the Chern-Simons-matter theory when the radius is large in string units and the string coupling is small. By looking at eqs. (2.3) and (2.4) this amounts to the condition .

The metric in (2.1) is the canonical Fubini-Study metric. In the context of the ABJM solution the space is usually represented as foliated by the manifold. Here we will write the metric in a form which is more convenient for our purposes. We will regard as an -bundle over , with the fibration constructed by using the self-dual instanton on the four-sphere. Explicitly, will be written as:

(2.6)

where is the standard metric for the unit four-sphere, () are cartesian coordinates that parametrize the unit two-sphere () and are the components of the non-abelian one-form connection corresponding to the instanton. Mathematically, the representation (2.6) is obtained when is constructed as the twistor space of the four-sphere. We shall now introduce a specific system of coordinates to represent the metric (2.6). First of all, let () be the left-invariant one-forms which satisfy (for an explicit representation of the ’s in terms of angular coordinates, see (4.1)). Together with a new coordinate , the ’s can be used to parameterize the metric of a four-sphere as:

(2.7)

where is a non-compact coordinate. The instanton one-forms can be written in these coordinates as:

(2.8)

Let us next parametrize the coordinates of the by means of two angles and (, ), namely:

(2.9)

Then, one can easily prove that:

(2.10)

where and are the following one-forms:

(2.11)

Therefore, the canonical Fubini-Study metric of can be written in terms of the one-forms defined above as:

(2.12)

As a check, one can verify that the volume of obtained from the above metric is . We shall now consider a rotated version of the forms by the two angles and . Accordingly, we define three new one-forms as:

(2.13)

In terms of the forms defined in (2.13) the line element of the four-sphere is just obtained by substituting in (2.7). Let us next define the one-forms and as:

(2.14)

in terms of which the metric of the four-sphere is just . The RR two-form can be written in terms of the one-forms defined in (2.11) and (2.14) as:

(2.15)

Notice that is a closed two-form due to the relation:

(2.16)

Notice that there is a non-trivial in the geometry, which in our coordinates is parametrized by the angles and at a fixed point in the base . As one can readily verify by a simple calculation from (2.15), the flux of the RR two-form through this is given by:

(2.17)

Eq. (2.17) is essential to understand the meaning of as the Chern-Simons level of the gauge theory. Indeed, let us consider a fractional D2-brane, i.e. a D4-brane wrapping a two-cycle and extended along the Minkowski directions. For such a brane there is a coupling to the worldvolume gauge field of the type which, taking into account (2.17), clearly induces a Chern-Simons coupling for the gauge field with level .

Some basic facts of the geometry of the ABJM solution in our coordinates are compiled in appendix A. In particular, it is shown how the uplifted solution in M-theory corresponds to the space , where the is realized as an -bundle over . Also, the non-trivial cycles of are displayed.

3 Deforming the ABJM background

We will now analyze the generalization of the ABJM background obtained by performing a certain deformation of the metric which preserves some amount of supersymmetry. Specifically, we shall consider a ten-dimensional string frame metric of the form:

(3.1)

where is a warp factor and is a seven-dimensional metric containing an fibered over an in the same way as in , namely:

(3.2)

with , and being functions of the radial variable . Notice that and determine the sizes of the and of the internal manifold. If we will say that the is squashed. We will verify below that this squashing is compatible with supersymmetry when the functions of the ansatz satisfy certain first-order BPS equations.

The type IIA supergravity solutions we are looking for are endowed with a RR two-form , for which we will adopt the same ansatz as in (2.15). In addition, as it is always the case for the solutions associated to D2-branes, there is a RR four-form given by:

(3.3)

where is a function of the radial coordinate . Moreover, we will assume that the dilaton depends only on .

Notice that the Bianchi identities are automatically satisfied. Moreover, the Hodge dual of is equal to:

(3.4)

and, thus, the equation of motion of the four-form (namely ), leads to:

(3.5)

where the constant should be determined from a quantization condition. Thus, it follows that can be written in terms of the other functions of the ansatz, namely:

(3.6)

Notice that is subjected to the following flux quantization condition:

(3.7)

where is the six-dimensional angular manifold enclosing the D2-brane. Using our ansatz this quantization condition is converted into:

(3.8)

where denotes the volume form of a unit -sphere and we are using string units. Therefore, the coefficient should be related to the number of D2-branes as:

(3.9)

and the function is related to the other functions in the ansatz as:

(3.10)

We will determine the functions entering our ansatz by requiring that our background preserves (at least) two supersymmetries. As shown in detail in appendix B, this requirement is fulfilled if the dilaton , the warp factor and the functions and satisfy the following system of first-order BPS equations:

(3.11)

In the first two equations of the system (3.11) the function should be understood as given by (3.10). This BPS system is obtained after imposing the following projections on the Killing spinors:

(3.12)

where the denote antisymmetrized products of flat Dirac matrices in the basis of one-forms written in (B.2).

The BPS system (3.11) can be rewritten in a compact fashion in terms of two calibration forms. In order to recast (3.11) in this way, let us next define the calibration seven-form as:

(3.13)

where the components are the fermionic bilinears:

(3.14)

with being a Killing spinor of the background and the prefactor in (3.14) is included to account for the proper normalization of (see (B.22)). By using the projections satisfied by , one can verify that is given by:

(3.15)

In a background generated by D2-branes, it is natural to have also a calibration three-form. Accordingly, we also define the three-form , as:

(3.16)

Using again the projections satisfied by the spinor , one can show that:

(3.17)

One can now verify that the BPS equations (3.11) can be compactly recast as:

(3.18)

3.1 Partial integration

Let us now carry out some simple manipulations of the BPS system (3.11), which will allow us to perform a partial integration. First of all, let us define the function as follows:

(3.19)

Clearly, from this definition one has:

(3.20)

Moreover, it is easy to prove that , and close the following system of first-order differential equations:

(3.21)

Notice that the function has disappeared from the system (3.21) and that and only appear through the combination . Actually, by combining the first two equations in (3.11) one proves that can be written as:

(3.22)

The warp factor and the dilaton can be obtained from the solution of the system (3.21). Indeed, by using again the first two equations in (3.11) together with (3.6) , one arrives at:

(3.23)

where we have used the definition (3.19). Eliminating between (3.20) and (3.23), we get:

(3.24)

Eq. (3.24) is a first-order differential equation for that can be solved by the method of variation of constants. The result is:

(3.25)

where is a constant of integration that should be adjusted appropriately. We have not been able to integrate the BPS system (3.21) in general. However, we have found some important particular solutions which we will discuss in the next two subsections and in appendix C. In some of these solutions the supersymmetry is enhanced with respect to the two supersymmetries preserved by the generic solution of (3.21).

3.2 Anti-de-Sitter solutions

We will be mostly interested in backgrounds with the Anti-de-Sitter geometry and in their corresponding deformations. In order to find these solutions systematically, let us now introduce a new radial variable , related to as follows:

(3.26)

If the dot denotes derivative with respect to , the system of equations (3.21) reduces to:

(3.27)

Let us next define the following combination of functions:

(3.28)

Notice that measures the squashing of the and internal directions. Actually, the right-hand-side of the equations in (3.27) depends only on and and it is straightforward to find the following system of two equations involving and :

(3.29)

One can take , and (say) as independent functions. In fact, can be obtained by simple integration once and are known, due to the equation:

(3.30)

We have thus reduced the full BPS system to a set of two coupled differential equations for the functions and .

We will now obtain some particular solutions of (3.30) in which the squashing factor is constant, as expected for an background. It follows from the second equation in (3.29) that, in this case, must be also constant. Actually, we can eliminate from the equations and get a simple algebraic equation for . In order to express this equation in simple terms, let us define the quantity as:

(3.31)

Then, implies the following quadratic equation for :

(3.32)

which has two solutions, namely:

(3.33)

Notice that corresponds to the ABJM background, while should correspond to a background of reduced SUSY of the form , with being a squashed version of . This background was proposed in ref. [15] to be the gravity dual of an superconformal Chern-Simons-matter gauge theory with global symmetry. We will describe this solution and some generalizations in appendix C. In the remaining of this subsection we will concentrate in showing how the ABJM background reviewed in section 2 is obtained in this formalism from the solution. First of all, we notice that when , the system (3.29) leads to the following solution for :

(3.34)

Using these values of and in (3.30) one readily gets that and, by using (3.26) one shows that the radial variables and are related as . Therefore, in the original variable , one has:

(3.35)

Taking into account that the function defined in (3.19) is , we get that and we can obtain , and from eqs. (3.25), (3.19) and (3.22) respectively. The dilaton obtained in this way is constant and is just the one written in (2.4), while and are given by:

(3.36)

By rescaling the Minkowski coordinates as with , one can verify that, indeed, the metric and RR four-form for this solution coincide with the ones in (2.1) and (2.5).

3.3 Running solutions

We will now solve the BPS system (3.21) in a power series expansion in the radial coordinate . The aim is to find new solutions that approach the background in the IR limit . We begin by rewriting the system (3.21) in a more convenient form. Let us define the new function as:

(3.37)

Then, one can recast (3.21) as:

(3.38)

The ABJM solution (without squashing ) can be simply written as . We will now solve (3.38) in a series expansion in powers of around this solution. We will look for a solution in which , and take the form:

(3.39)

By substituting this ansatz on the system (3.38) and solving for the different powers of up to third order, one can find the following solution:

(3.40)

where is an arbitrary constant (if we come back to the solution). Plugging the expansions (3.40) into the right-hand side of (3.25) and adjusting the integration constants in such a way that vanishes at , one gets the following expression of the warp factor :

(3.41)

Similarly, the dilaton runs as:

(3.42)

whereas the function is given by:

(3.43)

Notice that, when the constant is non-vanishing, the geometry is not Anti-de-Sitter and the internal space is squashed by an -dependent function.

4 SUSY embeddings of flavor D6-branes

In this section we will study the addition of flavor D6-branes to a background of the type studied in section 3. We will analyze certain configurations in which the D6-branes preserve some amount of supersymmetry of the background. In the present section we will work in the probe approximation, corresponding to having quenched quarks on the field theory side, in which the background supergravity solution is not affected by the presence of the flavor D6-branes. The effect of the backreaction will be considered in detail in sections 5 and 8.

Generically, we will consider configurations corresponding to massless quarks which extend along the three Minkowski directions , the radial coordinate and that wrap a three-dimensional cycle of . On general grounds [16, 17] it is expected that this three-cycle of is a special lagrangian cycle which can be identified with . Let us show how this arises in our coordinates. With this purpose let us parametrize the left invariant one-forms of the four-sphere metric (2.7) in terms of three angles , and , namely:

(4.1)

with , , . In order to write down a coordinate description of the D6-brane configuration, let us choose the following set of worldvolume coordinates:

(4.2)

In these coordinates our embedding is defined by the conditions:

(4.3)

Notice that and vary inside the , whereas varies inside the . Actually, and parametrize an , while is a maximal . The induced worldvolume metric is:

(4.4)

where the metric of the three-cycle is given by:

(4.5)

with being the squashing factor defined in (3.31). Let us verify that this three-dimensional metric corresponds to (a squashed) . We first perform the following change of variable from to a new angular variable , defined as:

(4.6)

In terms of the metric becomes:

(4.7)

Let us next define new angles and as:

(4.8)

Then, the metric (4.7) becomes:

(4.9)

It is clear from (4.8) that . Moreover, by comparing the volume obtained with the metric (4.7) with the one obtained with (4.9), one concludes that and that our three-cycle is indeed a squashed manifold inside .

Let us now verify that the cycle (4.3) preserves some amount of supersymmetry. First of all, let us note that the embedding (4.3) can be characterized by the three differential conditions:

(4.10)

which are integrable due to Frobenius theorem since when (4.10) holds and . From this result one can verify that the cycle is calibrated by the form for a metric given by the general form (3.1) and (3.2). Indeed, the only non-zero component of the pullback of (denoted by ) is the one containing in (3.15) and one has:

(4.11)

which one can easily show that coincides with the volume form derived from the worldvolume metric (4.4). In appendix D we have confiormed, by making use of kappa symmetry, that these embeddings preserve the supersymmetry of the background. Actually, in that appendix we generalize the embeddings (4.3) to the case in which is not constant. If , the supersymmetric configurations are those where the following first-order BPS equation is satisfied:

(4.12)

Notice that, indeed, the solutions of (4.12) with constant must necessarily have . Moreover, one can easily show by analyzing (4.12) that, when is not constant, the radial coordinate reaches a minimal value in the corresponding brane embedding. Therefore, one can interpret these D6-brane configurations with varying as flavor branes that add massive flavors to the Chern-Simons-matter theory. The corresponding quark mass is related to the minimal distance . Notice also that one can generate a whole continuous family of embeddings equivalent to the ones studied so far by acting with the different isometries of the metric. In the next sections we will construct supergravity solutions that incorporate the deformation of the geometry due to the presence of such a continuous set of branes.

5 Backreacted massless flavor

Let us now study the backreaction of the flavor branes on the backgrounds of the ABJM type. With this purpose in mind we first analyze the modification of the Bianchi identity introduced by D6-brane sources. This modification is determined by the WZ term of the D6-branes, which for a collection of of them is given by:

(5.1)

where the hat denotes the pullback to the D6-brane worldvolume. Let us rewrite this expression in terms of a charge distribution three-form as:

(5.2)

with being only non-vanishing at the location of the D6-branes. The term (5.2) induces a violation of the Bianchi identity of . Indeed, let us write the supergravity plus branes action in terms of the eight-form and its seven-form potential . This action contains a contribution of the form:

(5.3)

It is manifest from (5.3) that the D6-branes act as a source for the RR seven-form potential . Actually, the equation of motion of derived from (5.3) gives rise to the following Maxwell equation for :

(5.4)

which, as , is equivalent, as claimed, to the violation of the Bianchi identity of , namely:

(5.5)

where we have used the fact that, in our units, we have .

Following the general procedure reviewed in [28], we will consider a large number of flavor branes and we will substitute the discrete set of branes by a continuous distribution characterized by the three-form . We will assume that the flavor branes are delocalized in such a way that there are no Dirac -functions in the expression of . In this way we will be able to find solutions of the different field equations of supergravity with sources. Actually, it is easy to find an expression of which preserves both the two supersymmetries and the form of the metric of the deformed unflavored solutions. Notice, however, that the ansatz for must be modified in order to satisfy (5.5). In fact, by looking at (2.16) it is easy to find the appropriate modification of our ansatz (2.15). Indeed, the two-form written in (2.15) is closed because there is a precise balance between the term (along the fibered ) and the two other terms with components along the . Clearly, to get a non-closed two-form without distorting much the -