Some BPS configurations of the BLG Theory

We obtain BPS configurations of the BLG theory and its variant including mass terms for scalars and fermions in addition to a background field with different world-volume and R-symmetries. Three cases are considered, with world-volume symmetries SO(1,1) and SO(2) and preserving different amounts of supersymmetry. In the former case we obtain a singular configuration preserving N=(3,3) supersymmetry and an one-quarter BPS configuration corresponding to intersecting M2-M5-M5-branes. In the latter instance the BPS equations are reduced to those in the self-dual Chern-Simons theory with two complex scalars. In want of an exact solution, we find a topological vortex solution numerically in this case. Other solutions are given by combinations of domain walls.


Introduction
The Bagger-Lambert-Gustavsson(BLG) theory [1][2][3][4][5] is an N = 8, supersymmetric Chern-Simons type gauge theory based on a ternary gauge algebra coupled to matter in (2 + 1)-dimensions with SO(8) R-symmetry. The theory is deemed to have Osp(8|4) superconformal symmetry based on strong evidences [6] and is thus a candidate for a world-volume theory of M2-branes. It contains eight scalar fields interpreted as eight directions transverse to the world-volume of M2-branes in M-theory and eight corresponding fermions in addition to a gauge field. Imposing complete antisymmetry of the structure constant of the ternary gauge algebra along with the closure of the supersymmetry algebra constrains the gauge group to be SO (4). The theory has a sixteen-dimensional moduli space lending itself to the interpretation as a theory of two M2-branes. [7][8][9]. Various aspects and variants of the BLG theory have been considered [10][11][12][13][14][15][16][17][18][19][20][21].
In this article we shall be concened with BPS configurations in the BLG theory and a particular modification of it. This entails the inclusion of mass terms for the scalars and the fermion and a flux term [20]. BPS states of the BLG theory have been classified [22] according to world-volume symmetries, namely, SO(1, 2), SO(1, 1) and SO (2). A BPS configuration of the modified theory with SO(1, 2) world-volume symmetry and SO(4) R-symmetry has been studied earlier [9]. In this article we study BPS configurations in three cases with SO(1, 1) and SO(2) world-volume symmetries. We consider BPS configurations in the BLG theory preserving N = (3, 3) supersymmetry with SO(1, 1) world-volume symmetry and SO(3) × SO(3) × SO(2) R-symmetry and obtain a solution to the BPS equations. The solution has scalars diverging at a finite distance of a world-volume coordinate. We then consider the deformed variant with SO(1, 1) world-volume symmetry and SO(4) × SO(4) Rsymmetry preserving N = (4, 0) supersymmetry. Finally we deal with an N = 4 BPS configuration with SO(2) world-volume symmetry and SU (2) × SO(4) R-symmetry. In the former case we obtain a configuration which may be interpreted as a system of intersecting M2-M5-M5-branes, following the popular interpretation of the BLG theory. The other solutions turn out to be combination of domain walls interpolating between pairs of classical vacua. In the latter case the problem, upon choosing an appropriate ansatz, is mapped to the self-dual U (1) 2 Chern-Simons theory with two complex scalar fields. This problem has been studied earlier [27][28][29]. In want of an analytic solution to the BPS equations we present a numerical one corresponding to a single vortex. In considering these examples we find that casting the BPS equations in terms of gauge-invariant variables used earlier [9] furnishes a useful guideline for the choice of ansatze for scalars in the BPS equations.
The article is organized as follows. In the following section we recall some aspects of the modified BLG theory. In section 3 we obtain solution to the BPS configurations with SO(1, 1) symmetry in the world-volume and SO(3) × SO(3) × SO(2) R-symmetry. In section 4 we discuss the SO(1, 1)-invariant BPS configuration with SO(4) × SO(4) R-symmetry and its domain-wall solution. In section 5 we proceed to discuss the SO(2)-invariant BPS configuration having SU (2) × SO(4) R-symmetry, map it to the Chern-Simons theory with two complex scalars and present a numerical solution for an Abelian topological vortex, before concluding in section 6.

BLG theory
Let us begin with a brief description of the BLG theory and its deformation by a background four-form field. The modified BLG theory is an N = 8 supersymmetric theory in 2 + 1-dimensions, given by the Lagrangian where the first term is the original BLG Lagrangian. Here µ = 0, 1, 2 designates the world-volume directions, I = 1, . . . , 8 indexes the flavors and a = 1, 2, 3, 4 the gauge algebra. X I a , Ψ a and A µab are the scalars, the Majorana-Weyl spinor and the gauge field, respectively. The three-and eight-dimensional gamma matrices are denoted γ and Γ, respectively. The ternary bracket of the gauge algebra is denoted as , , , while its structure constants are denoted by f abcd . Repeated indices are summed over in the above expression and in the following unless stated otherwise. Denoting the generators of the ternary algebra as τ a , the metric tensor raising and lowering gauge indices is written as We use the generators to write the fields valued in the ternary algebra as Here D µ denotes the covariant derivative, In the presence of a four-form field G IJKL the BLG Lagrangian is augmented by a mass term and a flux term The four-form field satisfies a self-duality condition where the dual of the four-form field G is defined as The mass m is determined by the four-form field as m 2 = c 2 768 G 2 , with G 2 = G IJKL G IJKL and c is a parameter which is found to be equal to 2 [20]. Thus the BLG theory is recovered in the limit of vanishing c.
The action corresponding to the Lagrangian (2) is under the supersymmetry transformations [20] δX where φ in the transformation of the gauge field represents either a X I or Ψ and θ denotes the parameter of supersymmetry variation, satisfying The supersymmetry transformations close on-shell up to translation and local gauge transformations if the structure constant of the ternary algebra is the rank-four antisymmetric tensor, that is, so that the gauge group is SO(4). The scalars and the fermion transform as vectors of the gauge group SO(4). We thus choose, for example, for all I. For future convenience we have set the level of the Chern-Simons action to be unity, and the metric is taken to be Euclidean, The ternary bracket then reads We shall be concerned with the BPS configurations of the theory with Lagrangian (1). The BPS equation is obtained by setting the supersymmetry variation of the fermion to zero, that is Depending on the subgroup of the R-symmetry as well as the world-volume symmetry to be maintained, the supersymmetry parameter θ is restricted by means of a projector, Ω. Thus, the BPS equations are given by Let us note that only the anti-self-dual combination of the four-form field appears in the last term on the left hand side, linear in X. The R-symmetry in this formulation is realized explicitly in terms of the four-form field as where θ 1 and θ 2 are two parameters of supersymmetry variation. The conserved charged under the global SO(8) symmetry of the BLG theory is given by the R-charge, namely where R IJ is antisymmetric in I and J. We now proceed to study certain BPS configurations of the theory discussed above. Furthermore, the BPS configurations have to satisfy the Gauss constraint, namely where F denotes the field strength corresponding to the gauge fieldÃ.

BPS configuration with
In this section we present a solution to the BPS equations preserving N = (3, 3) supersymmetry in the BLG theory without the mass and the four-form terms, corresponding to the Lagrangian (2). The world-volume has SO(1, 1) symmetry and the R-symmetry is together with a set of Basu-Harvey equations with respect to the world-volume coordinate y, It will be useful to first write the BPS equations in terms of gauge-invariant variables [9]. This furnishes a guideline for the choice of ansatze for the X's. Let us introduce the gauge-invariant fields where indices are raised or lowered with the Euclidean bilinear (17). The gauge-invariants satisfy due to the antisymmetry of the gauge field. Using this and (27) and (28) we obtain from (29) a set of first-order equations for the gauge-invariants and ∂ y Y IJ = 0 for all other I, J. So far our analysis has been completely general. Now Let us assume that the constant Y 's, not appearing in (32), are zero. Also, from the above set of first-order equations we note that it is convenient to choose where i = 1, 2, 3, 4 and S i are the four mutually orthogonal canonical basis vectors of R 4 . Then from (32) it is apparent that X 5 and X 6 are linear combinations of S 1 and S 2 , while X 7 and X 8 are linear combinations of S 3 and S 4 . Using the relations (27) and (28) we can fix the coefficients of these linear combinations and obtain where f and θ are to be determined from the differential equations. From the equation for Y 11 , say, we obtain an equation for f , which is solved to obtain where c = 2 and we have chosen the integration constant to be vanishing. Now, comparing the y f (y) − 1 2 Figure 1: Plot of f (y) for the N = (3, 3) configuration expressions for D y X 5 , D y X 1 and D y X 2 , upon using (34), we obtain implying that θ is a constant. We have thus fixed the solution (34) completely. A sketch of the function f (y) is shown in Figure 1. For this solution all the scalars diverge at a finite distance in the y direction, namely, y = −1/2. Interpretation of this solution in terms of M2-and M5-branes is not clear.

BPS configurations with SO(1, 1) × SO(4) × SO(4) symmetry
In this section we study SO(1, 1)-invariant N = (4, 0) BPS configurations having SO(4) × SO(4) Rsymmetry. The BPS equations are derived for the modified BLG theory by applying the appropriate BPS projection operator on the supersymmetry variation of the fermion and equating it to zero by (20). The generic form of the SO(1, 1) BPS projector is given in terms of 32 × 32 gamma matrices as where α 0 , α 1 , α 2 , α 3 are sign factors assuming values ±1 and γ and Γ designates the gamma matrices defined on the world-volume and transverse to the world-volume, respectively. The chiral projection operator P is defined in terms of Γ 9 = Γ 12···8 as Different choices of the sign factors in (38) give the BPS projection matrix which correspond to breaking R-symmetry in a certain manner [22][23][24]. The projector preserving SO(4) × SO (4) By equation (14) this operator corresponds to the projections the last one being a dependent one. Applying the projection matrix (40) on (20), we obtain the BPS equations. These comprise of differential equations namely for all I = 1, 2, · · · , 8 and a set of modified Basu-Harvey equations, where we have split the flavor indices as i, j, k, l = 1, 2, 3, 4 and p, q, r, s = 5, 6, 7, 8. D t , D x and D y designate the covariant derivatives with respect to the world-volume coordinates (6). Coefficients of the terms linear in X's in equation (43) are determined in terms of the four-form field by η 1 = 3cG 1234 and η 2 = 3cG 5678 with c = 2, as discussed above. We assume η 1 and η 2 to be positive in the following. Our objective here is to find a topological solution to the BPS equations (42), (43) and (23). First let us write down the BPS equations in terms of the gauge-invariant variables Y as before. Multiplying both sides of (43) and (44) with an appropriate X and taking linear combinations, using (31), we obtain equations for the gauge-invariants for i, j = 1, · · · , 4 and r, s = 5, 6, 7, 8, where we defined the gauge-invariant four-form F IJKL = X I a X J b X K c X L d ǫ abcd . Further, using (42) in the Gauss constraint (23) we get while using (43) and (44) to eliminate the covariant y-derivatives in (23) we obtain The modified Basu-Harvey equations have been solved earlier to obtain a domain wall and a fuzzy funnel solution [3,25]. Let us point out that the equations for the scalars are the same for half and quarter BPS configurations with SO(4) × SO(4) R-symmetry. The gauge fields satisfy different equations in these two cases, however. For example, while we have the equation (42) for the quarter-BPS configuration, the half-BPS configurations satisfy D t X I = D x X I = 0 [22]. Hence, it is important to write down the gauge fields explicitly, even if in a special gauge. In order to obtain explicit expressions for gauge field configurations we shall choose simplifying ansatze. The scalar fields X i , i = 1, 2, 3, 4, are chosen to be mutually orthogonal SO(4) vectors, as are X p , p = 5, 6, 7, 8. However, if we choose the basis vectors to be the constant vectors S i as in the last section, the gauge fields will remain undetermined. Hence for the present case we choose a different set of mutually orthogonal basis vectors and express the scalars as where κ, κ ′ = ±1. We shall find solutions corresponding to both signs of κ, κ ′ . The gauge-invariant co-ordinates for this choice are Y ii = f 2 , i = 1, 2, 3, 4, Y pp = g 2 , p = 5, 6, 7, 8. The gauge field is chosen to be of the formÃ in accordance with the aove choice for the scalars, hence breaking the gauge group SO(4) to SO(2) × SO(2). In equation (48) the functions Θ and Φ correspond to the freedom of gauge choice for the residual SO(2) × SO(2) subgroup. For future convenience we shall leave these arbitrary. However, the stress-energy tensor does not depend on them, as we find later. We shall first obtain restrictions on f, g, Θ, Φ and then determine the components of the gauge field in terms of them. We restrict ourselves to stationary solutions. Then all the fields are independent of time. From (45) we obtain equations for f and g, namely We now relate the components of the gauge field to f, g, Θ, Φ. First, using (48) and the equation for X 1 from (43) we obtain, for the first component, and similarly for the first component of the equation for X 3 from (43). Comparing with (50) we obtaiñ This demonstrates the utility of the gauge-invariant equations and justifies keeping Θ and Φ arbitrary in (48). Setting the temporal derivative to zero in (42) we obtain Putting a = 1, 2, resp. b = 2, 1 and usingÃ µ a b = −Ã µ b a and equations (48), we obtain that is, f is independent of x. Similarly, from (44) and (50) we obtain ∂ x g = 0.
In other words, f and g are functions of y only. This leads toÃ Thus the gauge fields are determined in terms of Θ and Φ. Using (52), (42) and (55), the Gauss constraint (23) yields ∂ yÃt while (47) yields ∂ xÃt so thatÃ t is a function of y alone. Now, eliminating the combination f 2 +g 2 between the two equations in (56) we obtain ∂ y (Ã t leading to the conclusion that the squares of the components of the gauge field A t may differ by a constant only. By linearly combining the equations (56) we can cast them as first order differential equations for the combinationsÃ t which are solved to obtainÃ where A 0 and B 0 are constants. This solution satisfies (58). The solution to (50) is where a, a ′ are constants of integration. Different configurations ensue from the choices of κ, κ ′ . By allowing the supersymmetry variation of L to vanish [20], the mass parameter in the scalar term in L mass gets related to the four-form field G IJKL as resulting in m 2 = 9c 2 G 2 , where G 2 = G 1234 G 1234 = G 5678 G 5678 . Using this value of m and the ansatz for the scalars (48), the scalar potential obtained from the sextic term in L BLG , the scalar term in L mass along with L flux is with η 1 , η 2 > 0. The classical vacua are therefore, The solution (61) with κ = κ ′ = 1 interpolates between the vacua V I and V II as y varies from −∞ to ∞. Taking into account that the solution is independent of x, this is therefore a domain wall solution. By (60), the temporal component of the gauge field is However, in order to keepÃ t finite in the whole domain of y, we have to set A 0 to zero. Thus, finally, the two components ofÃ t are given bỹ where we have taken the difference of their squares to be vanishing. Having thus obtainedÃ t ,Ã x and A y are determined, up to gauge transformation, by (55) and (52), respectively. The enrgy-momemtum tensor is obtained by varying the Lagrangian L with respect to the world-volume metric, where the potential V is given by (63). Energy density of the configuration obtained above is given by T tt . Plugging in the solutions for the scalars, (61), and the gauge fields, (52), (55) and (66) in the expression for T tt , we obtain the energy density to be For κ = κ ′ = −1, the f and g and hence the gauge invariants Y ii and Y rr diverge at y = − 1 η1 ln a and y = − 1 η2 ln a ′ , respectively. The corresponding solutions for the gauage fieldsÃ t 1 2 andÃ t 3 4 are given byÃ where C 0 and D 0 are constants. In accordance with (41) the solution thus describes an M2-brane ending on two M5-branes with world-volumes spanning directions 1, 2, 3, 4 and 5, 6, 7, 8, respectively and sharing the directions x and t with the M2-brane [26]. Thus we obtained a quarter-BPS configuration of the mass deformed BLG theory given by a bound state of M2-M5-M5-branes.

BPS configuration with SO(2) × SU (2) × SO(4) symmetry
In this section we consider N = 4 BPS configurations in the modified BLG theory with world-volume symmetry SO (2). We find that upon choosing a certain ansatz the equations for the scalars reduce to the scalar equations of a self-dual U (1) 2 Chern-Simons theory with two complex scalars. Topological configurations in the latter case have been investigated in earlier [27][28][29]. However, no analytic solution to the equations appears to be known. We shall consider special cases in which we obtain certain solutions.
As before, to write down the SO(2)-invariant BPS equations we project the supersymmetry variation of the fermion with the SO(2) invariant BPS projector. In terms of the 32 × 32 gamma matrices the projector is where α 1 , α 2 and α 3 are sign factors ±1 and P denotes the chiral projection matrix as before (39). We shall consider the situation in which the R-symmetry is broken to SU (2) × SO(4). The projector (70) assumes the form corresponding to the choice of the α's as {+, +, +}, {+, +, −}. The projector corresponds to or, equivalently, Γ 1234 θ = −θ. For simplicity, we set the four scalars X 5 , X 6 , X 7 , X 8 to zero and write the BPS equations for the non-zero fields only. This reduction in the flavour degrees of freedom breaks the R-symmetry further to SU (2). The BPS equations in terms of the non-vanishing scalars, along with Using (31) these can be written in terms of the gauge-invariant variables Y . From (74) we obtain and These gauge-invariant equations provide restrictions on the choice of ansatz for the scalar fields. For stationary configurations the time derivatives are set to zero and these equations yield relations between the Y 's. In particular, they imply that Y 12 = Y 34 = 0, meaning, X 1 , X 2 are mutually orthogonal, as are X 3 , X 4 . They also require Y 11 = Y 22 and Y 33 = Y 44 . An ansatz satisfying these relations compatible with the remaining SU (2) R-symmetry is For this choice For this ansatz the gauge group breaks down to U (1) 2 . We retain the ansatz (49) for the gauge fieldÃ µ . Let us introduce complex combinations, We also define A z =Ã z 1 2 and B z =Ã z 3 4 for the spatial components of the gauge field and A t =Ã t 1 2 and B t =Ã t 3 4 for the temporal parts. The gauge-invariant variables are, then In terms of these complex quantities the BPS equations (73) are written as which can be solved to express the gauge fields to be expressed as in terms of the scalars as Similarly, assuming stationarity the equations (74) relate the temporal part of the gauge field to the complex scalars φ and χ, namely, Using (81) and (82) we can now rewrite the action (2) and the Gauss constraint (23) in terms of the complex scalars φ and χ. The Lagrangian reads while the Gauss constraint leads to two differential equations for the scalars, namely, where ∇ 2 = ∂ z ∂z. Defining rescaled fields these two equations take the form where λ = 2η 2 1 . These coupled elliptic partial differential equations have been studied in the context of U (1) 2 self-dual Chern-Simons theory with two complex scalar fields [27,28]. In particular, existence of topological vortex solutions, characterized by the boundary conditions |φ| → 1, |χ| → 1, as |z| → ∞, have been established [27]. For a single vortex at the origin, the solution is also proved to be unique [28]. However, no explicit analytic construction of vortex solutions seem to exist in literature. The conserved R-charges for the SO(2) × SU (2) BPS configuration are given by while the total energy of the configuration is given by

A special case
Let us consider the special case of the BLG theory without the four-form field, corresponding to the Lagranigan L BLG . Putting η 1 = 0 in (84) we obtain the Gauss constraint equations for this case as Subtracting these we obtain Adding the equations (89), on the other hand, we obtain a Liouville-like equation for ρ = |φχ| 2 , namely which is solved by where ξ(z) is an analytic function of z. From (90) and (92) we conclude that both |φ| 2 and |χ| 2 are proportional to dξ/dz 1−|ξ(z)| 2 , modulo analytic functions. Thus, |φ| 2 , |χ| 2 , hence Y II , I = 1, 2, 3, 4, are singular on the curve ξ(z) = 1. Given a ξ, this corresponds to two M2-brane spikes extended along 1-2 and 3-4 directions corresponding to the two U (1) factors of the gauge group on the original M2-brane of the BLG theory lying on the z-plane [26].

Numerical solution
While general closed form solutions to (86) are not known, we can solve the equations numerically. Here we present a numerical solution for the simplest case of a single vortex. Uniqueness of the solution for this case has been established earlier [28]. Using the SO(2) world-volume symmetry of the BPS configuration let us now write z = re iθ and drop the θ-dependence of all the functions. Writing |φ| 2 = e ρ(r) and |χ| 2 = e σ(r) the equations (86) become ρ ′′ (r) + 1 r ρ ′ (r) + 2e σ(r) (η 1 − e ρ(r) ) = 0, σ ′′ (r) + 1 r σ ′ (r) + 2e ρ(r) (η 1 − e σ(r) ) = 0, where a prime denotes a derivative with respect to r. To solve these equations numerically to obtain vortex solutions we have to impose two boundary conditions on each of ρ and σ. The first set of asymptotic boundary conditions are chosen as ρ(r) → 0 and σ(r) → 0 as r → ∞. The second set of boundary conditions arise from the quantization of magnetic fluxes corresponding to the gauge fields A z and B z , namely, dz ∧ dzF zz = 2πN 1 and dz ∧ dzF ′ zz = 2πN 2 , where N 1 and N 2 are integers representing vorticity. In the limit r → ∞ the flux quantization conditions written in terms of ρ(r) and σ(r) require Solutions for ρ and σ can now be obtained numerically. A plot of |φ| 2 and the corresponding gauge field A θ (in polar cordinates) is shown in Figure 2 with unit vorticity for both vortices and η 1 = 1. The plots of |χ| and B r are similar.

Summary
To summarize, we have studied BPS configurations of the BLG theory with and without the mass and four-form deformations. We considered three cases of interest. In the first case the solution with worldvolume symmetry SO(1, 1) preserving N = (3, 3) supersymmetry in the absence of any deformation has eight scalars which blow up at a finite value of the world-volume coordinate y. We then considered a quarter BPS configuration with SO(4) × SO(4) R-symmetry. In this case there are two types of solutions. One of them is a pair of domain walls each extending along four directions in agreement with the R-symmetry. The other solution features the M2-brane merging into two M5-branes at a finite distance in y, the latter intersecting along the x direction. This has been interpreted as a system of intersecting M2-M5-M5-branes [30]. Finally we considered a configuration with SO(2) symmetry in the world-volume and SU (2) × SO(4) R-symmetry. We chose to turn off the four scalar corresponding to the SO(4). By choosing an appropriate ansatz for the scalars and the gauge field, the system maps into the self-dual U (1) 2 Chern-Simons theory with two complex scalars. Existence of vortex solutions to these equations has been established earlier. We presented a solution for the special case with no deformation, giving rise to a Liouville-like equation. We also presented a numerical solution for the single topological vortex, which is known to be unique. In dealing with the system of BPS equations we found that expressing them in terms of the gauge-invariant variables introduced earlier [9] appears to be of immense help in the choice of ansatze for the solutions in all cases. Other cases in the classification of BPS configurations of the BLG theory may also be considered in a similar fashion. However, the solutions for those are given either by constant scalars or combinations of domain walls or the singular solutions of section 3.