Non-commutative M-branes from Open Pure Spinor Supermembrane

Open supermembrane with a constant three-form flux in the pure spinor formalism is examined. The BRST symmetry of the open supermembrane action leads to non-commutative (NC) M-branes. In addition to the NC M5-brane with a self-dual two-form flux, we find a NC M9-brane with an electric flux and a NC M9-brane with a magnetic flux. The former reduces in the critical electric flux limit to an M2-brane on the M9-brane, while the latter reduces in the strong magnetic flux limit to infinitely many Kaluza-Klein monopoles dissolved into the M9-brane. These NC M-branes are shown to preserve a half of 32 supersymmetries.

found that the self-duality of the two-form gauge field on the M5-brane world-volume follows from κ-symmetry of the open supermembrane [9]. In addition, intersecting NC M-branes are discussed in [10,11].
In this paper we examine an open supermembrane with a constant three-form flux in the pure spinor formalism [12]. In the pure spinor formalism, the κ-symmetry is replaced with the BRST symmetry. One of advantages of our approach is to consider BRST symmetry which is expected to survive quantum corrections. It implies that our analysis in this paper may give a quantum consistency check for the κ-symmetry arguments. In addition, we will derive a NC M9-brane with an electric flux and a NC M9-brane with a magnetic flux. The former reduces in the critical electric flux limit to an M2-brane on the M9-brane. The latter reduces in the strong magnetic flux limit to infinitely many KK monopoles dissolved into the M9-brane, and is identified with a bound state of an M9-brane and KK monopoles.
We will examine the supersymmetry variation of an open supermembrane in the pure spinor formalism, and find that the boundary condition for the BRST symmetry solves those for supersymmetry. This shows that the NC M-branes derived in this paper preserve a half of 32 supersymmetries and should be 1/2 BPS objects. This paper is organized as follows. In the next section, we introduce an open supermembrane action with constant fluxes in the pure spinor formalism, and give the BRST transformation law and the supersymmetry transformation law under which the action is invariant. We derive surface terms of the BRST transformation and deduce boundary conditions to eliminate them in section 3. In section 4, as solutions of the boundary conditions, we obtain NC M-branes; a NC M5-brane with self-dual three-form fluxes, a NC M9-brane with an electric flux and a NC M9-brane with a magnetic flux. In addition, the NC Mbranes are shown to be half supersymmetric, namely 1/2 BPS. The last section is devoted to summary and discussions. In the appendix A, we give a derivation of NC M5-branes, and we describe a derivation of surface terms for the supersymmetry transformation of the action in the appendix B.

Pure Spinor Supermembrane with constant threeform fluxes
Before introducing the pure spinor supermembrane action, we introduce the κ-symmetric supermembrane action in d = 11. It is composed of two parts [1] where x µ (µ = 0, 1, · · · , 9, 10) are flat spacetime coordinates, and τ i (i = 0, 1, 2, and I = 1, 2 are space indices) are coordinates on the world-volume Σ. We have introduced The Γ µ denote 32 × 32 gamma matrices and θ α is a 32-component Majorana spinor in d = 11. We defineθ byθ = θ T C with the charge conjugation matrix C so that (CΓ µ 1 ···µn ) αβ is symmetric under the exchange α ↔ β iff n = 1, 2 mod 4. The e 0 and e I are Lagrange multipliers for reparametrization constraints. By eliminating P µ by its equation of motion, L 0 reduces to the Nambu-Goto Lagrangian − det Π µ i Π jµ [13]. We have introduced a constant three-form flux H = C − db in L WZ where C and b denote the three-form gauge potential and the two-form gauge field on the boundary brane, respectively. It should be noted that the action (2.1) is κ-symmetric and supersymmetric even in the presence of H.
In the pure spinor formalism, κ-symmetry is replaced with BRST symmetry. The supermembrane action in the pure spinor formalism [12] is given as . Define the momentum conjugate to θ α by ‡ p α ≡ ∂ r L ∂θ α , and then we introduce d α as 32 fermionic constraints The Grassmann-even spinor fields (λ α , w α ) are pure spinor ghosts. The Lagrange multiplier e 0 has been set to −1/2.
The supermembrane action (2.4) is invariant under the supersymmetry transformations whereP µ and d α are defined to be invariant under supersymmetry transformations.
The BRST operator § is defined by Q ≡ λ α d α which acts on a field f by Qf = i{Q, f ].
The BRST invariance of S pure can be shown [12] by following the method used in [16].
First, we note that whereS denotes S in (2.1) with the replacements P µ →P µ and e 0 → −1/2. It is convenient for us to introduce a Grassmann-odd parameter ε to the BRST transformations: δf = εQf , As in [12], one may show that 10) ‡ The derivative with a superscript r denotes the right derivative. § The double spinor formalism [14] sheds some light on a derivation of the BRST operator for the pure spinor supermembrane.
¶ For further discussion on this issue see [12] [15]. and that Gathering these together, we may conclude that S pure is BRST invariant will restore e 0 as a Lagrange multiplier, and we consider the bosonic part of (2.1) which is classically equivalent to the Nambu-Goto action [13]. Varying it with respect to x µ , we obtain the surface term where n i is the unit vector normal to ∂Σ.
We will turn on a constant H along the world-volume of the Dirichlet p-brane of the open supermembrane. In order to eliminate the surface term, we must impose either of the following boundary conditions [17]: the Neumann boundary condition ∂ n xμ+Hμνρ∂ t xν∂ τ xρ = 0 for Neumann directions xμ a (a = 0, 1, · · · , p), or the Dirichlet boundary condition

BRST surface terms
Now we shall consider the surface terms of the BRST transformation. For the BRST symmetry to be unbroken in the presence of the boundary, these surface terms must be eliminated by appropriate boundary conditions on the fermionic variables.
We find that the surface terms of the BRST transformation of S pure come from the WZ term and take the form where we have introduced ξ by ξ ≡ ελ, and L (n) denotes the n-th order terms in ξ as well as θ. It is worth noting that if we set ξ = δ κ θ, the above surface term coincides with that of the κ-symmetry transformation of the κ-symmetric open supermembrane action examined in [3,9,10,11]. For the present paper to be self-contained, we will derive conditions on θ and ξ for the surface terms to be deleted.
The third equality follows from the anti-symmetricity of three indices of ǫ ijk .
Next we consider L (2) in (3.3). The bosonic boundary condition implies that L (2) = − i 2 (HμνρθΓρξ +θΓμνξ)∂ t xμ∂ τ xν, and then we require that We demand that the boundary condition on ξ is the same as that on θ. This is because the BRST transformation (2.9) relates them each other as δθ = ξ. It implies that the BRST symmetry is preserved even in the presence of the boundary. Before solving the boundary condition (3.8), we consider L (4) in (3.4). The bosonic boundary condition reduces it to The relation (3.8) makes the first line of the form 1 4 Hμνρ(θΓρξ ·θΓμ∂ t θ +θΓρ∂ t θ ·θΓμξ)∂ τ xν − (t ↔ τ ), which vanishes due to the anti-symmetricity of the three indices of Hμνρ. As a result, for L (4) = 0 we requireθ Summarizing, the surface terms should disappear if (3.8) and either of two equations in (3.10) are satisfied.

Non-commutative M-branes
In this section, we will fix the fermionic boundary conditions which solve (3.8) and (3.10).
We shall impose the same boundary condition on θ and ξ where M is the gluing matrix. This is because the BRST transformation relates them each other as δθ = ξ. In addition to the NC M5-brane obtained in [9], two kinds of NC M9-branes will be presented below.

Non-commutative M5-brane
Here we present the boundary condition for a NC M5-brane which solve (3.8) and (3.10).
Because the derivation of the boundary condition is similar to those given in [9], we put it in the appendix A.
A gluing matrix and fluxes for a NC M5-brane are M = e ϕΓ 345 Γ 01···5 , and H 012 = sin ϕ and H 345 = tan ϕ. As explained in the appendix A, the fluxes satisfy which is nothing but the self-duality condition [18] for the two-form gauge field on the M5brane world-volume [19]. We note that the self-duality condition for the two-form gauge fields has been derived from the BRST symmetry of an open supermembrane.
On the other hand for ϕ → π/2, the gluing matrix reduces to M → Γ 012 , while fluxes are H 012 → 1 and H 345 → ∞. It seems that this may describe an M2-brane with a critical flux H 012 = 1. However this limit is nothing but the OM limit [20], so that this M2-brane should The gluing matrix M = e ϕΓ 012 Γ 01···5 with fluxes H 345 = sinh ϕ and H 012 = tanh ϕ gives a different parametrization of the NC M5-brane above.
be one of infinitely many M2-branes dissolved into the M5-brane * * . This is consistent with the fact that there must be the M5-brane for the charge conservation [22]. Consequently the NC M5-brane should be regarded as a bound state of an M5-brane and M2-branes.

Non-commutative M9-branes
We will consider two types of non-commutative M9-branes. We may choose {0, 1, · · · , 9} as the M9-brane world-volume directions without loss of generality. The Dirichlet direction is the 10-th direction which is denoted as ♮ below to avoid confusion.

Non-commutative M9-brane with an electric flux
First we consider the following gluing matrix which reduces to the gluing matrix for an M9-brane when h 1 = 0. We shall turn on H 012 , and examine we deriveθ

Non-commutative M9-brane with a magnetic flux
Next, we shall consider the following gluing matrix and turn on H 789 . Sinceθ we deriveθ  and the flux diverges H 789 → ∞. It seems that this describes a 6-brane, but there is no seven-form gauge potential in eleven-dimensions. In ten-dimensions, however, we have a RR seven-form gauge potential C 7 which is dual to a RR one-form gauge potential C 1 . The gauge potential C 7 couples to a D6-brane which is characterized by a harmonic function H on the space E 3 transverse to the world-volume E 1,6 . An eleven-dimensional lift of the D6-brane is known as a KK monopole which is magnetically charged with respect to C 1 and takes the form [6] ds 2 =ds 2 (E 1,6 ) + H(y)dy i dy i + H(y) −1 (dz + dy i C i (y)) 2 ,  For an open supermembrane we are considering, the partial integration leaves a surface term which has to be deleted by an appropriate boundary condition for some amount of supersymmetry to be preserved. We find that the surface terms come from the Wess-Zumino term (2.3) and take the form

Supersymmetry of non-commutative M-branes
A derivation of them was given in the appendix B. Now, we shall compare these surface terms with the BRST surface terms in (3.3),(3.4) and (3.5). We found that It implies that the boundary conditions which eliminate the BRST surface terms also eliminate the surface terms of supersymmetry † † . As a result, we may conclude that the NC M-branes obtained in the sections 4.1 and 4.2 would preserve a half of 32 supersymmetries so that they are 1/2 BPS objects.

Summary and discussions
We arguments. This implies that our results may give a quantum consistency check for the previous ones. † † For the κ-symmetric open supermembrane, see [3], where it was shown that the boundary conditions to eliminate the κ-symmetry surface term will preserve a half of the supersymmetries.
For the NC M9-brane with an electric flux characterized by (4.9), we assumed that there should be an M9-brane behind the M2-brane in the critical flux limit. To confirm this point we need to know the M9-brane effective action which describes the coupling to the M2-brane.
It is interesting for us to clarify this point. constant fluxes in the pure spinor formalism. As for D-branes in AdS 5 ×S 5 , we examined them from the BRST symmetry of the pure spinor superstring in [28]. Furthermore recently world-sheet supersymmetries are examined in [29]. It is also interesting to examine NC D-branes by using the pure spinor superstring with two-form fluxes. We expect that this analysis may support the previous results obtained from the κ-symmetry arguments [30].
(A.8) This is nothing but the self-duality condition [18] for the two-form gauge field on the M5brane world-volume [19]. It is straightforward to see that the latter condition in

B Supersymmetry surface term
We shall derive the supersymmetry surface terms (4.19) and (4.20) in section 4.3 below.
The surface term L SUSY given in (4.19) may be derived from terms contained in δ ǫ L WZ which are the four-th order terms in fermions as follows In the second equality we have utilized the Fierz identity, and for the third equality partial integration has been performed for ∂ iθ . Similarly, examining terms contained in δ ǫ L WZ which are six-th order in fermions, we derive (4.20) as follows.