5.2 Intersecting M2-branes

As a first example of an excited configuration, consider the intersection of two M2-branes in a point corresponding to the array
M 2 : 1 2 x x x x x x x x (252 ) M 2 : x x 3 4 x x x x x x .
In the probe approximation, the M2-brane effective action describes the first M2-brane by fixing the static gauge and the second M2-brane as an excitation above this vacuum by turning on two scalar fields (X3, X4 ) according to the ansatz
X μ = σ μ , Xi = ci, X3 (σa) ≡ y(σa ) , X4 (σa) ≡ z(σa), (253 )
where a runs over the spatial world volume directions and i over the transverse directions not being excited.

Supersymmetry analysis:
Given the ansatz (253View Equation), the induced metric components equal 𝒒 = − 1, 𝒒 = 0, 𝒒 = δ + ∂ Xr ∂ Xs δ 00 0a ab ab a b rs (with r,s = 3,4), whereas its determinant and the induced gamma matrices reduce to

− det𝒒 = 1 + |βƒ—∇y |2 + |βƒ—∇z |2 + (βƒ—∇y × βƒ—∇z )2, (254 ) γ = Γ , γ = Γ + ∂ Xr Γ . (255 ) 0 0 a a a r
Altogether, the kappa symmetry preserving condition (214View Equation) is
√ -------- ( ) − det𝒒 πœ– = Γ 012 + πœ€ab∂ay ∂bzΓ 034 − πœ€ab∂axrΓ 0br πœ–. (256 )
If the excitation given in Eq. (253View Equation) must describe the array in Eq. (252View Equation), the subspace of Killing spinors πœ– spanned by the solutions to Eq. (256View Equation) must be characterised by two projection conditions
Γ πœ– = Γ πœ– = πœ–, (257 ) 012 034
one for each M2-brane in the array (252View Equation). Plugging these projections into Eq. (256View Equation)
(√ -------- ab ) ab r − det𝒒 − (1 + πœ€ ∂ay∂bz) πœ– = πœ€ ∂aX Γ 0brπœ–, (258 )
one obtains an identity involving two different Clifford-valued contributions: the left-hand side is proportional to the identity matrix acting on the Killing spinor, while the right-hand side involves some subset of antisymmetric products of gamma matrices. Since these Clifford valued matrices are independent, each term must vanish independently. This is equivalent to two partial differential equations
∂2y = − ∂1z , ∂1y = ∂2z. (259 )
Notice this is equivalent to the holomorphicity of the complex function U (σ+) = y + iz in terms of the complex world space coordinates σ± = σ1 ± iσ2, since Eqs. (259View Equation) are equivalent to the Cauchy–Riemann equations for + U(σ ).

When conditions (259View Equation) are used in the remaining left-hand side of Eq. (258View Equation), one recovers an identity. Thus, the solution to Eq. (214View Equation) in this particular case involves the two supersymmetry projections (257View Equation) and the BPS equations (259View Equation) satisfied by holomorphic functions U (σ+).

Hamiltonian analysis:
Since this is the first non-trivial example of a supersymmetric soliton discussed in this review, it is pedagogically constructive to rederive Eqs. (259View Equation) from a purely Hamiltonian point of view [225Jump To The Next Citation Point]. This will also convince the reader that holomorphicity is the only requirement to be on-shell. To ease notation below, rewrite Eq. (259View Equation) as

βƒ—∇y = ⋆∇βƒ—z , (260 )
where standard vector calculus notation for ℝ2 is used, i.e., βƒ—∇ = (∂1,∂2) and ⋆∇βƒ— = (∂2,− ∂1). Consider the phase space description for the M2-brane Lagrangian given in Eq. (226View Equation) in a Minkowski background. The Lagrange multiplier fields a s impose the world space diffeomorphism constraints. In the static gauge, these reduce to
Pa = PI ⋅ ∂aXI , (261 )
where PI are the conjugate momenta to the eight world volume scalars I X describing transverse fluctuations. For static configurations carrying no momentum, i.e., PI = 0, the world space momenta will also vanish, i.e., Pa = 0.

Solving the Hamiltonian constraint imposed by the Lagrange multiplier λ for the energy density β„° = P0, one obtains [225Jump To The Next Citation Point]

2 2 2 2 2 2 (β„°βˆ•TM2 ) = 1 + |βƒ—∇y | + |βƒ—∇z | + (βƒ—∇y × βƒ—∇z ) = (1 − βƒ—∇y × βƒ—∇z ) + |∇βƒ—y − ⋆∇βƒ—z | . (262 )
This already involves the computation of the induced world space metric determinant and its rewriting in a suggestive way to derive the bound
β„°βˆ•TM2 ≥ 1 + |βƒ—∇y × βƒ—∇z |. (263 )
The latter is saturated if and only if Eq. (260View Equation) is satisfied. This proves the BPS character of the constraint derived from solving Eq. (214View Equation) in this particular case and justifies that any solution to Eq. (260View Equation) is on-shell, since it extremises the energy and there are no further gauge field excitations.

Integrating over the world space of the M2 brane allows us to derive a bound on the charges carried by this subset of configurations

E ≥ E0 + |Z|. (264 )
E0 stands for the energy of the infinite M2-brane vacuum, whereas Z is the topological charge
∫ i ∫ Z = TM2 dy ∧ dz = TM2 -- dU ∧ dU¯ , (265 ) M2 2 M2
accounting for the second M2-brane in the system.

The bound (264View Equation) matches the spacetime supersymmetry algebra bound: the mass (E ) of the system is larger than the sum of the masses of the two M2-branes. Field theoretically, the first M2-brane charge corresponds to the vacuum energy (E ) 0, while the second corresponds to the topological charge (Z ) describing the excitation. When the system is supersymmetric, the energy saturates the bound E = E0 + |Z | and preserves 1/4 of the original supersymmetry. From the world volume superalgebra perspective, the energy is always measured with respect to the vacuum. Thus, the bound corresponds to the excitation energy E − E0 equalling |Z |. This preserves 1/2 of the world volume supersymmetry preserved by the vacuum, matching the spacetime 1/4 fraction.

For more examples of M2-brane solitons see [95] and for a related classification of D2-brane supersymmetric soltions see [33].

  Go to previous page Go up Go to next page