4.2 Hamiltonian formalism

In this subsection, I review the Hamiltonian formalism for brane effective actions. This will allow us not only to compute the energy of a given supersymmetric on-shell configuration solving Eq. (214View Equation), but also to interpret the constraints i fj(ϕ ) = 0 as BPS bounds [107Jump To The Next Citation Point, 429Jump To The Next Citation Point]. This will lead us to interpret these configurations as brane-like excitations supported on the original brane world volume.

The existence of energy bounds in supersymmetric theories can already be derived from purely superalgebra considerations. For example, consider the M-algebra (171View Equation). Due to the positivity of its left-hand side, one derives the energy bound

P0 ≥ f(Pi,Zij,Yi1...i5; Z0i,Y0i1...i4), (215 )
where the charge conjugation matrix was chosen to be 0 C = Γ and the spacetime indices were split as m = {0, i}. For simplicity, let us set the time components Y0i1...i4 and Z0i to zero. The superalgebra reduces to
0 ¯ ¯ 0 −1 0i 1- 0ij 1- 0i1...i5 {Q, Q } = P (1 + Γ ), with Γ = (P ) [Γ Pi + 2Γ Zij + 5!Γ Yi1...i5]. (216 )
The bound (215View Equation) is now equivalent to the statement that no eigenvalue of ¯2 Γ can exceed unity. Any bosonic charge (or distribution of them) for which the corresponding ¯Γ satisfies
¯Γ 2 = 𝟙 , (217 )
defines a projector 1 2(𝟙 + ¯Γ ). The eigenspace of ¯Γ with eigenvalue 1 coincides with the one spanned by the Killing spinors 𝜖∞ determining the supersymmetries preserved by supergravity configurations corresponding to individual brane states. In other words, there is a one-to-one map between half BPS branes, the charges they carry and the precise supersymmetries they preserve. This allows one to interpret all the charges appearing in ¯ Γ in terms of brane excitations: the 10-momentum Pi describes d = 11 massless superparticles [93], the 2-form charges Zij M2-branes [90, 91], whereas the 5-form charges Yi1...i5, M5-branes [464]. This correspondence extends to the time components {Y0i1...i4, Z0i}. These describe branes appearing in Kaluza–Klein vacua [311Jump To The Next Citation Point, 481]. Specifically, Y0i ...i 1 4 is carried by type IIA D6-branes (the M-theory KK monopole), while Z0i can be related to type IIA D8-branes.

That these algebraic energy bounds should allow a field theoretical realisation is a direct consequence of the brane effective action global symmetries and Noether’s theorem [406, 407]. If the system is invariant under time translations, energy will be preserved, and it can be computed using the Hamiltonian formalism, for example. Depending on the amount and nature of the charges turned on by the configuration, the general functional dependence of the bound (215View Equation) changes. This is because each charge appears in ¯Γ multiplied by different antisymmetric products of Clifford matrices. Depending on whether these commute or anticommute, the bound satisfied by the energy P0 changes, see for example a discussion on this point in [394Jump To The Next Citation Point]. Thus, one expects to be able to decompose the Hamiltonian density for these configurations as sums of the other charges and positive definite extra terms such that when they vanish, the bound is saturated. More precisely,

  1. For non-threshold bound states, or equivalently, when the associated Clifford matrices anticommute, one expects the energy density to satisfy
    2 2 2 ∑ (i j )2 ℰ = 𝒵 1 + 𝒵 2 + tfi(ϕ ) . (218) i
  2. For bound states at threshold, or equivalently, when the associated Clifford matrices commute, one expects
    2 2 ∑ ( i j )2 ℰ = (𝒵1 + 𝒵2 ) + tfi(ϕ ) . (219) i

In both cases, the set {ti} involves non-trivial dependence on the dynamical fields and their derivatives. Due to the positivity of the terms in the right-hand side, one can derive lower bounds on the energy, or BPS bounds,

∘ --------- ℰ ≥ 𝒵2 + 𝒵2 (220 ) 1 2 ℰ ≥ |𝒵1 | + |𝒵2| (221 )
being saturated precisely when f (ϕj) = 0 i are satisfied, justifying their interpretation as BPS equations [107, 429]. Thus, saturation of the bound matches the energy ℰ with some charges that may usually have some topological origin [165Jump To The Next Citation Point].

In the current presentation, I assumed the existence of two non-trivial charges, 𝒵1 and 𝒵2. The argument can be extended to any number of them. This will change the explicit saturating function in Eq. (215View Equation) (see [394]), but not the conceptual difference between the two cases outlined above. It is important to stress that, just as in supergravity, solving the gravitino/dilatino equations, i.e., δℱ = 0, does not guarantee the resulting configuration to be on-shell, the same is true in brane effective actions. In other words, not all configurations solving Eq. (214View Equation) and saturating a BPS bound are guaranteed to be on-shell. For example, in the presence of non-trivial gauge fields, one must still impose Gauss’ law independently.

After these general arguments, I review the relevant phase space reformulation of the effective brane Lagrangian dynamics discussed in Section 3.

4.2.1 D-brane Hamiltonian

As in any Hamiltonian formulation31, the first step consists in breaking covariance to allow a proper treatment of time evolution. Let me split the world volume coordinates as σμ = {t, σi} for i = 1,...,p and rewrite the bosonic D-brane Lagrangian by singling out all time derivatives using standard conjugate momenta variables

ℒ = X˙mPm + V˙iEi + ψ˙TDp − H . (222 )
Here Pm and i E are the conjugate momentum to m X and Vi, respectively, while H is the Hamiltonian density. ψ is the Hodge dual of a p-form potential introduced in [94Jump To The Next Citation Point] to generate the tension T Dp dynamically [86, 356]. It is convenient to study the tensionless limit in these actions as a generalisation of the massless particle action limit. It was shown in [94] that H can be written as a sum of constraints
H = ψi𝒯i + Vt𝒦 + siℋi + λ ℋ , (223 )
where
𝒯i = − ∂iTDp, &tidle; i p+1 ℱ 𝒦 = − ∂iE + (− 1) TDp𝒮 with 𝒮 = ∗(ℛe )p, ℋi = P&tidle;aEai + E&tidle;j ℱij with Eai = Eam ∂iXm, 1 ℋ = --[P &tidle;2 + &tidle;EiE&tidle;j 𝒢ij + T 2Dpe−2ϕdet (𝒢ij + ℱij)]. (224 ) 2
The first constraint is responsible for the constant tension of the brane. It generates abelian gauge transformations for the p-form potential generating the tension dynamically. The second generates gauge field transformations and it implements the Gauss’ law constraint 𝒦 = 0. Notice its dependence on ℛ, the pullback of the RR field strengths R = dC − C ∧ H3, coming from the WZ couplings and acting as sources in Gauss’ law. Finally, ℋa and ℋ generate world-space diffeomorphisms and time translations, respectively.

The modified conjugate momenta 𝒫a and &tidle;Ei determining all these constraints are defined in terms of the original conjugate momenta as

( ) &tidle;Pa = Eam (Pm + EiZ ⋆(imB )i + TDp𝒞m ), with 𝒞m = ∗ Z⋆(imC ) ∧ eℱ , p &tidle;Ei = Ei + T𝒞i , with 𝒞i = [∗ (𝒞e ℱ)p−1]i. (225 )
⋆ Z (imB )i stands for the pullback to the world volume of the contraction of B2 along the vector field m ∂ ∕∂X. Equivalently, ⋆ n Z (imB )i = ∂iX Bmn. ⋆ Z (imC ) is defined analogously. Notice ⋆ stands for the Hodge dual in the p-dimensional D-brane world space.

In practice, given the equivalence between the Lagrangian formulation and the one above, one solves the equations of motion on the subspace of configurations solving Eq. (214View Equation) in phase space variables and finally computes the energy density of the configuration P0 = ℰ by solving the Hamiltonian constraint, i.e., ℋ = 0, which is a quadratic expression in the conjugate momenta, as expected for a relativistic dynamical system.

4.2.2 M2-brane Hamiltonian

The Hamiltonian formulation for the M2-brane can be viewed as a particular case of the analysis provided above, but in the absence of gauge fields. It was originally studied in [88]. One can check that the full bosonic M2-brane Lagrangian is equivalent to

[ ] ℒ = ˙XmP − si &tidle;P Ea − 1λ &tidle;P2 + T 2 det𝒢 , (226 ) m a i 2 M2 ij
where the modified conjugate momentum P&tidle;a is related to the standard conjugate momentum Pm by
m ⋆ &tidle;Pa = E a (Pm + TM2 𝒞m) with 𝒞m = ∗(Z (imC3 )), (227 )
where ∗ describes the Hodge dual computed in the 2-dimensional world space spanned by i,j = 1,2. Notice no dynamically-generated tension was considered in the formulation above.

As before, one usually solves the equations of motion a δℒ ∕δs = δℒ ∕δv = 0 in the subspace of phase space configurations solving Eq. (214View Equation), and computes its energy by solving the Hamiltonian constraint, i.e., δℒ ∕δλ = 0.

4.2.3 M5-brane Hamiltonian

It turns out the Hamiltonian formulation for the M5-brane dynamics is more natural than its Lagrangian one since it is easier to deal with the self-duality condition in phase space [92Jump To The Next Citation Point]. One follows the same strategy and notation as above, splitting the world volume coordinates as σμ = {t, σi} with i = 1,...5. Since the Hamiltonian formulation is expected to break SO (1,5 ) into SO (5), one works in the gauge 0 a = σ = t. It is convenient to work with the world space metric 𝒢ij and its inverse ij 𝒢 532. Then, the following identities hold

ij 1 ijk1k2k3 H&tidle; = -√-------𝜀 ℋk1k2k3 , 6 det 𝒢5 ij det(𝒢μν + &tidle;H μν) = (𝒢00 − 𝒢0i𝒢5 𝒢0j)det 5(𝒢 + H&tidle;) , (228 )
where det𝒢5 is the determinant of the world space components 𝒢ij, 5 &tidle; &tidle; det (𝒢 + H ) = det(𝒢ij + Hij ) and H&tidle;ij = 𝒢ik𝒢jlH&tidle;kl.

It was shown in [92] that the full bosonic M5-brane Lagrangian in phase space equals

1 ℒ = X˙mPm + -ΠijV˙ij − λℋ − siℋi + σij𝒦ij , (229 ) 2
where Pm and Πij are the conjugate momenta to Xm and the 2-form Vij
Pm = Ema &tidle;Pa + TM5 ˆð’žm , 1 ˆð’žm = ∗[Z ⋆(imC6 ) − -Z ⋆(imC3 ) ∧ (𝒞3 + 2ℋ3 )], 2 Πij = 1T 𝜀ijk1k2k3∂ V . (230 ) 4 M5 k1 k2k3
Notice the last equation is equivalent to Π = 1T ⋆ (dV) 2 M5, from which we conclude d ⋆ Π = 0, using the Bianchi identify for dV2. The last three functionals appearing in Eq. (229View Equation)
1- 2 2 5 &tidle; ℋ = 2 [𝒫 + TM5 det (𝒢 + H )], m ˆ ℋi = ∂iX Pm + TM5 (Vi − 𝒞i), ij ij 1- ijk1k2k3 𝒦 = Π − 4 TM5𝜀 ∂k1Vk2k3, (231 )
correspond to constraints generating time translations, world space diffeomorphisms and the self-duality condition. The following definitions were used in the expressions above
1 Vi = --𝜀i1i2i3i4i5Hi3i4i5Hi1i2i, 24 𝒫a = EamPm + TM5 (V i∂iXmEmb ηba − 𝒞ˆa) , 𝒞ˆ = Em ˆð’ž . (232 ) a a m
As for D-branes and M2-branes, in practice one solves the equations of motion in the subspace of phase space configurations solving Eq. (214View Equation) and eventually computes the energy of the system by solving the quadratic constraint coming from the Hamiltonian constraint ℋ = 0.
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