4.3 Calibrations

In the absence of WZ couplings and brane gauge field excitations, the energy of a brane configuration equals its volume. The problem of identifying minimal energy configurations is equivalent to that of minimising the volumes of p-dimensional submanifolds embedded in an n-dimensional ambient space. The latter is a purely geometrical question that can, in principle, be mathematically formulated independently of supersymmetry, kappa symmetry or brane theory. This is what the notion of calibration achieves. In this subsection, I review the close relation between this mathematical topic and a subset of supersymmetric brane configurations [235, 228, 2]. I start with static brane solitons in ℝn, for which the connection is more manifest, leaving their generalisations to the appropriate literature quoted below.

Consider the space of oriented p dimensional subspaces of n ℝ, i.e., the Grassmannian n G (p,ℝ ). For any n ξ ∈ G (p,ℝ ), one can always find an orthonormal basis {e1,...,en} in n ℝ such that {e1,...,ep} is a basis in ξ so that its co-volume is

⃗ξ = e1 ∧ ...∧ ep . (233 )
A p-form φ on an open subset U of n ℝ is a calibration of degree p if

One of the applications of calibrations is to provide a bound for the volume of p-dimensional submanifolds of n ℝ. Indeed, the fundamental theorem of calibrations [289] states

Theorem:
Let φ be a calibration of degree p on ℝn. The p-dimensional submanifold N, for which

φ(⃗N ) = 1, (235 )
is volume minimising. One refers to such minimal submanifolds as calibrated submanifolds, or as calibrations for short, of degree p. The proof of this statement is fairly elementary. Choose an open subset U of N with boundary ∂U and assume the existence of a second open subset V in another subspace W of ℝn with the same boundary, i.e., ∂U = ∂V. By Stokes’ theorem,
∫ ∫ ∫ ∫ ⃗ vol(U ) = U φ = V φ = φ(V )μV ≤ V μV = vol(V) , (236 )
where μV = α1 ∧ ...αp is the volume form constructed out of the dual basis {α1,...,αp } to {e1,...,en}.

Two remarks can motivate why these considerations should have a relation to brane solitons and supersymmetry:

  1. For static brane configurations with no gauge field excitations and in the absence of WZ couplings, the energy of the brane soliton equals the volume of the brane submanifold embedded in ℝn. Thus, bounds on the volume correspond to brane energy bounds, which are related to supersymmetry saturation, as previously reviewed. Indeed, the dynamical field Xi(σ) does mathematically describe the map from the world volume p ℝ into n ℝ. The above bound can then be re-expressed as
    ∫ ∘------- ∫ dpσ det 𝒢μν ≥ X ∗φ, (237)
    where X ∗φ stands for the pullback of the p-form φ.
  2. There exists an explicit spinor construction of calibrations emphasising the connection between calibrated submanifolds, supersymmetry and kappa symmetry.

Let me review this spinor construction [159, 287]. For p = 1,2 mod 4, the p-form calibration takes the form

φ = dXi1 ∧ ...∧ dXip 𝜖T Γ 0i1...ip𝜖, (238 )
where the set i X (i = 1,...,n) stands for the transverse scalars to the brane parameterising n ℝ, 𝜖 is a constant real spinor normalised so that 𝜖T 𝜖 = 1 and Γ i ...i 1 k are antisymmetrised products of Clifford matrices in ℝn. Notice that, given a tangent p-plane ξ, one can write φ| ξ as
√ ----- φ |ξ = det 𝒢 𝜖T Γ ξ𝜖, (239 )
where the matrix
1 Γ ξ =--√------ 𝜀μ1...μp∂μ1Xi1 ⋅⋅⋅∂μpXipΓ 0i1...ip (240 ) p! det 𝒢
is evaluated at the point to which ξ is tangent. Given the restriction on the values of p,
2 Γ ξ = 𝟙. (241 )
It follows that φ|ξ ≤ volξ for all ξ. Since φ is also closed, one concludes it is a calibration. Its contact set is the set of p-planes for which this inequality is saturated. Using Eq. (239View Equation), the latter is equally characterised by the set of p-planes ξ for which
Γ ξ𝜖 = 𝜖. (242 )
Because of Eq. (241View Equation) and the fact that trΓ = 0 ξ, the solution space to this equation is always half the dimension of the spinor space spanned by 𝜖 for any given tangent p-plane ξ. However, this solution space generally varies as ξ varies over the contact set, so that the solution space of the set is generally smaller.

So far the discussion involved no explicit supersymmetry. Notice, however, that the matrix Γ ξ in Eq. (240View Equation) matches the kappa symmetry matrix Γ κ for branes in the static gauge with no gauge field excitations propagating in Minkowski. This observation allows us to identify the saturation of the calibration bound with the supersymmetry preserving condition (214View Equation) derived from the gauge fixing analysis of kappa symmetry.

Let me close the logic followed in Section 4 by pointing out a very close relation between the supersymmetry algebra and kappa symmetry that all my previous considerations suggest. Consider a single infinite flat M5-brane propagating in d = 11 Minkowski and fix the extra gauge symmetry of the PST formalism by a(σμ) = t (temporal gauge). The kappa symmetry matrix (157View Equation) reduces to

-------1-------- 0 i 1- 0 ij &tidle; 1- 0 i1...i5 Γ κ = ∘----------&tidle;---[Γ Γ it+ 2Γ Γ Hij − 5!Γ Γ i1,...,i5𝜀 ], (243 ) det(δij + Hij)
where all {i,j} indices stand for world space M5 indices. Notice that the structure of this matrix is equivalent to the one appearing in Eq. (216View Equation) for ¯ Γ by identifying
i1...i5 i1...i5 &tidle; Y = − 𝜀 , Hij = Zij,----------- i 1- ij1j2j3j4 0 ∘ P = 8 𝜀 Zj1j2Zj3j4 , P = det(δij + Zij). (244 )
Even though, this was only argued for the M5-brane and in a very particular background, it does provide some preliminary evidence for the existence of such connection. In fact, a stronger argument can be provided by developing a phase space formulation of the kappa symmetry transformations that allows one to write the supersymmetry anticommutator as [278Jump To The Next Citation Point]
∫ {Q, Q } = Γ 0 dpσ [Γ a&tidle;p + γ] , with γ = 1-𝜀a1...ap∂ Xi1 ⋅⋅⋅∂ XipΓ . (245 ) a p! a1 ap i1...ip
This result has not been established in full generality but it agrees with the flat space case [165] and those non-flat cases that have been analysed [438, 437]. I refer the reader to [278Jump To The Next Citation Point] where they connect the functional form in the right-hand side of Eq. (245View Equation) with the kappa symmetry transformations for fermions in its Hamiltonian form.

The connection between calibrations, supersymmetry and kappa symmetry goes beyond the arguments given above. The original mathematical notion of calibration was extended in [277, 278] relaxing its first condition dφ ⁄= 0. Physically, this allowed one to include the presence of non-trivial potential energies due to background fluxes coming from the WZ couplings. Some of the applications derived from this notion include [231, 229, 230, 373, 139]. Later, the notion of generalised calibration was introduced in [344Jump To The Next Citation Point], where it was shown to agree with the notion of calibration defined in generalised Calabi–Yau manifolds [267] following the seminal work in [298]. This general notion allows one to include the effect of non-trivial magnetic field excitations on the calibrated submanifold, but still assumes the background and the calibration to be static. Some applications of these notions in the physics literature can be found in [344, 377, 413]. More recently, this formalism was generalised to include electric field excitations [376], establishing a precise correspondence between generic supersymmetric brane configurations and generalised geometry.

Summary:
A necessary condition for a bosonic brane configuration to preserve supersymmetry is to solve the kappa symmetry preserving condition (214View Equation). In general, this is not sufficient for being an on-shell configuration, though it can be, if there are no gauge field excitations. Solutions to Eq. (214View Equation) typically impose a set of constraints on the field configuration, which can be interpreted as BPS equations by computing the Hamiltonian of the configuration, and a set of projection conditions on the constant parts 𝜖∞ of the background Killing spinors 𝜖. The energy bounds saturated when the BPS equations hold are a field theory realisation of the algebraic bounds derived from the supersymmetry algebra. An attempt to summarise the essence of these relations is illustrated in Figure 6View Image.


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