Quantization of P-branes, D-p-branes and M-branes *

Killing spinors of space-time BPS configurations play an important role in quantization of theories with the fermionic worldvolume local symmetry. We show here how it works for the GS superstring, BST supermembrane and M-5-brane. We show that the non-linear generalization of the (2,0) d=6 tensor supermultiplet action is the M-5-brane action in a Killing gauge. For D-p-branes the novel feature of quantization is that they can be quantized Lorentz covariantly, in particular, for D-0-brane a gauge exists where the action is covariant and free. We present a general condition on possible choice of gauges for the κ-symmetric branes. We review here some work on quantization of the local fermionic worldvolume symmetry presented in [1–4]. This is a generalized version of the talk presented at Strings 97 in Amsterdam in June 1997. We give a short summary of the new results in the end of this paper. The actions of Green-Schwarz string in D=10 target space and of the Bergshoeff-Sezgin-Townsend membrane in D=11 target space have local fermionic κ-symmetry, reparametrization invariance and manifest space-time supersym-metry. They depend on worldvolume fields Z M = (X m (ξ), θ α (ξ)) related to coordinates of the space-time superspace. For the string p = 1 and for the membrane p = 2. Both actions belong to a general class of κ-symmetric p-brane actions and upon gauge-fixing describe the scalar supermultiplets of worldvolume supersymmetry. D-p-brane actions [5–7] depend in addition on world-volume 1-form gauge potential A i and upon gauge-fixing describe the vector multiplets of worldvolume supersymmetry. Finally, the M-5-brane action [8,9] depends on a 2-form gauge potential A ij with the self-dual field strength. This action after gauge-fixing describes the tensor multiplets of worldvolume su-persymmetry. The class of actions we consider have 32-dimensional global space-time supersymmetry and the local fermionic κ-supersymmetry : δ ǫ θ = ǫ, δ ǫ X m = ¯ ǫΓ m θ , (1) δ κ θ = (1 + Γ) κ(ξ) , δ κ X m = ¯ Both ǫ and κ are 32-dimensional, but ǫ is global and κ depends on coordinates of the brane ξ. Here dots mean the transformations of Born-Infeld or a tensor field. Γ is a function of the fields of the brane Γ = Γ (Z(ξ), A(ξ)) and depends on ξ i therefore. The matrix Γ(ξ) squares to 1 and has a vanishing trace: therefore (1 + Γ)(1 − Γ) = 0, i.e. …