Pointing out a crucial relation with caustics of the eikonal equation we discuss the singularity formation of 2-dimensional surfaces that sweep out 3-manifolds of zero mean curvature in R 3,1 .
Explicit structure constants are calculated for certain Lie algebras of vectorfields on 2-dimensional compact manifolds.A central feature of M-brane theories is their diffeomorphism invariance, which in a partially gauge-fixed light-cone description (leading to a polynomial Hamiltonian and a Yang-Mills type matrix model for membranes [1,2]) 1 is reduced to volume preserving diffeomorphisms (VPD), i.e. diffeomorphisms (of the parameter space of the Mdimensional extended object) that have unit Jacobian, i.e. generated by divergence-free vector fields. Interestingly, volume non-preserving diffeomorphisms (VNPD) (re)appear in this partially gauge-fixed description [1,2,3] in the reconstruction of the longitudinal field ζ that (apart from the zero mode ζ 0 and its conjugate degree of freedom η) disappears in the Hamiltonian formulation. The relativistically invariant internal mass-squared involves only the (internal) transversal coordinates x α =0 and their conjugate momenta p α := pY α (ϕ)d M ϕ § mkhynek@kth.se 1 see e.g. [5,3] for reviews
We consider the problem of quantization of the bosonic membrane via the large N limit of its matrix regularizations H N in Fock space. We prove that there exists a choice of the Fock space frequency such that H N can be written as a sum of a non-interacting Hamiltonian H 0,N and the original normal ordered quartic potential. Using this decomposition we obtain upper and lower bounds for the ground state energy in the planar limit, we study a perturbative expansion about the spectrum of H 0,N , and show that the spectral gap remains finite at N = ∞ at least up to the second order. We also apply the method to the U (N )-invariant anharmonic oscillator, and demonstrate that our bounds agree with the exact result of Brezin et al. * mkhynek@kth.se 1 See [6] for a discussion of the spectrum, including the supersymmetric version of the model and related issues 2 The power of n for a quartic interaction it is usually argued to be −1, (the t' Hooft coupling [8]), however note that due to the definition of f (n) abc which contains an explicit factor of n 3 2 , the coupling constant in front of our potential should be multiplied by n −4 . In Section 4 we will show that this also follows from our construction
The emergence of a critical dimension is one of the most striking features of string theory. One way to obtain it is by demanding closure of the Lorentz algebra in the light-cone gauge quantisation, as discovered for bosonic strings more than fourty years ago. We give a detailed derivation of this classical result based on the operator product expansion on the Lorentzian world-sheet.
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