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, we study a perturbative expansion about the spectrum of $H_{0,N}$, and show that the spectral gap remains finite at $N=\infty$ at least up to the second order. We also apply the method to a toy model, the $U(N)$-invariant anharmonic oscillator, and compare our bounds with the exact values.

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 R3,1.

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.

Explicit structure constants are calculated for certain Lie algebras of vectorfields on 2-dimensional compact manifolds.

We explicitly prove, using some nontrivial identities involving gamma matrices, that there can be only one Spin(9)×SU(2) invariant state which depends only on fermionic variables.