Mass-Gaps and Spin Chains for (Super) Membranes

We present a method for computing the non-perturbative mass-gap in the theory of Bosonic membranes in flat background spacetimes. The analysis is extended to the study of membranes coupled to background fluxes as well. The computation of mass-gaps is carried out using a matrix regularization of the membrane Hamiltonians. The mass gap is shown to be naturally organized as an expansion in a ’hidden’ parameter, which turns out to be 1 d : d being the related to the dimensionality of the background space. We then proceed to develop a large N perturbation theory for the membrane/matrix-model Hamiltonians around the quantum/mass corrected effective potential. The same parameter that controls the perturbation theory for the mass gap is also shown to control the Hamiltonian perturbation theory around the effective potential. The large N perturbation theory is then translated into the language of quantum spin chains and the one loop spectra of various Bosonic matrix models are computed by applying the Bethe ansatz to the one-loop effective Hamiltonians for membranes in flat space times. The spin chains corresponding to the large N effective Hamiltonians for the relevant matrix models are generically not integrable. However, we are able to find large integrable sub-sectors for all the spin chains of interest. Moreover, the continuum limits of the spin chains are mapped to integrable Landau-Lifschitz models even if the underlying spin chains are not integrable. Apart from membranes in flat spacetimes, the recently proposed matrix models (hep-th/0607005) for non-critical membranes in plane wave type spacetimes are also analyzed within the paradigm of quantum spin chains. The Bosonic sectors of all the models proposed in (hep-th/0607005) are diagonalized at the one-loop level and an intriguing connection between the existence of supersymmetric vacua and one-loop integrability is also presented.