Brane Universality Classes in non-Minimal String Theory

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In this work non-minimal strings are defined through the world sheet picture to be Liouville gravity plus matter content which is non-minimal conformal. By contrast with minimal strings, whose boundary states - equivalently branes - have been thoroughly studied there are many unanswered questions when the matter has an extended algebra and there are extra conserved currents in the bulk system. Typically it appears that branes are classified by whether or not these extra currents are conserved at the boundary (ie by the brane) - of course the stress-tensor itself is always conserved to keep the particle content massless. Those branes that have the same current conservation structure are known in some cases to exhibit Seiberg-Shih equivalence (SSE). In work with previous STFC funded graduate student, Ben Niedner, we established that in the simplest case of a single extra current (the matter is essentially the critical point excitations of a Q=3 Potts model) there is a natural holomorphic structure which underlies the SSE and that this structure exists regardless of regularisation (it was constructed explicitly in terms of the planar random graph and spin-system formulation of Liouville gravity coupled to matter).In this project we will do two things. Firstly we will explore the boundary renormalization flows in the previously established holomorphic structure perturbed by the appropriate operators and examine whether the behaviour is consistent with the boundary c-theorem. This involves extending the calculations that lead to the holomorphic structure to include the perturbations which essentially take the form of generalised boundary magnetic fields. It is not clear whether these flows between fixed points sustain the holomorphic structure, although recent work on the simplest case suggests that they do. The second phase of the project will explore whether the holomorphic structure generalises to other cases of an extended system of conserved currents. Essentially we are investigating whether the possible sets of SSEs can be regarded as being classified by the possible holomorphic structures. Clearly to establish this in more generality would be a very clean way of understanding the relationships between the possible boundary states of these systems.