Brane Universality Classes in non-Minimal String Theory

本文通过世界图将非极小弦定义为刘维尔引力加上非极小保角物质含量。最小弦的边界状态——相当于膜——已经得到了彻底的研究，与之相反，当物质具有扩展代数并且在体系统中存在额外的守恒电流时，有许多未解的问题。通常情况下，膜是根据这些额外的电流是否在边界上守恒（即由膜）来分类的——当然，应力张量本身总是守恒的，以保持粒子含量无质量。在某些情况下，已知具有相同电流守恒结构的膜表现出Seiberg-Shih等效（SSE）。在与以前STFC资助的研究生Ben Niedner的合作中，我们建立了在单个额外电流的最简单情况下（物质本质上是Q=3波茨模型的临界点激发），存在一个天然的全纯结构，它是SSE的基础，并且无论正则化如何都存在（它是根据平面随机图和自旋系统公式明确构建的）。在这个项目中，我们将做两件事。首先，我们将探索先前建立的全纯结构中受适当算子摄动的边界重整化流，并检验其行为是否符合边界c定理。这涉及到扩展导致全纯结构的计算，以包括基本上采取广义边界磁场形式的微扰。不动点之间的这些流动是否能维持全纯结构尚不清楚，尽管最近对最简单情况的研究表明它们能。该项目的第二阶段将探索是否全纯结构推广到守恒电流扩展系统的其他情况。本质上，我们研究的是sse的可能集是否可以被看作是由可能全纯结构分类的。很明显，在更一般的情况下建立这一点，将是理解这些系统可能的边界状态之间关系的一种非常清晰的方式。

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.