膜相互作用和黎曼曲面的三维模拟
On membrane interactions and a three-dimensional analog of Riemann surfaces
摘要：
M理论中的膜预计通过分裂和连接过程相互作用。我们研究这些影响的PP-波矩阵模型，在其中，它们与不同数量的膜真空上建立的部门之间的状态转换。这种状态之间的过渡幅度接收来自BPS瞬子配置之间的不同真空插值的贡献。BPS瞬子的模空间的各种性质是已知的，但很少有显式解的已知例子。我们提出了一种新的方法来建设瞬子解决方案之间的状态包含任意数量的膜，连续近似的基础上有效的大尺寸矩阵插值。所提出的方案使用二维空间上的函数来近似矩阵，并且它依赖于M理论中膜自由度的矩阵正则化背后的相同思想。我们发现，BPS瞬子方程有一个连续对应，可以映射到三维拉普拉斯方程通过一系列的变量的变化。对应于膜分裂/接合过程的构型描述可以用三维黎曼曲面的拉普拉斯方程的解来给出，该黎曼曲面由多个R^3的副本组成，这些副本通过分支切割的推广来连接。我们讨论了我们的建议的各种一般特征，我们也提出了明确的解析解。
英文摘要：
Membranes in M-theory are expected to interact via splitting and joining processes. We study these effects in the pp-wave matrix model, in which they are associated with transitions between states in sectors built on vacua with different numbers of membranes. Transition amplitudes between such states receive contributions from BPS instanton configurations interpolating between the different vacua. Various properties of the moduli space of BPS instantons are known, but there are very few known examples of explicit solutions. We present a new approach to the construction of instanton solutions interpolating between states containing arbitrary numbers of membranes, based on a continuum approximation valid for matrices of large size. The proposed scheme uses functions on a two-dimensional space to approximate matrices and it relies on the same ideas behind the matrix regularisation of membrane degrees of freedom in M-theory. We show that the BPS instanton equations have a continuum counterpart which can be mapped to the three-dimensional Laplace equation through a sequence of changes of variables. A description of configurations corresponding to membrane splitting/joining processes can be given in terms of solutions to the Laplace equation in a three-dimensional analog of a Riemann surface, consisting of multiple copies of R^3 connected via a generalisation of branch cuts. We discuss various general features of our proposal and we also present explicit analytic solutions.
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