Modular index theory and spectral flow

“线性运算符”是一个数学对象，它(在许多情况下)可以像电子表格一样由数字的正方形数组(或“矩阵”)表示。通过一个称为“矩阵乘法”的过程，这些运算符对数字的垂直列表(称为“向量”)进行“运算”，以产生新的垂直列表(新的向量)。如果A是运算符的符号，X是垂直列表(向量)的符号，那么我们用AX表示新转换的列表(向量)。特别重要的是找到运算符A的特定向量，例如Y，并且特定的数字(取决于A和Y)例如是2，使得AY是Y的2倍，符号AY=2Y。在这种情况下，我们可以说2在A的“谱”中。也就是说，A的谱是所有这样的数n的集合，对这些数n有一个向量，比方说Z，使得AZ=NZ。当然，不同的运算符通常会有不同的谱。这类运算符及其谱的例子在科学和经济应用中经常出现，以至于在大学一年级(甚至高中)，学生们通常会被教授以《矩阵理论》或《线性代数》为幌子的这门学科的基础知识。在更深的层面上，这些算符对理解量子力学和该领域的所有无数应用都至关重要。我计划继续详细研究这些算符的光谱如何随着算符本身的不断变化而不断变化。这就是“流动光谱”或“光谱流动”的概念。尤其重要的是，能够计算有多少频谱从负值变为正值，而不必经历计算每个流动时刻的所有光谱的巨大耗时的过程。我和我的合著者一直在开发在许多不同情况下做到这一点的公式。警告：出于阐述的目的，上述描述过于简单化。

A "linear operator" is a mathematical object which (in many cases) can be represented by a square array (or"matrix") of numbers just like a spreadsheet. By a process called "matrix multiplication", such operators"operate" on vertical lists of numbers (called "vectors") to produce new vertical lists (new vectors). If A is the symbol for an operator and X is the symbol for a vertical list (vector), then we denote the newly transformedlist (vector) by AX. It is especially important to find particular vectors for the operator A, say Y, and particularnumbers (depending on A and Y) say 2 for example so that AY is 2 times Y, in symbols AY=2Y. In this case, we would say that 2 is in the "spectrum" of A. That is, the spectrum of A is the collection of all such numbers n for whichthere is some vector, say Z so that AZ=nZ. Of course different operators will usually have different spectrums.   Examples of such operators and their spectrums occur so often in scientific and economic applications thatstudents are routinely taught the rudiments of the subject in first year university (or even high school) in theguise of "Matrix Theory" or "Linear Algebra". At a deeper level, these operators are crucial to the understanding of quantum mechanics and all the myriad applications of that field.   I plan to continue my detailed study of how the spectrums of such operators change continuously as the operators themselves change continuously. This is the notion of "flowing spectrum" or "Spectral Flow". In particular, it is important to be able to calculate how much of the spectrum changes from negative to positive,without having to go through the hugely time-consuming process of calculating all the spectrums at each instant of the flow. My co-authors and I have been developing formulas which do this in many diverse cases.   WARNING: The above descriptions are highly over-simplified for the purposes of exposition.