标题： Dirac 算子在矩阵几何中的研究 显示英文标题

标题： Dirac Operator in Matrix Geometry

摘要： 我们回顾了黎曼几何中 Dirac 算子及其性质的构造，并展示了热核迹的渐近展开如何决定 Dirac 算子的谱不变量及其指标。我们还指出，爱因斯坦-希尔伯特泛函可以表示为 Dirac 算子前两个谱不变量的线性组合。接下来，我们报告了之前尝试将 Dirac 算子的概念推广到矩阵几何的情形，在这种情况下，没有黎曼度规，而是有一个矩阵值的自伴对称二阶张量扮演“非交换”的度规的角色。我们构造了不变的一阶和二阶自伴椭圆型偏微分算子，这些算子可以被称为“非交换”Dirac 算子和“非交换”Laplace 算子。我们为非交换 Laplace 型算子构造了相应的热核，并计算了它的前两个谱不变量。这两个谱不变量的线性组合给出一个泛函，它可以被视为爱因斯坦-希尔伯特作用量的一个非交换推广。 显示英文摘要

摘要： We review the construction of the Dirac operator and its properties in Riemannian geometry and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein-Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a ``non-commutative'' metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, that can be called ``non-commutative'' Dirac operator and non-commutative Laplace operator. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein-Hilbert action.