Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a $K_0$-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the even index pairing. This makes the invariant numerically accessible. The index-theoretic proof heavily uses fuzzy spheres.

中文翻译：

---

偶数索引配对的频谱定位器

偶数索引对是将偶数Fredholm模块与由投影指定的$ K_0 $类组合的整数值同态不变量。从微分和非交换几何学和物理学中已知许多经典例子。此处显示了如何构造一个称为频谱定位器的有限维自伴和可逆矩阵，以使其签名的一半等于偶数索引对。这使得不变量数值可访问。指数理论证明大量使用模糊球体。