Chern character

Definition

The Chern character is an important topological invariant associated with complex vector bundles, which provides a connection between K-theory and cohomology. It captures information about the curvature of the vector bundle and its underlying geometric structure, serving as a bridge in various applications, from fixed point theorems to differential geometry.

5 Must Know Facts For Your Next Test

1. The Chern character is defined as a polynomial in the Chern classes of a complex vector bundle, effectively linking K-theory to cohomological invariants.
2. It satisfies properties such as functoriality and naturality, making it a valuable tool for comparing vector bundles over different spaces.
3. The Chern character can be computed explicitly using the Chern-Weil theory, which expresses characteristic classes in terms of curvature forms.
4. In K-homology, the Chern character plays a critical role in relating topological data to differential geometric structures, specifically through the index theorem.
5. Applications of the Chern character extend to areas like noncommutative geometry and string theory, where it aids in understanding D-branes and their interactions.

Review Questions

• How does the Chern character relate to Chern classes and why is this relationship important?
  • The Chern character is directly derived from Chern classes, being expressed as a polynomial function of these classes. This relationship is significant because it allows us to translate geometric properties captured by Chern classes into topological invariants that can be utilized in K-theory. By studying the Chern character, we gain insights into both the topology of vector bundles and their curvature properties, making it easier to classify and analyze these bundles across different mathematical contexts.
• Discuss the role of the Chern character in K-homology and its connection to index theory.
  • In K-homology, the Chern character serves as a bridge between topological data and differential geometry through index theory. It connects the K-homology class of a manifold with its analytical data by establishing a correspondence with cohomology classes. This correspondence allows mathematicians to compute indices of elliptic operators on manifolds, leading to profound results such as the Atiyah-Singer index theorem, which relates geometric properties of manifolds to analytical characteristics of differential operators.
• Evaluate how the Chern character contributes to advancements in noncommutative geometry and string theory.
  • The Chern character plays a pivotal role in noncommutative geometry by providing a framework for understanding geometric structures on noncommutative spaces. It aids in defining invariants that characterize these spaces and their symmetries. In string theory, particularly in the context of D-branes, the Chern character helps describe how these branes interact with background fields through topological charges. This connection between topology and physics has led to deeper insights into dualities and consistency conditions within string theory models.

Related terms

Chern classes: Chern classes are characteristic classes associated with complex vector bundles, representing the curvature properties of the bundle and are used in the computation of the Chern character.

K-theory: K-theory is a branch of mathematics that studies vector bundles and their classes, providing a framework for understanding topological spaces through algebraic structures.

Atiyah-Hirzebruch spectral sequence: The Atiyah-Hirzebruch spectral sequence is a computational tool in algebraic topology that relates K-theory to cohomology, often using the Chern character as a key component.