🪡K-Theory Unit 7 – Index Theory and the Atiyah–Singer Index Theorem

Key Concepts and Definitions

• Index theory studies the relationship between analytical and topological invariants of operators on manifolds
• Analytical invariants include the dimension of the kernel and cokernel of an operator
• Topological invariants encompass characteristic classes, such as the Chern character and the Todd class
• Elliptic operators, which generalize the Laplacian operator, play a central role in index theory
  • Elliptic operators have a well-defined index, which is the difference between the dimensions of the kernel and cokernel
• The Atiyah-Singer index theorem relates the analytical index of an elliptic operator to a topological expression involving characteristic classes
• K-theory, a generalized cohomology theory, provides a natural framework for studying index theory
  • K-theory groups classify vector bundles and their associated operators on manifolds

Historical Context and Development

• Index theory has its roots in the work of Fritz Noether and Heinz Hopf in the 1920s and 1930s on the Lefschetz fixed-point theorem
• In the 1960s, Michael Atiyah and Isadore Singer developed the general framework of index theory, culminating in the Atiyah-Singer index theorem
• The Atiyah-Singer index theorem generalized and unified various special cases, such as the Riemann-Roch theorem and the Hirzebruch signature theorem
• The development of K-theory by Alexander Grothendieck and Michael Atiyah in the 1950s and 1960s provided a powerful tool for studying index theory
• The work of Raoul Bott on the periodicity theorem in homotopy theory and its applications to K-theory was crucial in the proof of the Atiyah-Singer index theorem
• Subsequent work by Atiyah, Singer, and others extended the index theorem to various settings, such as equivariant and families index theorems

Fundamental Principles of Index Theory

• The index of an operator is a measure of the obstruction to the existence of a solution to the equation $Pu = f$, where $P$ is the operator and $u$ and $f$ are functions
• For an elliptic operator $P$ on a compact manifold, the index is defined as $\text{ind}(P) = \dim \ker P - \dim \text{coker} P$
  • The kernel $\ker P$ consists of functions $u$ such that $Pu = 0$
  • The cokernel $\text{coker} P$ is the quotient space of the range of $P$ by the image of $P$
• The index is a homotopy invariant, meaning it remains constant under continuous deformations of the operator
• The Chern character and the Todd class are characteristic classes that encode topological information about vector bundles and manifolds
  • The Chern character is a ring homomorphism from K-theory to rational cohomology
  • The Todd class is a multiplicative characteristic class related to the Riemann-Roch theorem
• The analytical index and the topological index are equal for elliptic operators on compact manifolds, a fundamental principle of index theory

The Atiyah-Singer Index Theorem: Statement and Significance

• The Atiyah-Singer index theorem states that for an elliptic operator $P$ on a compact manifold $M$, the analytical index is equal to the topological index: $\text{ind}(P) = \int_M \text{ch}(\sigma(P)) \cdot \text{td}(TM)$
  • $\text{ch}(\sigma(P))$ is the Chern character of the symbol of $P$
  • $\text{td}(TM)$ is the Todd class of the tangent bundle of $M$
• The theorem provides a powerful link between analysis and topology, allowing the computation of the index using topological methods
• The Atiyah-Singer index theorem generalizes and unifies various special cases, such as the Riemann-Roch theorem for complex manifolds and the Hirzebruch signature theorem for oriented manifolds
• The theorem has far-reaching consequences in mathematics and physics, providing a framework for studying the geometry and topology of manifolds and their associated operators
• The proof of the Atiyah-Singer index theorem relies on deep results from various fields, including differential geometry, algebraic topology, and functional analysis
• The index theorem has been extended to various settings, such as equivariant index theory for group actions and families index theory for fibrations

Applications in Mathematics and Physics

• The Atiyah-Singer index theorem has numerous applications in mathematics, including:
  • The proof of the Riemann-Roch theorem for complex manifolds
  • The computation of the signature of oriented manifolds using the Hirzebruch signature theorem
  • The study of the geometry and topology of moduli spaces of instantons and monopoles in gauge theory
• In physics, the index theorem plays a crucial role in various areas, such as:
  • Anomaly cancellation in quantum field theory and string theory
  • The computation of the dimension of the moduli space of instantons in Yang-Mills theory
  • The study of the quantum Hall effect and topological insulators in condensed matter physics
• The index theorem provides a link between the zero modes of the Dirac operator and the topology of the underlying manifold, which is essential in the study of fermions in quantum field theory
• The equivariant index theorem has applications in the study of symmetries in physics, such as the computation of the dimension of the space of states in supersymmetric quantum mechanics
• The families index theorem is used in the study of anomalies in gauge theories and the computation of the effective action in string theory

Proof Techniques and Strategies

• The proof of the Atiyah-Singer index theorem relies on a combination of techniques from various fields, including differential geometry, algebraic topology, and functional analysis
• The heat kernel method is a key tool in the proof, which involves studying the asymptotic behavior of the heat kernel associated with the elliptic operator
  • The heat kernel is the fundamental solution to the heat equation, which describes the diffusion of heat on the manifold
  • The asymptotic expansion of the heat kernel near the diagonal contains information about the index of the operator
• The symbol of an elliptic operator, which encodes its highest order terms, plays a crucial role in the proof
  • The symbol defines a vector bundle on the cotangent bundle of the manifold, called the symbol bundle
  • The Chern character of the symbol bundle appears in the topological side of the index theorem
• The proof involves the construction of a parametrix, which is an approximate inverse to the elliptic operator
  • The parametrix is used to reduce the problem to the study of the index of a simpler operator, called the model operator
• K-theory and the Thom isomorphism theorem are used to relate the topological index to characteristic classes on the manifold
• The proof of the index theorem involves a combination of local and global arguments, using techniques such as partitions of unity and the Mayer-Vietoris sequence

Connections to K-Theory

• K-theory provides a natural framework for studying index theory, as it classifies vector bundles and their associated operators on manifolds
• The K-theory groups $K^0(X)$ and $K^1(X)$ of a compact space $X$ are defined using equivalence classes of vector bundles and maps between them
  • $K^0(X)$ classifies virtual vector bundles, which are formal differences of vector bundles
  • $K^1(X)$ classifies homotopy classes of maps from $X$ to the infinite unitary group $U(\infty)$
• The Atiyah-Singer index theorem can be formulated as a statement about the equality of two homomorphisms from K-theory to the integers:
  • The analytical index map, which sends an elliptic operator to its index
  • The topological index map, which sends a K-theory class to its pairing with characteristic classes
• The Bott periodicity theorem, which states that the K-theory groups are periodic with period 2, plays a crucial role in the proof of the index theorem
• The Thom isomorphism theorem in K-theory relates the K-theory of a vector bundle to the K-theory of its base space, which is used in the construction of the topological index
• The Chern character provides a link between K-theory and rational cohomology, allowing the computation of characteristic classes in terms of K-theory classes

Advanced Topics and Current Research

• Equivariant index theory studies the index of elliptic operators in the presence of group actions on the manifold
  • The equivariant index theorem relates the equivariant analytical index to equivariant characteristic classes
  • Equivariant index theory has applications in the study of symmetries in physics and the representation theory of Lie groups
• The families index theorem generalizes the Atiyah-Singer index theorem to families of elliptic operators parametrized by a space
  • The families index theorem has applications in the study of anomalies in gauge theories and the computation of the effective action in string theory
• The index theorem for foliations relates the index of a leafwise elliptic operator to characteristic classes of the foliation
  • The index theorem for foliations has applications in the study of the geometry and topology of foliated manifolds
• The index theorem for manifolds with boundary relates the index of an elliptic operator on a manifold with boundary to the eta invariant of the boundary operator
  • The eta invariant measures the asymmetry of the spectrum of the boundary operator
  • The index theorem for manifolds with boundary has applications in the study of the geometry and topology of manifolds with boundary and their associated operators
• Current research in index theory includes the study of index theorems for noncommutative spaces, such as the Connes-Moscovici index theorem for spectral triples
  • Noncommutative index theory has applications in the study of the geometry and topology of noncommutative spaces and their associated operators
  • The Connes-Moscovici index theorem generalizes the Atiyah-Singer index theorem to the setting of noncommutative geometry