🔢Noncommutative Geometry Unit 8 – Cyclic cohomology

Key Concepts and Definitions

• Cyclic cohomology extends the notion of de Rham cohomology to noncommutative algebras and provides a cohomological framework for studying noncommutative spaces
• Hochschild cohomology, a precursor to cyclic cohomology, describes the cohomology of associative algebras with coefficients in themselves
• Cyclic cohomology groups $HC^n(A)$ are defined for an associative algebra $A$ over a field $k$ and are invariant under cyclic permutations of the arguments
  • The cyclic cohomology groups form a graded module over the polynomial ring $k[u]$, where $u$ is an indeterminate of degree 2
• Connes' boundary map $B: HC^n(A) \to HC^{n+1}(A)$ connects the cyclic cohomology groups and satisfies $B^2 = 0$, leading to a complex called the cyclic complex
• Cyclic cocycles are elements of the cyclic cohomology groups and play a crucial role in noncommutative geometry by providing a way to integrate noncommutative differential forms
• The Chern character in cyclic cohomology associates cyclic cocycles to $K$-theory classes, establishing a connection between $K$-theory and cyclic cohomology
• The periodic cyclic cohomology $HP^*(A)$ is obtained by taking the direct limit of the cyclic cohomology groups under the action of the boundary map $B$

Historical Context and Development

• Cyclic cohomology was introduced by Alain Connes in the early 1980s as a noncommutative analog of de Rham cohomology, motivated by the study of foliations and operator algebras
• The development of cyclic cohomology was influenced by earlier work on Hochschild cohomology and the algebraic $K$-theory of operator algebras
• Connes' formulation of the Chern character in cyclic cohomology provided a powerful tool for studying the $K$-theory of noncommutative spaces
  • The Chern character establishes a bridge between $K$-theory and cyclic cohomology, allowing for the computation of $K$-theory groups using cyclic cocycles
• The cyclic homology theory, dual to cyclic cohomology, was developed by Tsygan and Loday-Quillen independently, providing a homological perspective on the subject
• The connection between cyclic cohomology and the noncommutative geometry of foliations and leaf spaces was a major driving force behind the development of the theory
• Cyclic cohomology has found applications in various areas of mathematics and mathematical physics, including index theory, quantum field theory, and the geometry of quantum groups
• The theory has been generalized and extended in various directions, such as the cyclic cohomology of Hopf algebras, equivariant cyclic cohomology, and twisted cyclic cohomology

Algebraic Foundations

• Cyclic cohomology is built on the algebraic framework of associative algebras over a field $k$, typically the complex numbers $\mathbb{C}$
• The Hochschild cohomology $HH^n(A,M)$ of an associative algebra $A$ with coefficients in an $A$-bimodule $M$ is defined using the Hochschild complex
  • The Hochschild complex consists of multilinear maps $f: A^{\otimes n} \to M$ satisfying a certain coboundary condition
• The cyclic cohomology $HC^n(A)$ is obtained by considering a subcomplex of the Hochschild complex invariant under cyclic permutations of the arguments
  • Cyclic permutations are generated by the operator $\tau(a_0 \otimes \cdots \otimes a_n) = (-1)^n a_n \otimes a_0 \otimes \cdots \otimes a_{n-1}$
• The cyclic complex $(C^\lambda(A), b, B)$ is a mixed complex, consisting of the cyclic cochains $C^\lambda(A)$, the Hochschild coboundary $b$, and Connes' boundary map $B$
  • The cohomology of the total complex associated with the mixed complex yields the cyclic cohomology groups $HC^n(A)$
• The cup product in Hochschild cohomology induces a graded commutative algebra structure on the cyclic cohomology groups $HC^*(A)$
• The cyclic cohomology of an algebra $A$ is closely related to its Hochschild homology $HH_*(A)$ via a long exact sequence involving the boundary map $B$
• The algebraic properties of cyclic cohomology, such as functoriality and Morita invariance, make it a powerful tool for studying noncommutative spaces

Cyclic Cohomology Theory

• The cyclic cohomology groups $HC^n(A)$ are defined as the cohomology of the cyclic complex $(C^\lambda(A), b, B)$, where $b$ is the Hochschild coboundary and $B$ is Connes' boundary map
  • The cyclic complex is a subcomplex of the Hochschild complex consisting of cyclic cochains, which are invariant under cyclic permutations
• The periodic cyclic cohomology $HP^*(A)$ is obtained by taking the direct limit of the cyclic cohomology groups under the action of the boundary map $B$
  • The periodic cyclic cohomology has a simpler structure compared to the cyclic cohomology, with only two groups $HP^0(A)$ and $HP^1(A)$
• The Connes-Tsygan long exact sequence relates the Hochschild homology, cyclic homology, and periodic cyclic homology of an algebra $A$
  • The boundary map $B$ plays a crucial role in connecting these homology theories
• The Chern character in cyclic cohomology associates cyclic cocycles to $K$-theory classes, providing a way to compute the $K$-theory of noncommutative spaces using cyclic cohomology
  • The Chern character is a graded trace on the $K$-theory algebra, taking values in the periodic cyclic cohomology
• The local index formula in cyclic cohomology expresses the index of an elliptic operator on a noncommutative space in terms of the pairing between its Chern character and a cyclic cocycle
• The cyclic cohomology of crossed product algebras, such as the group algebra of a discrete group, can be computed using the Feigin-Tsygan-Nistor spectral sequence
• The Cuntz-Quillen approach to cyclic cohomology uses the $X$-complex, a variant of the cyclic complex, to study the cyclic cohomology of algebras

Connections to Other Mathematical Areas

• Cyclic cohomology is closely related to $K$-theory, with the Chern character providing a bridge between the two theories
  • The Chern character in cyclic cohomology associates cyclic cocycles to $K$-theory classes, allowing for the computation of $K$-theory groups using cyclic cohomology
• The local index theorem in noncommutative geometry, formulated by Connes and Moscovici, expresses the index of an elliptic operator on a noncommutative space in terms of the pairing between its Chern character and a cyclic cocycle
  • This generalizes the classical Atiyah-Singer index theorem to the noncommutative setting
• Cyclic cohomology has applications in the study of foliations and leaf spaces, providing invariants for the transverse geometry of foliations
  • The Godbillon-Vey class, an important invariant in foliation theory, can be expressed as a cyclic cocycle
• The cyclic cohomology of Hopf algebras, developed by Connes and Moscovici, provides a cohomological framework for studying quantum groups and their actions on noncommutative spaces
• Cyclic cohomology has connections to algebraic topology, with the cyclic homology of commutative algebras being related to the $S^1$-equivariant homology of topological spaces
• The Deligne conjecture, proved by Kontsevich, establishes a deep connection between the Hochschild cohomology of an associative algebra and the deformation theory of the algebra
  • This result has implications for the study of deformation quantization and the quantization of Poisson manifolds
• Cyclic cohomology has found applications in mathematical physics, particularly in the study of quantum field theories and the geometry of quantum spaces
  • The noncommutative geometry approach to the Standard Model of particle physics, developed by Connes and collaborators, heavily relies on cyclic cohomology and the local index theorem

Applications in Noncommutative Geometry

• Cyclic cohomology provides a cohomological framework for studying noncommutative spaces, generalizing the role of de Rham cohomology in classical geometry
• The Chern character in cyclic cohomology allows for the computation of the $K$-theory of noncommutative spaces, such as $C^*$-algebras and crossed product algebras
  • The pairing between $K$-theory classes and cyclic cocycles yields numerical invariants of noncommutative spaces
• The local index theorem in noncommutative geometry expresses the index of elliptic operators on noncommutative spaces in terms of the pairing between their Chern characters and cyclic cocycles
  • This result has applications in the study of the geometry and topology of noncommutative spaces, such as the noncommutative tori and quantum spheres
• Cyclic cohomology provides invariants for the transverse geometry of foliations, with the Godbillon-Vey class being a notable example of a cyclic cocycle arising in this context
• The cyclic cohomology of quantum groups and Hopf algebras plays a crucial role in understanding their symmetries and actions on noncommutative spaces
  • The modular class of a Hopf algebra, which governs its trace properties, can be expressed as a cyclic cocycle
• The noncommutative geometry approach to particle physics, pioneered by Connes and collaborators, heavily relies on cyclic cohomology and the local index theorem
  • The spectral action principle, which derives the Standard Model Lagrangian from a noncommutative spacetime, involves the pairing between a cyclic cocycle and the Chern character of a Dirac operator
• Cyclic cohomology has been used to study the geometry of noncommutative quotient spaces, such as the noncommutative tori and the quantum spheres, providing invariants and computational tools for these spaces

Computational Techniques and Examples

• The cyclic cohomology groups of commutative algebras can be computed using the Hochschild-Kostant-Rosenberg theorem, which relates them to the exterior algebra of differential forms
  • For a smooth manifold $M$, the cyclic cohomology $HC^*(C^\infty(M))$ is isomorphic to the direct sum of the de Rham cohomology groups of $M$
• The cyclic cohomology of group algebras can be computed using the group homology and the Feigin-Tsygan-Nistor spectral sequence
  • For a discrete group $G$, the cyclic cohomology $HC^*(k[G])$ is related to the group homology $H_*(G, k)$ via the spectral sequence
• The cyclic cohomology of crossed product algebras, such as the noncommutative tori, can be computed using the Pimsner-Voiculescu exact sequence and its cyclic version
  • The Pimsner-Voiculescu sequence relates the $K$-theory of a crossed product algebra to the $K$-theory of the base algebra and the action of the group
• The Chern character in cyclic cohomology can be computed explicitly for certain classes of algebras, such as the noncommutative tori and the quantum spheres
  • For the noncommutative torus $A_\theta$, the Chern character of a projection $p \in A_\theta$ is given by a cyclic 2-cocycle involving the trace of powers of $p$
• The local index formula in noncommutative geometry can be used to compute the index of elliptic operators on noncommutative spaces
  • For the Dirac operator on the noncommutative torus, the local index formula expresses its index in terms of a pairing between its Chern character and a cyclic 2-cocycle
• Computational tools from homological algebra, such as spectral sequences and exact sequences, are widely used in the study of cyclic cohomology
  • The Connes-Tsygan long exact sequence and the Feigin-Tsygan-Nistor spectral sequence are examples of such computational tools
• Computer algebra systems, such as Mathematica and Sage, have been used to perform computations in cyclic cohomology and noncommutative geometry
  • These systems can handle symbolic computations involving noncommutative algebras and their cyclic complexes

Advanced Topics and Current Research

• The cyclic cohomology of Hopf algebras and quantum groups is an active area of research, with connections to the geometry of quantum spaces and the theory of integrable systems
  • The Hopf-cyclic cohomology, introduced by Connes and Moscovici, provides a framework for studying the cyclic cohomology of Hopf algebras and their actions on noncommutative spaces
• Equivariant cyclic cohomology, which incorporates the action of a group on an algebra, has been developed to study the geometry of noncommutative spaces with symmetries
  • Equivariant cyclic cohomology has applications in the study of orbifolds, quotient spaces, and the geometry of quantum groups
• The cyclic cohomology of algebras with coefficients in a module or a sheaf has been studied, leading to the notion of local cyclic cohomology
  • Local cyclic cohomology provides a sheaf-theoretic approach to the study of noncommutative spaces and has connections to the geometry of foliations
• The relationship between cyclic cohomology and the noncommutative geometry of quantum fields is an active area of research
  • The renormalization of quantum field theories has been studied using the Connes-Kreimer Hopf algebra of Feynman graphs and its cyclic cohomology
• The cyclic cohomology of algebras with additional structures, such as Poisson algebras and homotopy algebras, has been investigated
  • The cyclic cohomology of Poisson algebras is related to the deformation quantization of Poisson manifolds and the study of symplectic invariants
• The connection between cyclic cohomology and the geometry of loop spaces has been explored, leading to the development of loop space methods in noncommutative geometry
  • The cyclic homology of the free loop space of a manifold has been shown to be related to the cyclic homology of the manifold via the Jones isomorphism
• The categorification of cyclic cohomology, which lifts the theory to the level of categories and functors, is an emerging area of research
  • The categorified cyclic cohomology, also known as cyclic homology, has connections to the geometry of topological quantum field theories and the theory of motives