extend classical calculus to noncommutative algebras and spaces. This framework provides tools to analyze geometric structures in settings where traditional methods fall short, enabling the development of concepts like and in noncommutative realms.
At its core, noncommutative differential calculi consist of a and a . These elements allow for the formulation of key concepts such as , connections, and , which are essential for understanding the geometry of noncommutative spaces and their applications in physics and mathematics.
Noncommutative differential calculi
Noncommutative differential calculi extends the concepts of classical to noncommutative algebras and spaces
Plays a fundamental role in the study of noncommutative geometry, providing tools to analyze and manipulate geometric structures in this setting
Allows for the development of notions such as differential forms, connections, and curvature in the noncommutative realm
Definition of differential calculi
A differential calculus on an algebra A consists of a graded algebra Ω(A)=⨁n≥0Ωn(A) and a differential operator d:Ωn(A)→Ωn+1(A)
The elements of Ωn(A) are called n-forms and represent generalized differential forms in the noncommutative setting
The differential operator d satisfies the Leibniz rule: d(ω∧η)=dω∧η+(−1)∣ω∣ω∧dη, where ω and η are forms and ∣ω∣ denotes the degree of ω
Axioms of differential calculi
The algebra Ω(A) is associative and graded-commutative, meaning that for forms ω and η, we have ω∧η=(−1)∣ω∣∣η∣η∧ω
The differential operator d satisfies d2=0, which is known as the nilpotency condition
There exists a canonical map π:Ω1(A)→A⊗A that relates the 1-forms to the tensor product of the algebra, capturing the noncommutativity of the space
Differential vs integral calculi
Differential calculi focuses on the local properties of noncommutative spaces, such as derivatives, forms, and connections
Integral calculi, on the other hand, deals with global aspects like integration, cohomology, and characteristic classes
Both differential and integral calculi play crucial roles in the study of noncommutative geometry, providing complementary tools for understanding the structure and properties of noncommutative spaces
Differential forms
Differential forms are fundamental objects in noncommutative differential calculi, generalizing the notion of classical differential forms to the noncommutative setting
They provide a way to describe and study the geometry of noncommutative spaces, capturing information about derivatives, integration, and cohomology
Definition of differential forms
In a differential calculus (Ω(A),d) on an algebra A, a differential form of degree n is an element of Ωn(A)
0-forms are elements of the algebra A itself, 1-forms are elements of Ω1(A), and higher-degree forms are obtained by taking wedge products of 1-forms
Differential forms can be thought of as generalized functions on the noncommutative space that take values in the
Exterior algebra of forms
The exterior algebra Ω(A) is the graded algebra generated by the differential forms
It is equipped with a ∧ that combines forms of different degrees to produce higher-degree forms
The wedge product is graded-commutative, satisfying ω∧η=(−1)∣ω∣∣η∣η∧ω for forms ω and η
Wedge product of forms
The wedge product ∧ is a bilinear map Ωn(A)×Ωm(A)→Ωn+m(A) that combines forms of degrees n and m to produce a form of degree n+m
It is associative, meaning that (ω∧η)∧ξ=ω∧(η∧ξ) for forms ω, η, and ξ
The wedge product satisfies the graded-commutativity property, which introduces signs when exchanging the order of forms
Differential of forms
The differential d is a linear map Ωn(A)→Ωn+1(A) that increases the degree of forms by one
It satisfies the Leibniz rule, d(ω∧η)=dω∧η+(−1)∣ω∣ω∧dη, which describes how the differential interacts with the wedge product
The nilpotency condition d2=0 ensures that applying the differential twice always yields zero, leading to the notion of cohomology
Connections
Connections are additional structures on noncommutative spaces that allow for the notion of parallel transport and the comparison of geometric objects at different points
They provide a way to define , which are generalizations of the usual derivative that take into account the noncommutativity of the space
Definition of connections
A connection on a noncommutative space is a linear map ∇:Ω1(A)→Ω1(A)⊗AΩ1(A) satisfying certain properties
It can be extended to act on higher-degree forms using the Leibniz rule, ∇(ω∧η)=∇(ω)∧η+ω∧∇(η)
Connections allow for the definition of parallel transport along curves and the comparison of geometric objects at different points of the noncommutative space
Covariant derivatives
Given a connection ∇, the covariant derivative ∇X along a vector field X is a linear map Ωn(A)→Ωn(A) that satisfies the Leibniz rule
It measures the rate of change of forms along the direction specified by the vector field X
Covariant derivatives provide a way to differentiate forms and other geometric objects in a manner that is compatible with the noncommutative structure of the space
Torsion of connections
The torsion of a connection ∇ is a measure of its non-commutativity and is defined as T(X,Y)=∇XY−∇YX−[X,Y] for vector fields X and Y
It quantifies the extent to which parallel transport around infinitesimal loops fails to close
A connection is called torsion-free if its torsion vanishes identically, which is an important property in many geometric contexts
Curvature of connections
The curvature of a connection ∇ is a 2-form valued in the endomorphism algebra of the tangent bundle, measuring the non-integrability of the connection
It is defined as R(X,Y)=∇X∇Y−∇Y∇X−∇[X,Y] for vector fields X and Y
The curvature provides information about the local geometry of the noncommutative space and plays a crucial role in the formulation of geometric theories such as and gravity
Gauge theory
Gauge theory is a framework that describes the dynamics of connections and curvature on a noncommutative space, generalizing the classical gauge theories of particle physics
It provides a way to formulate physical theories in terms of geometric objects, such as gauge fields and , in the noncommutative setting
Gauge groups and bundles
A gauge group is a group of transformations acting on the connections and curvature of a noncommutative space, typically given by a matrix group (such as U(n) or SU(n))
A gauge bundle is a principal bundle with a gauge group as its structure group, encoding the symmetries and transformations of the gauge fields
provide the geometric framework for describing gauge theories on noncommutative spaces
Gauge transformations
A gauge transformation is a local symmetry transformation that acts on the connections and curvature of a noncommutative space
It is described by a function g:A→G taking values in the gauge group G, acting on connections by ∇↦g−1∇g+g−1dg
Gauge transformations leave the physical observables of the theory invariant and play a crucial role in the formulation of and equations of motion
Gauge-invariant actions
A gauge-invariant action is a functional of the connections and curvature that is invariant under gauge transformations
It is typically constructed using traces of products of curvature forms and other geometric quantities, such as the Hodge star operator
Gauge-invariant actions provide the dynamics for the gauge fields and determine the equations of motion for the theory
Chern-Simons theory
is a particular type of gauge theory in three dimensions, defined by the Chern-Simons action SCS=∫MTr(A∧dA+32A∧A∧A), where A is the gauge connection
It is a topological field theory, meaning that its observables depend only on the global topology of the underlying manifold and not on the local geometry
Chern-Simons theory has deep connections to knot theory, quantum groups, and the geometric quantization of noncommutative spaces
Cyclic cohomology
is a cohomology theory for noncommutative algebras that generalizes the classical de Rham cohomology of manifolds
It provides a way to study the global properties of noncommutative spaces, such as characteristic classes and index theorems, using algebraic and homological methods
Cyclic cocycles and cycles
A cyclic n-cocycle on an algebra A is a multilinear functional φ:A⊗(n+1)→C satisfying certain symmetry and cohomological conditions
A cyclic n-cycle is a triple (V,ρ,W) consisting of a vector space V, a representation ρ:A→End(V), and a closed graded trace W on V
and cycles provide the basic building blocks for the construction of cyclic cohomology and its dual theory, cyclic homology
Connes' spectral sequence
is a powerful tool in cyclic cohomology that relates the Hochschild cohomology of an algebra to its cyclic cohomology
It is a spectral sequence with E2 term given by the Hochschild cohomology of A with coefficients in the dual algebra A∗, converging to the periodic cyclic cohomology of A
The spectral sequence provides a way to compute cyclic cohomology groups in terms of more familiar Hochschild cohomology groups
Noncommutative de Rham theory
is a generalization of the classical de Rham theory of differential forms and cohomology to the noncommutative setting
It is based on the notion of and the noncommutative analog of the de Rham complex
The noncommutative de Rham cohomology groups provide invariants of noncommutative spaces and can be related to cyclic cohomology via the Connes-Chern character
Chern character in cyclic cohomology
The is a homomorphism from the K-theory of a noncommutative algebra A to its periodic cyclic cohomology
It generalizes the classical Chern character, which maps vector bundles to their characteristic classes in de Rham cohomology
The Chern character provides a way to relate the algebraic K-theory of noncommutative spaces to their global geometric properties captured by cyclic cohomology
Examples and applications
Noncommutative geometry has found numerous applications in various areas of mathematics and physics, providing new insights and techniques for studying a wide range of phenomena
The following examples showcase some of the key instances where noncommutative geometry has been successfully applied
Noncommutative tori
Noncommutative tori are a class of noncommutative spaces obtained by deforming the algebra of functions on the ordinary torus using a noncommutative parameter θ
They provide a simple yet rich example of a noncommutative space, exhibiting features such as irrational rotation algebras and Morita equivalence
Noncommutative tori have been studied extensively in the context of noncommutative geometry and have connections to various areas, including quantum field theory and string theory
Quantum groups and spaces
Quantum groups are noncommutative analogs of classical Lie groups, obtained by deforming the algebra of functions on the group using a quantum parameter q
They give rise to noncommutative spaces known as quantum spaces, such as the quantum plane and quantum spheres
Quantum groups and spaces have been widely studied in the context of noncommutative geometry and have applications in areas such as knot theory, integrable systems, and conformal field theory
Matrix geometries
Matrix geometries are a class of noncommutative spaces obtained by replacing the algebra of functions on a classical space with a matrix algebra
They provide a way to construct finite-dimensional approximations of continuous spaces and have been used to study the geometry of quantum systems and matrix models
Matrix geometries have been applied in various contexts, including the study of D-branes in string theory and the formulation of noncommutative gauge theories
Fuzzy spaces and spheres
Fuzzy spaces are a type of noncommutative space obtained by quantizing the algebra of functions on a classical space using a truncated harmonic expansion
Fuzzy spheres are a particular example, obtained by quantizing the algebra of functions on the ordinary sphere using a finite-dimensional matrix algebra
Fuzzy spaces and spheres have been studied as regularizations of continuous spaces and have found applications in areas such as quantum field theory and matrix models
Key Terms to Review (36)
Alain Connes: Alain Connes is a French mathematician known for his foundational work in noncommutative geometry, a field that extends classical geometry to accommodate the behavior of spaces where commutativity fails. His contributions have led to new understandings of various mathematical structures and their applications, bridging concepts from algebra, topology, and physics.
Bimodule: A bimodule is a mathematical structure that serves as a module for two different rings simultaneously, allowing for interaction between them. This concept is crucial in noncommutative algebra, particularly as it facilitates the study of representations and dualities of algebraic structures. Bimodules provide a way to connect different algebraic systems and enable the exploration of their properties in a unified manner.
C*-algebra: A c*-algebra is a complex algebra of bounded linear operators on a Hilbert space that is closed under the operation of taking adjoints and is also closed in the norm topology. This structure allows the integration of algebraic, topological, and analytical properties, making it essential in both functional analysis and noncommutative geometry.
Chern Character in Cyclic Cohomology: The Chern character in cyclic cohomology is a topological invariant that relates the geometry of a vector bundle to cohomological data, particularly in noncommutative geometry. It serves as a crucial tool for bridging the gap between differential geometry and algebraic topology, allowing for the study of the invariants of noncommutative spaces through cyclic cohomology, which generalizes de Rham cohomology for noncommutative algebras.
Chern-Simons Theory: Chern-Simons theory is a topological quantum field theory that is primarily concerned with 3-dimensional manifolds and the mathematical structures associated with gauge fields. It provides a framework to study the properties of knots and links through the lens of quantum mechanics and is deeply connected to the concept of differential forms and connections in differential geometry.
Connections: In the context of differential calculi, connections refer to mathematical structures that enable the comparison of tangent spaces at different points of a manifold. They provide a systematic way to define how vectors can be transported along curves on the manifold, which is essential for understanding geometric properties and curvature.
Connes' spectral sequence: Connes' spectral sequence is a powerful tool in noncommutative geometry that arises from the study of differential calculi on noncommutative algebras. It generalizes the classical spectral sequences used in algebraic topology to analyze the structure and properties of spaces defined by noncommutative geometries, allowing for the computation of homology and cohomology groups. This approach helps in understanding how differential forms can be defined and manipulated in the context of noncommutative spaces.
Covariant Derivatives: Covariant derivatives are a generalization of the concept of differentiation that accounts for the curvature of the underlying space. They provide a way to differentiate vector fields and tensors on a manifold while preserving the geometric structure, ensuring that the derivative remains a tensor of the same type. This is crucial in noncommutative geometry as it allows for the analysis of geometric and topological properties in a consistent manner.
Curvature: Curvature measures how much a geometric object deviates from being flat or straight, encapsulating the notion of bending in various dimensions. In the context of differential geometry and noncommutative geometry, curvature provides essential insights into the properties of spaces and shapes, influencing structures like vector bundles and connections. It plays a crucial role in understanding how geometric data can be abstracted and studied in noncommutative settings, allowing for the exploration of physical theories such as Yang-Mills theory.
Curvature of Connections: The curvature of connections is a geometric concept that measures the failure of a connection to be flat, providing insight into the local geometry of a manifold. It generalizes the idea of curvature from Riemannian geometry to a broader context, where connections describe how to transport vectors along curves on a manifold, allowing us to understand how space is curved in a noncommutative setting.
Cyclic cocycles: Cyclic cocycles are a type of mathematical structure used in noncommutative geometry, particularly in the context of differential calculi. They generalize the notion of differential forms and serve as a tool for defining and analyzing cohomological properties in noncommutative spaces. Cyclic cocycles are characterized by their invariance under cyclic permutations, making them essential for understanding the interplay between geometry and algebra in noncommutative settings.
Cyclic cohomology: Cyclic cohomology is a mathematical framework used to study noncommutative algebras, providing a way to compute invariants and establish connections between geometry and topology. This concept links differential forms on noncommutative spaces with the idea of cyclicity, where one can relate cycles and boundaries in a cohomological sense, paving the way for deep results in areas like noncommutative geometry and index theory.
Cyclic cycles: Cyclic cycles are algebraic structures that capture the essence of periodicity in noncommutative geometry, providing a framework for understanding differential calculus on noncommutative spaces. These cycles are important for defining various types of differential forms and operators that operate within the context of noncommutative algebras, enabling the examination of geometrical and topological properties in a novel way. Cyclic cycles allow us to explore how certain algebraic operations can reflect geometric concepts, enriching our understanding of both mathematics and physics.
Differential Calculus: Differential calculus is a branch of mathematics focused on the concept of the derivative, which measures how a function changes as its input changes. It helps in understanding rates of change and slopes of curves, providing tools for analyzing functions in various contexts such as physics, engineering, and economics. In noncommutative geometry, differential calculus allows for the extension of these ideas to more abstract mathematical structures.
Differential Forms: Differential forms are mathematical objects used in calculus on manifolds, representing a generalization of functions and vector fields. They allow for the integration over curves, surfaces, and higher-dimensional manifolds, enabling a unified approach to various concepts like gradients, divergence, and curl. Their importance lies in providing a framework for the development of Stokes' theorem and other integral theorems in advanced mathematics.
Differential operator: A differential operator is a mathematical operator defined as a function of the differentiation operator, which acts on functions to produce another function. It encapsulates the process of differentiation, allowing for the analysis of how functions change, and is fundamental in formulating and solving differential equations. Differential operators can be extended to work in various mathematical structures, including those found in noncommutative geometry.
Exterior Algebra: Exterior algebra is a mathematical framework that extends the concept of vector spaces to include the operations of exterior products, allowing for the construction of new algebraic objects called exterior forms. This framework provides powerful tools for analyzing geometric and topological properties, particularly in the context of graded algebras and differential calculus, where it helps in expressing multi-linear forms and capturing the essence of integration and differentiation in higher dimensions.
Forms: Forms are mathematical objects that generalize the notion of functions and can be thought of as tools for measuring and integrating over spaces. In the context of differential calculi, forms provide a framework for defining integrals on manifolds, enabling the analysis of geometric and topological properties through differential structures.
Gauge Bundles: Gauge bundles are mathematical structures that allow the description of fields and their symmetries in the framework of gauge theory, often used in the context of physics and noncommutative geometry. They provide a way to encode both the geometric and topological features of a system, facilitating the study of connections and curvature on these bundles. This is crucial for understanding the behavior of particles and forces in various physical theories.
Gauge groups: Gauge groups are mathematical structures that describe symmetries in physical systems, particularly in the context of gauge theories. These groups play a crucial role in defining the interactions between fundamental particles, as they dictate how the fields associated with these particles transform under certain symmetries. Gauge groups are essential for formulating the laws of physics, especially in noncommutative geometry where differential calculi are used to study these transformations.
Gauge theory: Gauge theory is a framework in physics that describes how certain symmetries dictate the interactions of fundamental particles and fields. It is crucial for understanding the forces of nature, as these theories explain how particles like electrons interact with gauge bosons, which are force carriers, through local symmetries associated with gauge groups.
Gauge transformations: Gauge transformations are changes in the fields or parameters of a physical theory that do not alter the observable predictions of that theory. They play a crucial role in ensuring the consistency of the mathematical framework used to describe physical systems, particularly in gauge theories where certain symmetries dictate the interactions and behavior of fields. These transformations are essential for understanding how different representations of the same physical situation can yield equivalent results.
Gauge-invariant actions: Gauge-invariant actions are mathematical formulations in physics that remain unchanged under local transformations of the fields involved. This concept is crucial because it ensures that the physical laws derived from these actions do not depend on arbitrary choices of local parameters, promoting consistency and symmetry in the theory. Gauge invariance often leads to the identification of conserved quantities and plays a central role in formulating gauge theories, which describe fundamental interactions.
Graded algebra: A graded algebra is an algebraic structure that is decomposed into a direct sum of subspaces, each associated with a non-negative integer grading. This grading allows for the classification of elements based on their degree, enabling a systematic approach to algebraic operations and the study of geometric and topological properties. Graded algebras play a crucial role in differential calculi by facilitating the construction of differential forms and defining operations like the exterior derivative.
Haar measure: Haar measure is a way to define a consistent notion of 'size' or 'volume' for sets in topological groups, ensuring that this measure is invariant under group operations. This concept is crucial for analyzing properties of various algebraic structures, particularly in the context of noncommutative geometry and functional analysis, allowing for integration over groups and the study of their representations.
Mikhail Gromov: Mikhail Gromov is a prominent mathematician known for his contributions to geometry, topology, and noncommutative geometry. His work has significantly influenced various areas of mathematics, especially through concepts such as Gromov-Wasserstein distances and geometric group theory. Gromov's ideas connect deeply with differential calculus, K-theory, and noncommutative probability, providing a framework to understand geometric structures in more abstract settings.
Module over a noncommutative algebra: A module over a noncommutative algebra is a mathematical structure that generalizes the notion of vector spaces, allowing for scalar multiplication by elements from a noncommutative algebra instead of just a field. This means that the multiplication in the algebra does not necessarily commute, which leads to interesting and complex behaviors. Modules can be thought of as a way to study representations of algebras and can connect with other important concepts such as differential calculi and vector bundles in noncommutative geometry.
Multivariable differential operator: A multivariable differential operator is a mathematical construct that acts on functions of multiple variables, allowing for the computation of derivatives with respect to those variables. This operator generalizes the concept of differentiation beyond single-variable calculus, facilitating operations such as gradient, divergence, and Laplacian in higher-dimensional spaces. It is essential for analyzing functions in fields like physics and engineering, where phenomena depend on several variables simultaneously.
Noncommutative de Rham theory: Noncommutative de Rham theory is a framework that extends classical differential geometry into the realm of noncommutative spaces, focusing on the study of differential forms and cohomology in contexts where the underlying algebra of functions does not commute. This theory allows for the exploration of geometric and topological properties in noncommutative settings, bridging algebraic structures and differential calculus.
Noncommutative differential calculi: Noncommutative differential calculi refers to mathematical frameworks that generalize the concepts of differential calculus to settings where the multiplication of functions is not commutative. In this context, noncommutative spaces are considered, allowing for the study of derivatives and differentials in a way that extends classical differential calculus while taking into account the underlying algebraic structure. This approach is crucial for understanding geometry and analysis in noncommutative spaces.
Noncommutative differential forms: Noncommutative differential forms are algebraic structures that extend the classical notion of differential forms to the noncommutative setting, where the multiplication of functions does not commute. This concept allows for the exploration of geometry and calculus in spaces that are defined by noncommutative algebras, enriching the framework of mathematical analysis in contexts such as quantum mechanics and operator algebras.
Noncommutative Manifold: A noncommutative manifold is a generalization of a traditional manifold where the coordinates do not commute, capturing the essence of spaces in quantum geometry. This concept allows for the exploration of geometric structures that arise in quantum physics, where the usual rules of classical geometry are modified, leading to the study of spaces that are inherently noncommutative. These manifolds serve as a foundation for various advanced topics in noncommutative geometry, linking concepts like differential structures and vector bundles.
Spectral Triple: A spectral triple is a fundamental construct in noncommutative geometry that consists of an algebra, a Hilbert space, and a Dirac operator. This structure provides a way to study geometric and topological properties of spaces that are not necessarily well-behaved in the classical sense. Spectral triples allow for the extension of geometrical concepts to noncommutative algebras, facilitating the analysis of quantum spaces, vector bundles, and various physical theories.
Torsion of connections: Torsion of connections refers to a measure of how a connection on a manifold fails to be symmetric when comparing parallel transport around infinitesimally small loops. It captures the failure of the connection to commute with itself when parallel transporting vectors along different paths. This concept is crucial in understanding the geometrical structure of spaces, especially when exploring differential calculi and their role in defining geometrical properties.
Von Neumann algebra: A von Neumann algebra is a type of operator algebra that is defined as a *-subalgebra of bounded operators on a Hilbert space which is closed in the weak operator topology and contains the identity operator. This structure plays a crucial role in the study of quantum mechanics and noncommutative geometry, particularly when discussing representations, integration, and differential calculus in infinite-dimensional spaces.
Wedge Product: The wedge product is an operation in differential geometry that combines two differential forms to produce a new differential form of higher degree. It is an essential tool for constructing exterior algebra and plays a crucial role in the context of differential calculi, where it helps to describe geometric and topological properties of manifolds.