🔢Noncommutative Geometry Unit 9 – Spectral triples

Key Concepts and Definitions

• Spectral triples consist of a triple $(A, H, D)$ where $A$ is an involutive algebra represented on a Hilbert space $H$, and $D$ is an unbounded self-adjoint operator on $H$
• Involutive algebra $A$ represents the noncommutative space and encodes its algebraic structure
  • Elements of $A$ are typically bounded operators on $H$
  • Involution on $A$ is compatible with the adjoint operation on $H$
• Hilbert space $H$ serves as the space of spinors or fermions in the noncommutative setting
  • $H$ is a complex vector space equipped with an inner product
  • Completeness of $H$ ensures well-defined limits and convergence properties
• Dirac operator $D$ is an unbounded self-adjoint operator on $H$ that encodes the metric and differential structure of the noncommutative space
  • $D$ satisfies the compact resolvent condition, meaning $(D - \lambda)^{-1}$ is compact for $\lambda \notin \mathbb{R}$
  • Commutator $[D, a]$ is bounded for all $a \in A$, capturing the notion of a derivative
• Spectral action principle allows the extraction of geometric information from the spectrum of the Dirac operator
• Real structure $J$ is an antilinear isometry on $H$ satisfying certain compatibility conditions with $A$ and $D$
  • $J$ encodes the reality condition and charge conjugation in noncommutative geometry

Historical Context and Development

• Noncommutative geometry emerged as a generalization of classical geometry to spaces where coordinates do not commute
• Alain Connes pioneered the field in the 1980s, developing the theory of noncommutative spaces and cyclic cohomology
• Spectral triples were introduced by Connes as a way to encode the geometry of noncommutative spaces
  • Motivated by the Atiyah-Singer index theorem and its noncommutative analogues
  • Generalize the notion of Riemannian manifolds to the noncommutative setting
• Early examples of spectral triples included the noncommutative torus and the quantum group $SU_q(2)$
• Connes' work on the Standard Model of particle physics using spectral triples sparked further interest in the field
  • Demonstrated the potential of noncommutative geometry in physics
• Developments in cyclic cohomology and K-theory provided algebraic tools for studying spectral triples
• Spectral triples have been applied to various areas of mathematics and physics, including quantum field theory, quantum gravity, and index theory

Mathematical Foundations

• Spectral triples are built upon the framework of functional analysis and operator algebras
  • Hilbert spaces provide the underlying structure for representing noncommutative spaces
  • Bounded and unbounded operators on Hilbert spaces are central objects of study
• C*-algebras, which are norm-closed self-adjoint algebras of bounded operators on a Hilbert space, serve as the algebras of observables in noncommutative geometry
  • Gelfand-Naimark theorem establishes a correspondence between commutative C*-algebras and locally compact Hausdorff spaces
• Von Neumann algebras, which are weakly closed self-adjoint algebras of bounded operators, provide a framework for studying measure theory and topology in the noncommutative setting
• Kasparov's KK-theory, a bivariant version of K-theory, plays a crucial role in the study of spectral triples
  • KK-theory allows the construction of maps between K-theory groups of C*-algebras
  • Provides a powerful tool for analyzing the index theory of spectral triples
• Cyclic cohomology, a noncommutative analogue of de Rham cohomology, is used to define and study the Chern character of spectral triples
  • Connes-Chern character maps K-theory classes to cyclic cohomology classes
  • Allows the computation of index formulas and the study of geometric invariants

Components of Spectral Triples

• Involutive algebra $A$ represents the algebra of observables or functions on the noncommutative space
  • $A$ is typically a dense subalgebra of a C*-algebra, allowing for smooth or continuous functions
  • Involution on $A$ is given by the adjoint operation, satisfying $(ab)^* = b^*a^*$ and $(a^*)^* = a$
• Hilbert space $H$ serves as the space of square-integrable spinors or fermions
  • Elements of $H$ are typically represented as column vectors or functions
  • Inner product on $H$ is denoted by $\langle \cdot, \cdot \rangle$ and satisfies conjugate symmetry and linearity in the second argument
• Dirac operator $D$ is an unbounded self-adjoint operator on $H$ that encodes the metric and differential structure
  • $D$ is typically an elliptic differential operator, such as the Dirac operator on a spin manifold
  • Commutator $[D, a] = Da - aD$ is bounded for all $a \in A$, capturing the notion of a derivative
• Real structure $J$ is an antilinear isometry on $H$ satisfying $J^2 = \pm 1$ and $JD = \pm DJ$
  • $J$ implements a reality condition and relates the algebra $A$ to its commutant $A'$
  • Sign of $J^2$ and commutation with $D$ depend on the dimension and signature of the noncommutative space
• Grading operator $\gamma$ is a self-adjoint unitary operator on $H$ satisfying $\gamma^2 = 1$ and $\gamma D = -D\gamma$
  • $\gamma$ provides a $\mathbb{Z}_2$-grading on $H$, decomposing it into positive and negative eigenspaces
  • Existence of $\gamma$ depends on the dimension of the noncommutative space (even or odd)

Properties and Theorems

• First-order condition: The commutator $[D, a]$ is bounded for all $a \in A$
  • Ensures that the Dirac operator encodes a differential structure on the noncommutative space
  • Allows for the construction of differential forms and the study of noncommutative geometry
• Compact resolvent condition: The resolvent $(D - \lambda)^{-1}$ is a compact operator for $\lambda \notin \mathbb{R}$
  • Guarantees that the spectrum of $D$ is discrete and has finite multiplicities
  • Enables the use of spectral theory and the study of summability properties
• Regularity condition: The elements of $A$ and $[D, A]$ are smooth with respect to the derivation $\delta(a) = [|D|, a]$
  • Ensures that the algebra $A$ consists of sufficiently smooth functions
  • Allows for the construction of smooth structures and the study of differential geometry
• Reality condition: The real structure $J$ satisfies $J^2 = \pm 1$, $JD = \pm DJ$, and $Ja^*J^{-1} = a^{op}$ for all $a \in A$
  • Implements a notion of charge conjugation and relates the algebra $A$ to its opposite algebra $A^{op}$
  • Sign conditions depend on the dimension and signature of the noncommutative space
• Poincaré duality: The K-theory and K-homology of the spectral triple are isomorphic
  • Establishes a noncommutative analogue of Poincaré duality in classical geometry
  • Relates the algebraic and analytic aspects of the noncommutative space
• Connes' trace theorem: The Dixmier trace of a suitable power of the Dirac operator computes the volume of the noncommutative space
  • Provides a noncommutative analogue of the classical volume form
  • Allows for the computation of geometric invariants and the study of noncommutative integration theory

Applications in Noncommutative Geometry

• Noncommutative tori: Spectral triples on irrational rotation algebras provide examples of noncommutative spaces
  • Encode the geometry of tori with noncommutative coordinates
  • Used to study the geometry and topology of noncommutative tori, such as the Gauss-Bonnet theorem and the Connes-Rieffel projective modules
• Quantum groups: Spectral triples on quantum groups, such as $SU_q(2)$, provide examples of noncommutative spaces with additional symmetry
  • Encode the geometry of quantum homogeneous spaces and quantum flag manifolds
  • Used to study the representation theory and harmonic analysis on quantum groups
• Noncommutative Standard Model: Spectral triples have been used to construct noncommutative geometries that unify gravity and the Standard Model of particle physics
  • Connes-Chamseddine spectral action principle allows for the derivation of the Standard Model Lagrangian from a spectral triple
  • Provides a geometric interpretation of the Higgs mechanism and the origin of the Higgs boson
• Index theory: Spectral triples provide a framework for studying index theory in noncommutative geometry
  • Connes-Moscovici local index formula computes the index of twisted Dirac operators using cyclic cohomology
  • Allows for the computation of topological invariants and the study of the geometry of noncommutative spaces
• Quantum field theory: Spectral triples have been used to construct noncommutative quantum field theories
  • Provide a framework for regularizing and renormalizing quantum field theories on noncommutative spaces
  • Used to study the UV/IR mixing and the noncommutative Euclidean field theory

Computational Techniques

• Spectral action computation: The spectral action functional can be computed using heat kernel techniques and pseudodifferential calculus
  • Involves the asymptotic expansion of the heat kernel trace and the computation of Seeley-DeWitt coefficients
  • Allows for the derivation of the Standard Model Lagrangian and the study of gravity-matter coupling
• Dirac spectrum computation: The spectrum of the Dirac operator can be computed using various techniques, such as the Fourier transform and the Poisson summation formula
  • Involves the study of the eigenvalues and eigenfunctions of the Dirac operator
  • Allows for the computation of geometric invariants and the study of the spectral geometry of noncommutative spaces
• Cyclic cohomology computations: Cyclic cohomology groups can be computed using various techniques, such as the Connes-Moscovici residue cocycle and the Chern character
  • Involves the study of cyclic cocycles and their pairing with K-theory classes
  • Allows for the computation of index formulas and the study of the noncommutative geometry of algebras
• Noncommutative differential forms: Differential forms on noncommutative spaces can be constructed using the spectral triple data
  • Involves the study of the algebra generated by $A$ and $[D, A]$, known as the algebra of differential forms
  • Allows for the study of the de Rham complex and the computation of cohomology groups
• Spectral zeta functions: Zeta functions associated to the Dirac operator can be used to study the spectral geometry of noncommutative spaces
  • Involves the study of the analytic continuation and the poles of the zeta function
  • Allows for the computation of geometric invariants and the study of the heat kernel asymptotics

Advanced Topics and Current Research

• Lorentzian noncommutative geometry: Extension of spectral triples to Lorentzian signature and the study of noncommutative spacetimes
  • Involves the use of Krein spaces and the notion of Wick rotation
  • Used to study the causal structure and the noncommutative geometry of spacetime
• Quantum gravity: Application of noncommutative geometry to the construction of quantum theories of gravity
  • Involves the use of spectral triples and the spectral action principle to derive gravitational theories
  • Used to study the noncommutative structure of spacetime at the Planck scale and the possible resolution of singularities
• Noncommutative gauge theories: Construction of gauge theories on noncommutative spaces using spectral triples
  • Involves the study of connections and curvature on noncommutative vector bundles
  • Used to study the noncommutative geometry of gauge fields and the possible unification of fundamental interactions
• Noncommutative index theory: Extension of index theory to noncommutative spaces and the study of higher index formulas
  • Involves the use of cyclic cohomology and the local index formula of Connes-Moscovici
  • Used to study the geometry and topology of noncommutative spaces and the computation of invariants
• Quantum groups and noncommutative geometry: Study of the geometry of quantum groups using spectral triples and the theory of covariant differential calculi
  • Involves the use of Hopf algebras and the notion of quantum homogeneous spaces
  • Used to study the noncommutative geometry of quantum flag manifolds and the representation theory of quantum groups
• Noncommutative arithmetic geometry: Application of noncommutative geometry to the study of arithmetic and number-theoretic objects
  • Involves the use of spectral triples and the theory of noncommutative motives
  • Used to study the geometry of algebraic varieties over noncommutative rings and the possible unification of geometry and arithmetic