Nets of Local Algebras in QFT

• Nets of local algebras are mathematical structures that assign C*- or von Neumann algebras to spacetime regions, obeying axioms such as isotony, locality, and covariance.
• They are constructed using methods like free field realizations, perturbative models, lattice techniques, and categorical approaches including Q-systems and modular theory.
• These nets underpin rigorous quantum field theory models, elucidating connections to topological phases, bulk-boundary dualities, and representation theory of Lie groups.

Nets of local algebras are mathematical structures central to the formulation of quantum field theory in the algebraic (AQFT) and operator-algebraic frameworks. Fundamentally, a net of local algebras assigns to each physically meaningful region of spacetime (or an analogous index set) an operator algebra, subject to axioms encoding isotony (inclusion/map monotonicity), locality (commutativity for spacelike or disjoint regions), covariance under symmetry groups, and additional physically motivated properties such as the existence of vacuum vectors or spectrum conditions. These nets provide a rigorous foundation for quantum field theory, topological phases of matter, and categorical approaches to defects and dualities.

1. Formal Definition and Variants

For a fixed index set $K$ (often a poset of open regions in a spacetime manifold $M$), a net of local algebras is a covariant functor

$O \mapsto \mathcal{A}(O)$

from $K$ into a category of operator algebras (e.g., unital C$^*$-algebras, von Neumann algebras), together with morphisms $\iota_{O_1,O_2}$ for $O_1\subset O_2$ that are injective $*$-homomorphisms or inclusions. The axioms are:

• Isotony: $$O_1\subseteq O_2 \implies \mathcal{A}(O_1)\subseteq \mathcal{A}(O_2)$$.
• Locality: If $O_1,O_2$ are spacelike separated (or mutually disjoint, depending on context), then $[\mathcal{A}(O_1), \mathcal{A}(O_2)] = 0$ ($*$-commutation).
• Covariance: A representation $U:G\to \mathrm{Aut}(H)$ implements $g\cdot\mathcal{A}(O) = \mathcal{A}(gO)$.
• Vacuum Cyclicity/Reeh–Schlieder: The vacuum vector is cyclic and often separating for each $\mathcal{A}(O)$.
• Haag Duality (optional): $\mathcal{A}(O)' = \mathcal{A}(O')$.

Variants exist for (i) von Neumann algebras (Vecchio et al., 2017, Bahns et al., 2017, Carpi et al., 2015), (ii) C$^*$-algebraic AQFT (Forger et al., 2013, Ciolli et al., 2011, Neeb, 12 Nov 2025), (iii) finite-dimensional boundary/bulk nets (Jones et al., 2023), and (iv) algebraic conformal nets without topology (Moreau, 8 Aug 2025).

2. Construction Techniques and Examples

2.1 Free and Interacting Quantum Field Nets

• Free scalar field: Constructed via Weyl operators $W(f)$ on the symplectic space of test functions, satisfying the canonical commutation relations (Bahns et al., 2017).
• Interacting models: The Sine-Gordon net is constructed by perturbative AQFT via relative S-matrices $S_g(f)$; automorphic properties and algebraic adiabatic limits yield well-defined interacting nets (Bahns et al., 2017).
• Lattice and loop group models: Nets from real subspaces $E(I)$ are second-quantized to Fock space algebras $R(E(I))$; lattice extensions adjoin charge-shift operators for even lattices, yielding completely rational conformal nets, including level-1 loop group nets for simply-laced groups (Bischoff, 2011, Neeb, 12 Nov 2025).

2.2 Defect and Extension Nets

Extensions can be formalized via Q-systems (or generalized Q-systems for infinite index) of intertwiners, leading to new local nets via crossed products, infinite index inclusions, and braided products for defect and phase boundary constructions (Vecchio et al., 2017).

2.3 Topological and Categorical Models

Nets can be defined in purely algebraic settings (without topologies), leading to algebraic conformal nets as functors from interval categories to associative algebras (Moreau, 8 Aug 2025). In the locally constant case, these correspond to commutative algebras, with defects, sectors, and intertwiners forming a tricategory equivalent to the tricategory of commutative algebras, bimodules, and their morphisms.

2.4 Boundary and Bulk-Boundary Correspondence

For topologically ordered spin systems, nets of ground state projections and their associated local boundary algebras provide an operator-algebraic formulation of the bulk-boundary correspondence. The DHR category of the boundary net categorifies the bulk modular tensor category of topological order (Jones et al., 2023).

3. Structural and Classification Properties

3.1 Modular Theory and Bisognano–Wichmann Property

Wedge regions and their modular objects (e.g., modular unitaries and conjugations) satisfy the Bisognano–Wichmann property, relating modular flows to geometrically meaningful symmetry transformations (boosts, dilations). The presence of Euler elements (elements $h$ with $\mathrm{ad}\, h$ diagonalizable with eigenvalues in $\{-1, 0, 1\}$) is both necessary and sufficient for the modular group to be geometrically implemented in this way; this yields standard subspace nets satisfying all AQFT axioms for a wide class of homogeneous spaces (Morinelli et al., 2023, Neeb, 12 Nov 2025).

3.2 Factor Types and Index Theory

In vacuum representations, local von Neumann algebras associated to wedge regions in standard AQFT models are type III$_1$ factors, a structure closely related to thermodynamic and ergodicity properties of the modular flow (Morinelli et al., 2023). Extensions and inclusions of local nets, including infinite index cases, are described by Jones index theory and generalized Q-systems, with implications for classification and the structure of sector categories (Vecchio et al., 2017).

3.3 Rationality and Categorical Aspects

Completely rational nets (split property, strong additivity, finite $\mu$-index) possess modular representation categories with finitely many sectors—e.g., lattice nets, holomorphic nets constructed via simple-current or code extensions (Kawahigashi et al., 2012, Bischoff, 2011). In the categorical setting, the equivalence classes of defects and sectors are organized via bimodule categories and fusion rules, with deep connections to modular tensor categories and higher-categorical structures (Moreau, 8 Aug 2025).

4. Sheaf, Topos, and Bundle Theoretic Formulations

4.1 Bohrification and Sheaf Conditions

A net of C$^*$-algebras $O\mapsto \mathcal{A}(O)$ defines a presheaf of ringed toposes $(T_U, \underline{A}_U)$ by Bohrification: at each region, $C(\mathcal{A}(U))$ is the poset of commutative subalgebras, and $T_U$ the corresponding topos. Locality of the net corresponds to the descent property (sheaf condition) for this presheaf: the net is local if and only if, for any spacelike cover, the presheaf satisfies descent by a local geometric morphism (Nuiten, 2012, Nuiten, 2011, Wolters et al., 2013).

4.2 Noncommutative Sheaves and Bundles

Given a net of local C$^*$-algebras indexed over a “good base” in a locally compact Hausdorff space $X$, the universal enveloping algebra is a C$_0(X)$-algebra $A$ whose continuous sections $\Gamma_0(X, \mathcal{A})$ recover the global algebra. The local algebras are quotients by kernel ideals vanishing outside open sets. This association gives rise to a sheaf of locally C$^*$-algebras or a “noncommutative space” in the sense of Forger–Paulinot, encompassing both the net and its bundle representation (Forger et al., 2013).

4.3 Independence and Topos-theoretic Axioms

Independence properties (microcausality, extended locality, C$^*$-independence) of a net can be recast as sheaf-theoretic descent and locality properties. Full C$^*$-independence plus a counit law among commutative subalgebras is equivalent to the sheaf condition for the net as a presheaf of C$^*$-spaces (Wolters et al., 2013).

5. Representation Theory, Homotopy, and Extensions

5.1 Net Representations and Homotopy Theory

The representation theory of nets (assigning left modules over each algebra, respecting inclusion maps) admits a homotopy-theoretic model structure: under mild conditions (e.g., for nets valued in differential-graded algebras or cochain complexes), categories of representations form model categories, and morphisms between nets induce Quillen equivalences when they are objectwise weak equivalences (Anastopoulos et al., 2022).

5.2 Extensions, Defects, and Tricategories

Algebraic conformal nets, along with their defects, sectors, and intertwiners, can be organized into a strict tricategory. In the locally constant case, this tricategory corresponds precisely to the tricategory of commutative algebras, bimodules, and bimodule morphisms. The fusion of defects is realized as tensor product over intermediate algebras (Moreau, 8 Aug 2025).

5.3 DHR Categories and Topological Order

For nets in topological phases, the Doplicher–Haag–Roberts (DHR) category of boundary algebra bimodules realizes the modular tensor category of bulk topological order. Boundary net fusion rules, bimodule categories, and the canonical ucp channel from bulk to boundary encode the bulk-boundary correspondence in a Hamiltonian-independent, operator-algebraic framework (Jones et al., 2023).

6. Connections to Geometry and Representation Theory

Nets of local algebras are intricately linked to the geometry and representation theory of Lie groups. Critical features such as the existence of Euler elements, three-grading structure on the Lie algebra, and invariant cone fields (encoding causal structure) control whether standard nets (of real subspaces or operator algebras) can be constructed via second quantization from unitary representations. The classification and existence theory for nets on homogeneous and causal symmetric spaces follows from these representation-theoretic constraints (Neeb, 12 Nov 2025, Morinelli et al., 2023).

7. Summary Table: Net Structures and Features

Setting	Algebras	Index Set	Key Properties/Axioms
Minkowski AQFT (Haag–Kastler, etc.)	von Neumann/C$^*$	Causally complete opens	Isotony, locality, covariance, etc.
Lattice/topological boundary nets	finite C$^*$	Rectangles/intervals	Axioms LTO1–4, bulk-boundary channel
Categorical conformal nets	associative K-algebras	Intervals, diagrams	Isotony, locality, additivity
Second quant. of real subspaces	von Neumann	Regions in G/H	Standardness, symmetry, modularity
Generalized Q-system extensions	von Neumann	Regions	Infinite index, braided fusion

These nets provide the scaffolding for a deep interplay between algebraic, geometric, topological, and categorical aspects of quantum field theory, operator algebras, and condensed matter/quantum information theory.