Abelian TQFT: Structures & Dualities

• Abelian TQFT is a class of topological field theories characterized by commutative state spaces and operators defined over vector spaces and abelian groups.
• It employs symmetric monoidal functors to map bordism categories to abelian categories, with 2D models linked to diagonal Frobenius algebras and 3D examples including Dijkgraaf–Witten and Chern–Simons theories.
• The framework underpins bulk-boundary dualities by connecting 3D topological data with 2D conformal field theories, enabling explicit computations of partition functions and quantum gravity states.

Abelian Topological Quantum Field Theory (TQFT) is a class of topological quantum field theories in which all state spaces and operators are linear objects over abelian categories—typically vector spaces, modules, or abelian groups—and all physical and algebraic data respect abelian (commutative) structures. Abelian TQFTs include, among others, Chern–Simons TQFTs for finite abelian gauge groups, state-sum models with abelian fusion rules, and continuum gauge-theoretic or homotopy-theoretic constructions where gauge groups or symmetry groups are abelian. These theories are mathematically tractable, often admit explicit computation via combinatorial, categorical, or homological techniques, and serve as model settings for understanding topological phases of matter, quantum gravity, and dualities in physics.

1. Fundamental Structure of Abelian TQFT

The core mathematical structure of an Abelian TQFT is a symmetric monoidal functor

$Z : \mathrm{Bord}_n \to \mathrm{Vect}_k$

from the category of $n$-dimensional bordisms (objects are closed $(n-1)$-manifolds, morphisms are $n$-dimensional bordisms) to the category of vector spaces over a field $k$. Unlike the general case, in abelian theories all state spaces and operator algebras are commutative, and often semisimple.

In 2D, the axiomatic classification by Abrams and others establishes that TQFTs correspond to commutative Frobenius algebras, and for abelian TQFTs these algebras are often diagonalizable into one-dimensional summands, mirroring the structure of abelian groups (Carqueville et al., 2017). In 3D, abelian TQFTs often arise from finite abelian gauge theory (Dijkgraaf–Witten models), abelian Chern–Simons theory, and their state-sum, lattice, or Hamiltonian realizations.

The assignment of partition functions and state spaces typically depends only on topological homological data such as $H_1(M,G)$ for a finite abelian group $G$, making these theories tractable via algebraic topology.

2. Higher-Categorical and Functorial Formulation

Abelian TQFTs extend naturally to the higher-categorical setting. The pseudo $n$-fold category formalism (Feshbach et al., 2011) generalizes monoidal and braided structures to encode multiple directions of composition (gluing) of bordisms, governed by associators $\alpha_i$ and interchangers $\beta_{ij}$ that satisfy coherence diagrams (pentagon, hexagon axioms). In this framework:

• An extended abelian TQFT is a symmetric monoidal functor between a pseudo $n$-fold category of bordisms and a pseudo $n$-fold linear (abelian) category.
• State spaces associated to manifolds (e.g., vector spaces or chain complexes) reflect the abelian structure, with composition and tensor product handled via categorical "extra directions," ensuring commutativity and associativity up to canonical isomorphism.

For example, $U(1)$ Chern–Simons theory, abelian BF theory, and Dijkgraaf–Witten gauge theories fit into this formalism, assigning to manifolds state spaces determined by flat abelian connections and to cobordisms explicit linear maps reflecting gluing.

This higher-categorical perspective facilitates explicit computations of partition functions, gluing rules, and dualities, and is particularly advantageous when comparing with the simplicial (quasi-category) approach of Lurie.

3. Stabilizer States, Equivalence Classes, and Topological Gravity

In the context of 3D abelian TQFT and quantum gravity (Angelinos, 30 Sep 2025), the path integral over all 3-manifolds $\mathcal{V}$ with fixed boundary $\Sigma_g$ prepares a quantum state in the Hilbert space $\mathcal{H}_{\Sigma_g}$. Crucially:

• Each 3-manifold is associated with a "stabilizer state" determined by a maximally isotropic (Lagrangian) submodule of $H_1(\Sigma_g, \mathbb Z_N)$, leading to a classification of manifolds into finitely many equivalence classes labeled by log tuples (e.g., $d=(d_1, ..., d_g)$ with $d_1 \mid ... \mid d_g \mid N$).
• The gravitational state is a sum over these classes, each represented by a stabilizer state and symplectically (mapping class) averaged for $Sp(2g, \mathbb Z)$ invariance. This can be expressed as

$$\mathcal{Z}_{\text{gravity}}[\mathcal{T}, \Sigma_g; w] = \sum_{d \in \mathbb{T}} w_d\, \mathrm{Sp}\{|\Omega_d\rangle\}$$

where $w_d$ are bulk weights, and $|\Omega_d\rangle$ are the stabilizer states.

Handlebody topologies correspond to specific classes, and their average under symplectic orbits reproduces the "Poincaré series of the vacuum", while other orbits correspond to non-handlebody geometries. All topologies within a class produce the same stabilizer state, linking the topology directly to quantum information-theoretic data.

4. Duality Between Bulk Geometries and Boundary CFTs

A prominent development in abelian 3D TQFT gravity is the emergence of a bulk-boundary duality (Angelinos, 30 Sep 2025):

• The gravitational state $\mathcal{Z}_{\text{gravity}}$ can be re-expressed as a weighted sum over boundary states—more precisely, as a weighted average of 2D conformal field theory partition functions (TBC states) on $\Sigma_g$.
• Each TBC state is associated with a "boundary Lagrangian" (a maximally isotropic submodule compatible with the quadratic/refinement structure), and orbit averages yield $Sp(2g,\mathbb Z)$-invariant states labeled by $q \in \mathbb{S}$.
• The dual description is

$$\mathcal{Z}_{\text{gravity}}[\mathcal{T}, \Sigma_g; w] = \sum_{q \in \mathbb{S}} m_q\, |\overline{\mathcal{L}_q}\rangle$$

where $m_q$ are the boundary weights, and $|\overline{\mathcal{L}_q}\rangle$ are the orbit-averaged boundary states, each interpreted as a CFT partition function.

This structure establishes a rigorous bulk-boundary dictionary: a duality between a weighted sum over bulk 3D topologies and an ensemble (weighted sum) of 2D CFTs. The duality is governed by the set of admissible topological boundary conditions (surface operators), which may be systematically classified using groupoidal and modular techniques.

5. The $\lambda$-Matrix and Mapping of Bulk to Boundary Data

The $\lambda$-matrix encapsulates the precise linear transformation between the bulk and boundary weights in this duality (Angelinos, 30 Sep 2025):

• The seed vector $v_q^d = \langle \mathcal{L}_q | \Omega_d \rangle$ provides overlaps between bulk and boundary stabilizer states.
• The intersection (overlap) matrix $D_{ij} = |\mathcal{L}_i \cap \mathcal{L}_j|^g$ is inverted to obtain a matrix $Y$.
• The $\lambda$-matrix is then given by

$\lambda_{d, q} = \sum_{q'} v_{q'}^{\,d}\, Y_{q',q}$

and the two sets of weights relate as $m_q = \sum_{d \in \mathbb{T}} w_d\, \lambda_{d,q}$.

The set of all topological boundary conditions (TBCs) admissible in the TQFT fully determines the $\lambda$-matrix, and there is a systematic, algorithmic classification of TBCs via Lagrangian submodules and compatibility with quadratic forms.

Explicit examples—such as cyclic and product symmetry groups—demonstrate that the $\lambda$-matrix need not be square (the number of classes $\mathbb{T}$ and $\mathbb{S}$ may differ). The construction extends to the computation of ensemble-averaged gravity partition functions and provides a practical bridge between bulk topological data and emergent boundary CFT states.

6. Algebraic and Physical Applications

Abelian TQFTs find diverse applications in both mathematics and physics:

• The algebraic structure (Frobenius algebras, braided fusion categories, and groupoid formalisms) enables explicit computations of partition functions, mapping class group representations, and state-sum invariants (Carqueville et al., 2017, Andreev et al., 2023).
• Abelian TQFTs underlie the classification and detection of topological phases of matter, SPT phases, and symmetry-enriched topological phases, with direct implications for the understanding of exotic statistics (such as three-loop braiding), duality defects, and emergent symmetry phenomena (Ye et al., 2015, Wang et al., 2018, Oğuz, 6 May 2025).
• They provide models for quantum gravity in low dimensions, where the sum over 3D topologies and their mapping to 2D CFTs realizes concrete ensemble holography (Angelinos, 30 Sep 2025).
• The explicit control over bulk-boundary correspondence via the $\lambda$-matrix and stabilizer states connects abelian TQFTs with stabilizer code theory and quantum information.

7. Representative Examples

Setting	Key Structure	Reference
$U(1)$ Chern–Simons TQFTs	Modular category, stabilizer states	(Angelinos, 30 Sep 2025, Andreev et al., 2023)
Dijkgraaf–Witten theories (Abelian)	State-sum, finite gauge groups	(Feshbach et al., 2011, Carqueville et al., 2017)
3D gravity via TQFT ensemble	Sum over topologies → stabilizer	(Angelinos, 30 Sep 2025)
Symmetry/duality manipulations	BF couplings, noncompact groups	(Oğuz, 6 May 2025)
Loop braiding in 3+1D	Higher-form TQFT, three-loop stats	(Wang et al., 2018)

These exemplars showcase the structural richness, algebraic tractability, and physical breadth of Abelian TQFT.

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In summary, Abelian TQFTs constitute a rich domain at the intersection of algebra, topology, quantum field theory, and quantum information, offering both foundational insight and wide applicability. Their classification via homological and categorical data, explicit dualities between bulk topologies and boundary conformal field theories, and computability make them fundamental tools in contemporary mathematical physics.