Topological Quantum Field Theory

• Topological Quantum Field Theory (TQFT) is a framework that assigns algebraic invariants to topological manifolds using symmetric monoidal functors.
• It employs functorial, state-sum, and extended approaches to encode manifold invariants and captures dualities relevant to quantum fields.
• TQFT bridges mathematics and physics, underpinning advances in topology, quantum computation, and the study of quantum phases of matter.

Topological Quantum Field Theory (TQFT) is a rigorous mathematical formalism and physical framework that assigns algebraic invariants to topological manifolds in a way that is insensitive to their geometric or metric structure. TQFTs are symmetric monoidal functors from suitable categories of cobordisms to algebraic categories such as vector spaces, modules, or more general categorical structures, encoding the “cutting and gluing” properties of manifolds by corresponding algebraic operations. Emerging at the interface of mathematics and theoretical physics, TQFT has led to deep developments in topology, category theory, representation theory, and the paper of quantum phases of matter.

1. Functorial Foundations and Axiomatic Formalism

The modern formulation of TQFT follows Atiyah’s axiomatic perspective: an $n$-dimensional TQFT is a symmetric monoidal functor

$Z : \mathrm{Bord}_n \to \mathcal{A}$

where $\mathrm{Bord}_n$ is the category (or higher category) whose objects are closed oriented $(n-1)$-manifolds and whose morphisms are oriented $n$-dimensional cobordisms (considered up to diffeomorphism or piecewise linear homeomorphism), and $\mathcal{A}$ is typically a category of vector spaces, Hilbert spaces, or modules over a ring. The functoriality encodes that gluing of manifolds corresponds to composition of the assigned linear maps, and the monoidal structure mirrors the behavior under disjoint union (e.g., $Z(E \sqcup F) \cong Z(E) \otimes Z(F)$, $Z(\varnothing) = k$ for base ring $k$) (Carqueville et al., 2017).

In this setting, to each closed $(n-1)$-manifold $E$ one assigns a vector space $Z(E)$, and to each $n$-dimensional cobordism $M: E_1 \to E_2$ a linear map $Z(M): Z(E_1) \rightarrow Z(E_2)$, such that all relations induced by gluing and rescaling by diffeomorphism class are reflected at the algebraic level. This structure is not merely formal: it mirrors the properties expected of quantum field theory path integrals specialized to topological contexts (partition functions, state spaces, amplitude composition) [(Carqueville et al., 2017); (Zhang et al., 2011)].

2. Categorification, Extended TQFT, and Higher Categories

Standard n-dimensional TQFTs capture invariants of closed $(n-1)$-manifolds and $n$-manifold cobordisms, but do not encode local or stratified information (e.g., surfaces with boundary, corners). This motivates extended TQFT, where one allows assignment of data even to manifolds of lower dimension (points, edges, faces), leading to symmetric monoidal 2- or $n$-functors from higher categories of stratified cobordisms (with corners, boundaries, etc.) to higher-categorical algebraic targets (e.g., bicategories, n-categories) [(Feshbach et al., 2011); (Cui, 2016)].

In the Ehresmann approach, the $n$-fold category of cobordisms with corners has $k$-morphisms corresponding to $k$-dimensional submanifolds and associators/interchangers governing gluing in orthogonal directions. In the formalism of (Feshbach et al., 2011), an extended TQFT is a symmetric monoidal functor from the pseudo-$n$-fold category of cobordisms with corners to a pseudo-$n$-fold category of sets, vector spaces, or more structured objects. This higher-categorical framework encompasses local-to-global phenomena, accommodates locality of field theory, and is crucial for realizing dualities and extended operators (lines, surfaces, etc.). Notably, it provides a foundational basis for the Baez-Dolan/Lurie "cobordism hypothesis," relating fully extended TQFTs to algebraic objects with invertible duals.

3. State-Sum Constructions and Invariants of Manifolds

A constructive and computationally vital perspective on TQFT is provided by state-sum models, where one triangulates or cellulates an $n$-manifold and assigns algebraic data (e.g., group elements, fusion category labels, representation-theoretic data) to cells, with weights determined by local rules. Partition functions (manifold invariants) and state spaces (vector spaces assigned to boundaries) are computed as finite sums over colorings, with weights built from associators, braiding, or cohomological data, and are required to be invariant under changes of decomposition (Pachner moves).

Prominent examples include:

• Dijkgraaf–Witten theory: For finite group $G$, assigns to a closed $n$-manifold $M$ the invariant

$$Z(M) = \frac{1}{|G|}\sum_{\phi: \pi_1(M)\rightarrow G} \omega(\phi)$$

with a weighting by a $n$-cocycle $\omega \in H^n(G, U(1))$, realizing group cohomology as a classification of 2D and 3D TQFTs [(Feshbach et al., 2011); (Cui, 2016)].

• Crane–Yetter (4D) and Turaev–Viro (3D) TQFTs: Based on (spherical, ribbon, or even $G$-crossed braided) fusion categories, define invariants of 3- and 4-manifolds (and alterfolds, see (Liu et al., 2023)) via state sums that depend only on the topology up to piecewise-linear homeomorphism.
• G-crossed Braided Spherical Fusion Category (BSFC) Construction: For a finite group $G$, a $G$-BSFC $\mathcal{C}^G$ defines a $(3+1)$-TQFT by triangulating the 4-manifold, labeling edges with $G$, and triangles with simple objects in $\mathcal{C}^G$, and forming a total state-sum as in (Cui, 2016): $$Z_{\mathcal{C}^G}(M;T)= \sum_{F=(g,f)} \frac{\left(\frac{D^2}{|G|}\right)^{|T^0|}\left(\prod_{\beta\in T^2} d_{f(\beta)}\right)\mathrm{Tr}\left(\bigotimes_{\sigma\in T^4} Z_F^{\epsilon(\sigma)}(\sigma)\right)}{(D^2)^{|T^1|}}$$ where $D^2$ is the total quantum dimension.

These models generalize and unify a broad collection of known manifold invariants, interpolate between group-theoretic, categorical, and cohomological data, and establish exact models of topological order in condensed matter physics.

4. Examples and Physical Motivation

TQFTs arise as the low-energy effective field theories describing topological phases of matter and encode universal data such as ground state degeneracy, particle and loop braiding statistics, and protected edge modes. Important examples include:

• Chern–Simons theory: For a compact Lie group $G$ and an integer (level), the Chern–Simons path integral yields 3D TQFTs whose state spaces and manifold invariants underpin Witten–Reshetikhin–Turaev and Turaev–Viro models, modular tensor categories, and the structure of quantum Hall fluids, knot invariants, and quantum computing primitives (Zhang et al., 2011).
• SPT Phases and Abelian TQFTs: In 3D bosonic SPT phases with Abelian symmetry $G = \mathbb{Z}_{N_1} \times ...$, low energy physics is described by TQFTs with multiple $BF$ terms and higher-order topological terms such as $a^1\wedge a^2 \wedge da^2$ (Ye et al., 2015). The quantization of coefficients and nontrivial implementation of symmetry in the partition function recovers the group cohomology classification, while coupling to background fields or “gauging” leads to higher-dimensional analogs of Dijkgraaf–Witten theories.
• Higher-Dimensional Phases/Alterfold TQFT: The construction of 3+1 and 4D TQFTs from functionals on stratified piecewise linear spheres (the "alterfold" framework) yields invariants sensitive to embedded surfaces, with the (unitary) local symmetry encoded in spherical 3-categories of Ising type and explicit 20j-symbols (Liu, 25 Sep 2024, Liu et al., 2023).
• Graph Coloring and Homological TQFTs: TQFT-based homology theories, arising from functors on cobordisms, have been shown to categorify chromatic (and Penrose) graph polynomials, and to reformulate the four-color theorem and flow conjectures in terms of the non-vanishing of TQFT homology groups (Baldridge et al., 2023).

5. Algebraic Structures: Frobenius Algebras, Fusion Categories, and Higher Categories

The algebraic input to TQFTs is now understood to be:

• 1D, 2D TQFTs: Entirely classified by finite-dimensional vector spaces and commutative Frobenius algebras, respectively (following a generators-and-relations description of the bordism category) (Carqueville et al., 2017).
• 3D TQFTs: Governed by modular tensor categories (fusion categories with compatible braiding and twist structure), providing state-sum rules and modular functors. These are connected to representation categories of quantum groups at roots of unity; the algebraic data is physically realized in anyon models.
• Extended and Higher-Dimensional Theories: Require higher categories, such as fusion 2-categories, for a fully local description; see discussions in (Cui, 2016). The precise axiomatic definition of spherical fusion 2-categories remains open, as coherence and sphericity conditions are subtle and not always matched by existing concrete examples.
• Internal and Non-semisimple TQFTs: TQFTs constructed using modular, ribbon, or premodular categories with additional structure (e.g., coend objects, transparent sectors), enabling generalizations such as the internal Reshetikhin–Turaev TQFT (Lallouche, 2023) or non-semisimple TQFTs modeling B-twisted abelian gauge theories (Garner et al., 29 Jan 2024).

6. Applications and Subfields

TQFT methods have had transformative impact in several domains:

• Low-dimensional topology: Construction and computation of manifold invariants, Rigorous proofs and advances in knot theory, development of skein modules.
• Algebraic and Symplectic Geometry: Quantum cohomology, Gromov–Witten theory, and the structure of moduli spaces are encoded by TQFTs, often via “Frobenius manifolds” and semisimple virtual fundamental classes (Wakabayashi, 2017, Goller, 2017).
• Representation Theory and Geometric Group Theory: TQFT methods yield explicit formulas for modular representations of symplectic groups and for dimensions and characters of highest weight modules in positive characteristic (Gilmer et al., 2016).
• Quantum Information and Topological Phases: Explicit TQFTs model anyonic systems, bosonic/fermionic SPT phases, and topological quantum computation proposals, encoding robust ground states and braiding statistics (Wang et al., 2018, Ye et al., 2015).

7. Open Questions and Ongoing Research

Despite broad success, several directions remain under active investigation:

• Refined Classification of Higher TQFTs: The definition, construction, and classification of fully extended $(n+1)$-TQFTs via $n$-categories and the unification of function-theoretic and state-sum approaches in higher dimensions (see alterfold and higher Ising type constructions) (Liu, 25 Sep 2024, Liu et al., 2023, Cui, 2016).
• Non-semisimple and Relative TQFTs: Modeling of non-unitary and non-semisimple TQFTs, necessary for a rigorous description of B-twisted supersymmetric gauge theories, Rozansky–Witten/CSRW models, and new types of relative modular categories (Garner et al., 29 Jan 2024).
• Physical Realizability and Experimental Implications: Realizing and manipulating topological phases governed by specific TQFTs in condensed matter systems, especially in (3+1)D where non-Abelian loop braiding statistics remain largely theoretical (Wang et al., 2018).
• Refinements and Generalizations: Incorporation of defects, orbifold/alterfold and Morita equivalence phenomena, and implications for quantum Fourier analysis and homological invariants in topology (Liu et al., 2023, Fock et al., 2021, Baldridge et al., 2023).

TQFT remains a central organizing principle for topological and quantum phenomena across mathematics and physics, offering a robust bridge between abstract categorical structures and concrete computational frameworks.