Topological Kleene Field Theories (TKFT)

• TKFT is a framework that recasts computability as smooth flows on manifolds, integrating dynamical bordisms and reaching functions to capture algebraic and topological complexity.
• It employs clean dynamical bordisms and categorical constructions to model computational processes, reflecting nontrivial topology through handles and defects.
• The theory extends traditional TQFTs by incorporating involutions, defects, and higher symmetries, offering new paradigms for understanding computation beyond classical models.

Topological Kleene Field Theories (TKFT) are a recently developed framework that reinterprets fundamental notions of computability, symmetry, and algebraic structure in terms of geometry and topology. Rooted in the interplay of field theory, topology, automata, and computation, TKFT brings together classical and modern perspectives by encoding computable processes, algebraic invariants, and categorical structures via the flow of smooth vector fields on manifolds with boundary, extended to include topological and involutive symmetries. This synthesis positions TKFT as both a generalization and enrichment of standard topological quantum field theory (TQFT), with particular emphasis on the combinatorial and categorical phenomena underlying recursion, defect, and nonorientability.

1. Foundations and Definition of Topological Kleene Field Theories

TKFT builds on Stephen Kleene's theory of partial recursive functions, but recasts computational processes as dynamical systems: smooth flows on manifolds with boundary (bordisms), whose topological structure is essential in realizing the full complexity of computation (González-Prieto et al., 20 Mar 2025). In TKFT, the basic computational objects are "clean dynamical bordisms"—pairs (W, X) where W is a manifold with boundary and X is a smooth (typically volume-preserving) vector field that is transverse to the boundary and satisfies suitable cleanness conditions. The key computational invariant is the reaching function $Z_0(W, X)$: for a Cantor-set–based encoding $\kappa: \mathbb{N}^n \rightarrow I^n$ and a point $n$ in the encoded discrete set, $Z_0(W, X)(n)$ is defined by following the flow of X from $n$ in the incoming boundary until it hits the outgoing boundary.

TKFT establishes that every partial recursive (that is, computable) function $f: \mathbb{N} \rightharpoonup \mathbb{N}$ is realized as the reaching function of a clean dynamical bordism, and conversely, every such reaching function is computable. This realization is promoted to a symmetric monoidal functor from a suitable bordism category to the category of partial recursive functions.

A central innovation is the necessity and exploitation of nontrivial topology in the manifolds and their gluings. While simple computations correspond to trivial bordisms (akin to identity, mapping cylinders, or pure cylinders), more complex functions require the introduction of handles, pants, or nontrivial gluings, reflecting structural complexity via topological invariants (such as Betti numbers) (González-Prieto et al., 20 Mar 2025).

2. Algebraic and Categorical Structures: Involution, Defects, and Symmetries

TKFT generalizes structures from oriented TQFTs by incorporating symmetry and involution, allowing for the paper of geometries and algebraic invariants on nonorientable or “Klein-type” surfaces. The classification of open Klein topological conformal field theories (TCFTs), and by extension TKFTs, is achieved via Calabi–Yau $A_\infty$-categories endowed with an involution. This involution, which acts as a contravariant functor $(\star)$ respecting $(f\circ g)^\star = g^\star \circ f^\star$ and $(f^\star)^\star = f$, models orientation-reversing processes in the topological setting (1503.02465).

The universal open–closed extension in this context constructs the closed sector from the involutive Hochschild chain complex:

$$C^{\text{inv}}_\bullet(\mathcal{C}) = \bigoplus_n \bigoplus_{a_0,\ldots,a_{n-1}} \text{Hom}(a_0,a_1) \otimes \cdots \otimes \text{Hom}(a_{n-1},a_0)[1-n]/\sim$$

with equivalence induced by the involution. The closed part of the theory (modelling, for instance, closed string states or extended observables) is quasi-isomorphic to this chain complex. Many phenomena in the closed sector, such as deformation theory or mirror symmetries with orientation reversal, are thus encoded in the open sector’s involutive algebraic data, revealing deep connections between geometry, symmetry, and categorification (1503.02465).

TKFTs are naturally situated within the language of bicategories and higher categories. The functorial formulation of field theories with defects, as described by [Benabou]'s bicategory framework, supports the assignment of defect data to bordisms, with objects, 1-morphisms, and 2-morphisms satisfying associativity and unitality up to coherent isomorphism (1107.0495).

3. Universal Construction, Foam Evaluation, and Algebraic Combinatorics

A central construction for 2D topological theories (and by extension, TKFTs) is the universal construction based on the evaluation of closed surfaces by a power series $Z(T) = \sum_{n=0}^\infty \alpha_n T^n$, with $\alpha_n$ an invariant of a surface of genus $n$ (Khovanov, 2020). The state space $A(1)$ is determined via a pairing whose Gram matrix $H = (\alpha_{i+j})$ is a Hankel matrix, and the dimension of $A(1)$ is finite exactly when $Z(T)$ is rational.

The universal construction reveals that state space structure, recurrence relations, and evaluation of cobordisms are governed by symmetric functions, specifically Schur and supersymmetric Schur polynomials. Overlapping foam evaluation (as in the Robert–Wagner formalism)—where two families of foam pieces can intersect, contributing terms $(x_i + y_j)$ or $(x_i - y_j)$ to evaluations—yields the Sergeev–Pragacz formula for supersymmetric Schur functions:

$$s_\lambda(x/y) = D_0^{-1}\sum_{\sigma \in S_M \times S_N} \text{sgn}(\sigma)\; \sigma\left(x^{\tau+\delta_M} y^{\eta'+\delta_N} \prod_{(i,j)\in\kappa} (x_i+y_j)\right)$$

as well as Day’s formula for Toeplitz determinants. These formulas connect topological theory with classical symmetric function theory, determinant evaluations, and categorification, providing a bridge to link homology and combinatorial invariants (Khovanov, 2020). Foam evaluation also extends the theory to cases that are not purely multiplicative, demonstrating that TKFTs possess robust algebraic frameworks even when they go beyond standard Frobenius algebra-based TQFTs.

4. Computational Model and Alternative Computation Paradigms

TKFT provides an alternative to Turing machine models by encoding computation as smooth flows on bordisms rather than as discrete state transitions. The key objects are:

• Clean dynamical bordisms: Smooth manifolds with boundary equipped with robust, locally well-behaved vector fields.
• Reaching functions: Maps $Z_0(W, X): \mathbb{N}^n \to \mathbb{N}^n$ encoding the outcome of flowing under $X$ from specified encoded points (typically via Cantor set embeddings).
• Gluing and topology: The complexity and non-triviality of computable functions are reflected in the nontrivial topologies required to glue elementary “basic” bordisms (read–write–shift tubes) into more complex computational flows.

The categorical structure is formalized as a symmetric monoidal functor from the category of clean dynamical bordisms to the category of partial recursive functions. The framework establishes the equivalence of computable functions and reaching functions, and claims that the topological structure required in the gluings directly reflects the complexity of the represented function (González-Prieto et al., 20 Mar 2025).

A notable claim is that TKFT may enable forms of computation beyond classical Turing and even quantum computation, by leveraging continuous flows, volume preservation, and the possibility of computation “in a single flow,” potentially leading to new paradigms in circuit and parallel computation (González-Prieto et al., 20 Mar 2025). A plausible implication is that linking topological invariants (such as Betti numbers) with computational complexity could provide new metrics for resource usage and algorithmic efficiency.

5. One-Dimensional and Boolean TKFTs: Defects, Automata, and Semirings

One-dimensional TKFTs with defects provide a bridge between categorical topology, algebra, and automata theory (Im et al., 2022, Im et al., 2022). Starting with intervals decorated by words in a finite alphabet $\Sigma$ and allowing for zero-dimensional defects at inner endpoints, the resulting linear category $C_\alpha$ can be completed (via the Karoubi envelope) to capture its essential algebraic information. The defect component is encoded in a symmetric Frobenius algebra $K$; the Karoubi envelope is equivalent to the quotient of the Frobenius–Brauer category of $K$ by negligible morphisms, revealing that all nontrivial invariants arise solely from the algebra $K$.

In the Boolean-valued case, the state spaces constructed via universal methods relate closely to the algebraic invariants of automata theory—specifically, the syntactic semiring of regular languages as defined by Polák. The state space $A(+-)$, built from Boolean linear combinations of decorated intervals and equipped with natural composition, realizes a semiring structure refining the classic syntactic monoid. This categorification connects the universal construction of topological field theories with classical ideas from formal languages and automata (Im et al., 2022).

6. Locality, Factorization Algebras, and Extended Structures

TKFTs are compatible with modern, extended, and local formulations of field theory. Results on codescent and classifying spaces show that global field theory data can be uniquely reconstructed from local restrictions on open sets, under the general axioms of locality and codescent for bordism categories (Grady et al., 2020). This matches the expectation that quantum observables in field theories should be packaged as factorization algebras over manifolds.

In hybrid settings (topological–holomorphic field theories), rigorous constructions of factorization algebras and proof of UV finiteness and anomaly cancellation enable the correct assembly of quantum observables in perturbative TKFTs (Wang et al., 11 Jul 2024). The factorization property guarantees that local quantum corrections combine consistently to yield global observables, and the vanishing of potential anomalies is established via careful analysis of Feynman graph integrals and boundary contributions.

7. Symmetry, Generalized Symmetry, and Anomalies

TKFT inherits and extends the robust symmetry frameworks developed for fully-extended topological field theories (Gripaios et al., 2022). In this context, symmetries are organized not just as ordinary group actions, but as a hierarchy of higher (q-form) symmetries encoded in ∞-groupoids. The automorphism group of a TKFT thus includes not only the usual 0-form symmetries but also higher categorical symmetries, with automorphisms interpreted as loops in the space of field theories (the loop space).

A refined analysis distinguishes between global and gauge symmetries and classifies anomalies (obstructions to gauging) into metaphysical (structural incompatibility) and unphysical (failures that do not appear in fully-extended settings, due to homotopical triviality). The correspondence between extended symmetry groups and automorphism groups of field theories, and the explanation of classical–quantum discrepancies in symmetry content, is applicable to TKFTs, especially in their involutive and nonorientable regimes.

Table: Key Constructions and Invariants in TKFT

Structure	Mathematical Realization	Relevance in TKFT
Clean dynamical bordism	(W, X) with X a smooth vector field on bordism W	Computational modeling of functions
Calabi-Yau $A_\infty$-category + involution	Algebraic model for open/closed sectors	Encoding of defect/involutive data
Hankel/Supersymmetric Schur polynomials	Evaluation of state spaces and foam intersections	Connections to combinatorics
Syntactic semiring (Polák)	Boolean semiring of words modulo evaluation	Automata-theoretic invariants
Symmetric Frobenius algebra K	Karoubi envelope of 1d theories with defects	Encodes open–closed structure

Conclusion

Topological Kleene Field Theories provide a comprehensive, unifying framework merging the fields of computation, topology, field theory, and category theory. By encoding computable processes as flows on topological bordisms, classifying algebraic and categorical structures via defects and involutions, and leveraging construction techniques from universal TQFTs and factorization algebras, TKFT connects recursion theory, algebraic combinatorics, and higher symmetries within a common geometric language. A plausible implication is that further exploration of the topological features of computation within TKFT will yield new insights into computational complexity, symmetry, and the categorification of classical invariants, as well as potential applications in quantum computation, fluids, and parallel algorithm design.