Dynamical Bordisms: Interpolating Geometry & Dynamics
- Dynamical bordisms are mathematical and physical constructs that extend classical bordism by incorporating time-dependent, causal, and process-oriented data.
- They bridge diverse fields—topology, quantum field theory, string theory, and computation—by rigorously interpolating between different geometric, algebraic, or physical regimes.
- Their applications range from modeling cosmological evolution and classifying quantum anomalies to enabling computational simulations through topological methods.
Dynamical bordisms are a class of mathematical and physical constructs that generalize the classical notion of bordism to settings that encode temporal, causal, or dynamical evolution. Originating at the intersection of topology, quantum field theory, string theory, and the theory of computation, dynamical bordisms offer a rigorous framework for interpolating between distinct geometric, algebraic, or physical regimes via manifolds or combinatorial structures equipped with additional dynamical or process-oriented data. In contemporary research, dynamical bordism underlies advances in cosmological model building, the classification of quantum anomalies, the design of computational field theories, and approaches to the swampland program in quantum gravity.
1. Conceptual Foundations of Dynamical Bordisms
Dynamical bordisms extend the classical notion of a bordism—a manifold interpolating between two boundary components—by endowing it with time dependence, causal structure, or process-like features that represent physical or computational evolution. Classical bordism, as found in topology and homotopy theory, is concerned with classifying manifolds up to the existence of an interpolating manifold with prescribed tangential structure. Dynamical bordisms, in contrast, may encode:
- Time-dependent geometries: These interpolate between spatial regions for varying cosmological, field-theoretic, or computational backgrounds (1009.3277).
- Causal structure: As in causal dynamical triangulations or combinatorial cell complexes, where sequences of operations represent discrete or continuous evolutions (Savoy, 2022).
- Process-theoretic morphisms: In category-theoretic or field-theoretic contexts, where a bordism models a transition or computation—an idea formalized in topological quantum field theories (TQFTs) and their generalizations (González-Prieto et al., 20 Mar 2025, González-Prieto et al., 14 Jul 2025).
A central intuition is that the bordism does not merely connect boundary data but models a “dynamical channel” through which analytic, geometric, or quantum information is transported, transformed, or computed.
2. Dynamical Bordisms in String Theory and Cosmology
One of the earliest and most influential appearances of dynamical bordisms is in time-dependent solutions in string and M-theory that bridge different effective descriptions across spatial domains. These “dynamical cobordisms” are constructed using spatial slices of negative curvature (often hyperbolic Riemann surfaces) with nontrivial topology, where different regions support different string-theoretic compactifications, orientifold structures, or spin conditions (1009.3277).
A canonical example involves constructing an expanding spacetime via a metric
where the part encodes a hyperbolic geometry with nontrivial topology (garters, cusps), and represents Ricci-flat compactification factors. By quotienting the hyperbolic space by a suitable discrete group, regions with different physical properties (such as type II versus heterotic strings, or oriented versus unoriented theories) are smoothly connected via expanding “domain walls.” Notably, these interpolating regions are causally connected—there are no global horizons separating them—and physical excitations can propagate between patches of distinct low-energy physics.
Key features include:
- Exactness with respect to α′ and curvature corrections—the Milne-type metrics used are obtained via quotienting flat Minkowski space, keeping the solutions under perturbative control.
- Tunable moduli—parameters such as garter length can be scaled to control the strength of corrections and maintain validity everywhere.
- Cosmological implications—such constructions allow for causally connected universes with coexisting domains governed by distinct string vacua, providing explicit dynamical “bridges” across the string theory landscape.
3. Categorical and Combinatorial Models
Beyond smooth manifolds, dynamical bordisms have been realized in discrete or combinatorial settings to capture the dynamics of field theories and geometric structures without assuming underlying manifold topology (Savoy, 2022).
- Combinatorial Cell Complexes: In this approach, geometry is replaced by a cell complex with a rank function, closure/intersection properties, and an inclusion-reversing duality operation. “Cobordisms” are built as inclusions and removals of cell complexes (denoted ), and composition is achieved via sequences of reduction and collapse maps.
- Causal Structure: Such frameworks capture causal evolution by decomposing bordisms into midsections (slices), akin to discrete time steps. This allows the modeling of dynamics similar to Causal Dynamical Triangulations, but in a more combinatorial and categorical context.
- Functorial Field Theories: The combinatorial category, equipped with a well-defined composition and duality, serves as a domain for field-theoretic functors, generalizing Atiyah’s TQFT axioms without recourse to smooth or continuous manifolds.
These constructions provide tools for building discrete quantum gravity models and understanding how causal structure and duality entangle with the possibility of dynamical evolution in a broader class of mathematical universes.
4. Dynamical Bordisms in Computation: Topological Kleene Field Theories
A recent paradigm leverages dynamical bordisms to “geometrize” computation, encoding partial recursive (computable) functions as flows on smooth manifolds with boundary, termed Topological Kleene Field Theories (TKFTs) (González-Prieto et al., 20 Mar 2025, González-Prieto et al., 14 Jul 2025).
- Dynamical Bordisms as Computations: In these frameworks, input and output data are embedded in boundary discs via Cantor set encodings. A smooth (volume-preserving) vector field is constructed on the bordism so that its flow computes the desired function—each trajectory exiting the input boundary determines the output according to the reaching function (or exit map). The main result is that any computable function can be simulated by the reaching function of a suitably constructed clean dynamical bordism.
- Role of Topology: Nontrivial bordism topologies (e.g., handles, loops, pair-of-pants structures) are essential. Trivial (cylinder-like) bordisms cannot realize generic computable functions, owing to topological constraints on continuity when restricted to the Cantor embedding.
- Categorical Formulation: The equivalence between clean dynamical bordisms and computable functions is realized as a monoidal functor from a category of bordisms to the category of partial recursive functions.
These results provide a bridge between discrete and continuous models of computation, demonstrating that computation can be fully encoded within the smooth and topological dynamics of a bordism. This approach also suggests new measures of computational complexity defined via topological invariants.
5. Dynamical Bordisms in Quantum Field Theory and Topology
The role of dynamical bordisms extends into areas such as topological field theory, representations of bordism categories, and the paper of quantum anomalies.
- Bordism Categories and Unitary Representations: In the context of pseudomanifolds, bordisms with additional coloring or structure serve as morphisms in a category. Representations of infinite symmetric groups produce functors from this bordism category to the category of Hilbert spaces and operators, turning the composition of geometric or combinatorial bordisms into the passage of states through evolutions described by bounded operators (1501.04062).
- Propagation of Orientations and Floer Gradings: In index theory, orientations and Floer gradings on elliptic differential operators can be propagated through bordisms. This categorical structure allows the index data (traditionally a number) to “evolve” functorially along a sequence of bordisms, crucial for applications in gauge theory and moduli space constructions (Upmeier, 2023).
- Invertible Topological Field Theories: Dynamical aspects also appear in the paper of extended and invertible topological field theories, where the symmetric monoidal bordism category—possibly equipped with tangential or duality structure—serves as the source of a functor landing in a Picard category. Technical refinements (e.g., in the topology on parameter spaces) ensure that the categorical and geometric realization of bordism is stable under transitions to “discrete time,” maintaining the essential dynamical and homotopy-theoretic information (Schommer-Pries, 2017).
- Cobordism Models for Topological Invariants: In low dimensions, dynamical decompositions of the bordism category have provided classifying spaces that model topological cyclic homology (TC) and explicit cocycles for Miller–Morita–Mumford classes, showing how dynamical bordism structure encodes information about invariants relevant to algebraic -theory and moduli problems (Steinebrunner, 2020).
6. Dynamical Bordisms and the Swampland
Dynamical bordisms are instrumental in recent work on the swampland program and quantum gravity.
- Cobordism Conjecture and Boundaries: In the context of the cobordism conjecture, it is argued that consistent quantum gravity theories must allow for the dynamical ending of spacetime—formally, every compact spacetime should be cobordant to nothing. In explicit constructions in AdS/CFT, dynamical bordism solutions realize ends-of-the-world (ETW) branes connecting asymptotic AdS vacua to boundaries via a smooth geometric transition, manifesting localization of gravity and the necessity of boundaries for anomaly cancellation (Huertas et al., 2023).
- String Theory, U-Dualities, and Global Symmetry Breaking: Novel computations of bordism groups associated to duality bundles in string and M-theory settings identify necessary defects and singular backgrounds required to break otherwise global symmetries, following the predictions of the cobordism conjecture. These include cases where non-geometric twists or singular objects are essential for compatibility with swampland constraints (Braeger et al., 21 May 2025, Debray, 2023, Debray et al., 2023).
7. Current Trends and Open Directions
Dynamical bordisms remain an active area of research, with ongoing developments in both mathematical formalism and physical applications:
- Refinements in Algebraic Topology: Progress continues in understanding twisted, equivariant, or spectral-sequence based computations of bordism groups with sophisticated tangential and duality structures, relevant for the classification of anomalies, global symmetries, and consistent field-theoretic backgrounds (Debray, 2023, Braeger et al., 21 May 2025).
- Dynamical Bordisms in Computation and Complexity: Connections between the topology of the underlying bordism and computational complexity measures offer prospects for new models of computation and may inform both the limitations and potential of geometric computation relative to Turing and quantum models (González-Prieto et al., 20 Mar 2025, González-Prieto et al., 14 Jul 2025).
- Categorical and Combinatorial Generalizations: Ongoing studies seek to extend dynamical bordism ideas to purely combinatorial frameworks, enabling field theories and dynamical evolution to be defined without reference to manifold structures, with potential implications for discrete quantum gravity and lattice models (Savoy, 2022).
- Physical Realizability and Universality: There are open questions concerning the physical realization of dynamical bordisms in continuous systems such as fluid dynamics, celestial mechanics, and condensed matter, and whether features like universality and computational supremacy can be realized or observed in nature (González-Prieto et al., 14 Jul 2025).
Dynamical bordisms thus represent both a powerful abstraction and a concrete toolkit for unifying topological, geometric, physical, and computational phenomena in modern mathematical physics.