Convergence of Sums Over Abelian Groups Determines 3-Manifold Partition Functions

The challenge of understanding three-dimensional space lies at the heart of modern mathematics and physics, and researchers continually seek new ways to characterise its complex structures. Thomas Nicosanti from SISSA, along with Thomas Nicosanti from INFN and Pavel Putrov from ICTP, investigate a simplified model of three-dimensional topology, focusing on how to systematically sum over all possible three-dimensional shapes with a fixed boundary. Their work demonstrates that this seemingly intractable problem can be recast as a more manageable sum over fundamental algebraic groups, offering a pathway to calculate meaningful quantities and establish convergence criteria for this summation. Significantly, the team’s analysis suggests the existence of a diverse collection of two-dimensional topological quantum field theories, and that summing over three-dimensional shapes effectively calculates the average behaviour of these theories on their boundaries, potentially bridging a gap between abstract mathematics and physical reality.

Researchers consider a toy model of three-dimensional topological quantum gravity, where the contribution of each space is determined by its partition function within a specific type of quantum field theory. This theory incorporates a crucial topological boundary condition, and the investigation focuses on summing over the homology groups associated with these spaces, providing a mathematical framework for understanding their topological properties. This approach allows exploration of how topological features influence the system’s behaviour and contributes to a deeper understanding of quantum gravity models.

At the boundary of these spaces, the quantum field theory’s partition function depends solely on the first homology group of the three-dimensional space, alongside some additional mathematical structure. This allows researchers to rewrite the sum over all three-dimensional spaces with a fixed boundary as a sum over finitely generated abelian groups, and the associated extra structure. The team presents bounds on the weights within this sum, ensuring the sum converges. Furthermore, under specific assumptions, they demonstrate the existence of a distribution of two-dimensional quantum field theories, such that the sum equates to the ensemble average of their partition functions.

TQFTs Construct 3-Manifold Invariants and Holography

This research explores the construction and study of topological quantum field theories in three dimensions, mathematical frameworks that assign vector spaces to spaces and linear maps to embeddings, satisfying specific consistency conditions. A key goal is to use these theories to define invariants of three-dimensional spaces, quantities that remain unchanged under smooth deformations. The work also investigates the role of finite groups in these theories, exploring how they can be constructed using finite groups as the symmetry group, leading to invariants sensitive to the group structure and its representations. This research connects these theories to ensemble theory and to holographic principles, aiming to understand the statistical properties of universes.

Cobordisms, manifolds with boundaries, are fundamental to this work, providing a way to relate different spaces. The research emphasizes the importance of factorization properties of cobordisms, crucial for defining consistent quantum field theories. The study also delves into global symmetries within these theories, particularly q-form symmetries, related to the existence of non-trivial cycles in the space and playing a crucial role in defining invariants. The work explores the connection between factorization properties and ensemble theory, and also investigates the connection to holographic principles.

The research discusses the relationship between these theories and Chern-Simons theory, a quantum field theory closely related to knot theory and three-manifold invariants, and also mentions the Dijkgraaf-Witten invariant, a specific three-manifold invariant constructed using Chern-Simons theory. The work delves into the use of finite group cohomology to construct three-manifold invariants, explaining how the cohomology groups of the finite group can be used to define the vector spaces and linear maps. The research explores the concept of reciprocity in the context of three-manifold invariants, discussing how reciprocity can be used to relate different invariants. The work references recent research on related topics, such as holographic duality, ensemble theory, and finite group symmetries, suggesting that the ideas presented could further develop these areas of research. This represents a sophisticated exploration of the connections between topological quantum field theory, finite group symmetries, and holographic principles, aiming to provide a mathematical framework for understanding the statistical properties of universes and for constructing invariants of three-dimensional spaces.

Topological Sums and Quantum Field Theory Correspondence

This research investigates a mathematical framework for summing over all possible three-dimensional topological spaces, analogous to a path integral in quantum gravity. The work demonstrates that this summation can be reformulated as a sum over finitely generated abelian groups, offering a more manageable approach. Importantly, the authors establish conditions under which this sum converges, providing a crucial mathematical foundation. The study further reveals a connection between this summation over three-dimensional spaces and a distribution of two-dimensional quantum field theories, suggesting that the sum over topologies can be interpreted as an ensemble average of these two-dimensional theories, establishing a “bulk-boundary correspondence” linking the properties of the higher-dimensional topological spaces to those of their lower-dimensional boundaries.