Analytic Torsion and Geometric Properties of Manifolds

Analytic torsion is a spectral invariant that bridges differential geometry, topology and global analysis by encoding subtle information about the spectrum of Laplace‐type operators on a manifold. Originally introduced to provide an analytic counterpart to Reidemeister torsion, it has evolved into a multifaceted tool for probing the geometry of fibrations, arithmetic quotients and spaces with singular or noncompact ends. Through heat‐kernel techniques and index‐theoretic methods, analytic torsion reveals how curvature, topology and group actions intertwine, yielding refined invariants that detect subtle geometric features invisible to classical cohomological invariants. Recent advances have expanded its reach into Arakelov geometry, the study of locally symmetric spaces and fibred boundary metrics, while deepening connections with zeta‐function regularisation, index theorems and automorphic forms. These developments have not only enriched our theoretical understanding but also provided concrete formulae for torsion in families of projective curves, arithmetic lattices and manifolds with nontrivial boundary structure.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Analytic Torsion and Geometric Properties of Manifolds publication trend

The graph below shows the total number of articles in analytic torsion and geometric properties of manifolds across all publications each year (not limited to Nature Index journals).

Technical terms

Analytic torsion: A spectral invariant defined by a zeta‐regularised product of nonzero eigenvalues of Laplace‐type operators, measuring the asymmetry in the spectrum and relating analytic data to topological torsion.

Reidemeister torsion: A combinatorial invariant of a manifold or CW‐complex capturing subtle topological information beyond homology, originally defined via bases in chain complexes.

Heat kernel: The fundamental solution to the heat equation on a manifold; its short‐time expansion encodes geometric information such as curvature and contributes to spectral invariants.

L2–analytic torsion: The limit of analytic torsion invariants for towers of covering spaces or congruence subgroups, defined using L2‐methods and often capturing asymptotic spectral behaviour.

Fibred boundary metric: A Riemannian metric on a manifold with boundary whose structure near the boundary decomposes into a product or fibration, enabling refined analysis of differential operators in noncompact settings.

References

1. Analytic torsion forms for fibrations by projective curves. Mathematische Zeitschrift (2023).
2. Analytic torsion for arithmetic locally symmetric manifolds and approximation of L 2-torsion. Journal of Functional Analysis (2023).
3. Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics. Analysis and Mathematical Physics (2022).