What question did this study set out to answer?

The research aims to define a cobordism category and classify topological quantum field theories relevant to arithmetic topology.

March 6, 2026Open Access

Arithmetic field theory via pro-p duality groups

Key Points

The research aims to define a cobordism category and classify topological quantum field theories relevant to arithmetic topology.
Utilized pro-p groups and relative Poincaré duality to define a cobordism category.
Classified two-dimensional topological quantum field theories on the created cobordism categories.
Investigated Frobenius algebras tied to automorphisms of p-adic integers.
Focused on arithmetic Dijkgraff–Witten theory for finite gauge p-groups.
Successfully classified topological quantum field theories on the defined cobordism categories.
Derived formulae for counting Galois extensions of local p-adic fields with specific Galois groups.

Abstract

Abstract Using the theory of pro- p groups and relative Poincaré duality, we define a type of cobordism category well suited to arithmetic topology. We completely classify topological quantum field theories on these two-dimensional versions of our cobordism categories. This classification uses Frobenius algebras with extra operations corresponding to automorphisms of the p-adic integers. We look in more detail at the example of arithmetic Dijkgraff–Witten theory for a finite gauge p-group in this setting. This allows us to deduce formulae counting Galois extensions of local p-adic fields whose Galois groups are the given gauge group.

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