Integrality for TQFTs

TL;DR

This paper explores how restricting cobordisms affects the coefficient rings in TQFTs, leading to new functors and invariants related to cyclotomic integers and prime power covers of 3-manifolds.

Contribution

It introduces methods to reduce coefficient rings in TQFTs, constructs new functors to modules over cyclotomic integers, and defines novel invariants from strong shift equivalence and integrality.

Findings

Reduced coefficient rings for TQFTs associated with SO(3) at odd primes.

Construction of functors to modules over cyclotomic integer rings.

New invariants for prime power cyclic covers of 3-manifolds.

Abstract

We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers :Z [zeta_{2p}], if p \equiv -1 mod{4}, and Z [zeta_{4p}], if p \equiv 1 \pmod{4}, where zeta_k is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs but the modules are only guaranteed to be projective.