Non-Semisimple Arithmetic TQFT: A Cobordism Functor from Arithmetic Topology to Defect-Sector Quantum Invariants

This paper constructs a non-semisimple arithmetic topological quantum field theory by extending the arithmetic cobordism program to categories with projective objects. The starting point is the scalar defect invariant Z_\varphi(M) of closed arithmetic 3-manifolds introduced previously from opaque bad-prime strata in the modular spectral Langlands setting. The present work upgrades that scalar invariant to a full 2+1-dimensional TQFT functor on arithmetic cobordisms. The source is the arithmetic cobordism category \mathrm{ArithCob}_3(F), whose objects are arithmetic surfaces \mathrm{Spec}\,F_v and whose morphisms are arithmetic 3-manifolds \mathrm{Spec}\,\mathcal O_{F,S}, with composition governed by Poitou–Tate duality. The target is the non-semisimple TQFT of Costantino–Geer–Patureau-Mirand and De Renzi–Geer–Patureau-Mirand (DGGPR), applied to the Drinfeld center Z(kG\text{-mod}) \simeq D(kG)\text{-mod} at primes \ell \mid |G|. For the explicit model G=S_3 and \ell=3, the paper computes the structure of the modular category D(\mathbf F_3S_3), including its simple objects, projective indecomposables, Cartan matrix, modified trace, Higman ideal, and modified S-matrix. It proves that the DGGPR admissibility condition has a direct arithmetic interpretation: an arithmetic 3-manifold is admissible precisely when each connected component contains a prime whose decomposition group has order divisible by \ell. This identifies bad-prime ramification as the source of the non-semisimple defect sector. The resulting symmetric monoidal construction assigns admissible decorated cobordisms to arithmetic cobordisms, and evaluation by the DGGPR functor yields the non-semisimple arithmetic TQFT. On closed arithmetic 3-manifolds, the partition function decomposes into a semisimple contribution and a defect correction arising from projective objects. The defect term is invisible to the ordinary categorical trace and is detected only by the modified trace. The paper works out the explicit example of S_3-extensions of \mathbf Q at p=31, where the defect sector contributes nontrivially. To the author’s knowledge, this is the first explicit construction of a surgery-based 3-manifold invariant whose input category is Langlands-type in this sense, and the first arithmetic cobordism functor landing in a non-semisimple quantum field theory of DGGPR type.