Title: A formalism for Arithmetic Quantum Field Theory
Abstract:

Arithmetic Topology, first pioneered by Mazur in 1963, draws analogies between number theory and low dimensional topology, primes and knots, and surface and p-adic fields.
On one hand, quantum field theory can be expressed in terms of the geometry and topology of low-dimensional manifolds, on the level of states (via the Atiyah-Segal) and on the level of observables (via the Beilinson–Drinfeld). Thus, as first proposed by Minhyong Kim (in his Arithmetic Chern-Simons Theory), one can try and find arithmetic versions of quantum field theoretic ideas.

In the talk, I will introduce a new general framework for (d+1)-dimensional arithmetic TQFT.
I will explain the classification of such TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure, this enables us to study pro-p cobordisms and TQFTs for p-adic fields and surfaces at the same time.

If time permits, I will outline how we use the above to compute a Dijkgraaf-Witten like invariants for G, a finite p-group, to get formulas for counting covers of Surfaces/p-adic fields with Galois group G (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach).

The talk is based on joint work with Oren Ben-Bassat.

No prior knowledge of Topological Quantum Field Theory or Arithmetic Topology will be assumed.


Zoom link
https://huji.zoom.us/j/88037279712?pwd=N3MwWW5RYzRTZHg4K0U2bS80Rmxjdz09