AP Theory IV: Intrinsic Topological Quantum Langlands Theory

TL;DR

This paper introduces AP Theory, a model-independent, topological framework for Langlands theory in four dimensions, emphasizing discrete group structures and topological dualities without reliance on traditional geometric or categorical tools.

Contribution

It presents a novel, purely topological, model-free approach to Langlands theory in 4D, connecting discrete group presentations with topological quantum field theories.

Findings

AP Theory is a universal 2D sigma-model for 4-manifolds.

Establishes a topological S-duality in a model-independent context.

Provides a discrete group-theoretic analogue of Donaldson/Seiberg-Witten theory.

Abstract

Without using any moduli, sheaves, stacks, nor any analytic, nor category-type arguments, we exhibit an analogue to Geometric Langlands Theory in an entirely model-independent, non-perturbative,purely smooth topological context in Artin Presentation Theory. A basic initial feature is that AP Theory, as a whole, is already, ab initio, a universal canonical 2D sigma-model, targeting smooth, compact, simply-connected 4-manifolds with a connected boundary, and its topological Planckian quantum starting point, as well as its cone-like, infinitely-generated at each stage, graded group of homology-preserving, but topology-changing transitions/interactions, exhibit the most general qualitative S-duality. We first point out the numerous mathematically rigorous, model-free, (i.e., intrinsic), topological AP analogues with the heuristic Kapustin-Witten version of Geometric Langlands theory, as well as the crucial differences between the two theories. The latter have to exist since AP Theory deals, a priori, essentially only with discrete group-theoretic presentation theory, not Lie group representation theory, not with category nor moduli theory, does not need classical SUSY, nor Feynman integrals, lattice models, etc., and furthermore is model-independent and characteristic of real dimension four. It will become clear that AP Langlands theory is a model-independent, cone-like, graded, compact completion and topological 'envelope' of 4D N=4 SUSY YM theory, their starting point, anchored, kept in place, so to speak, by a purely discrete group-theoretic analogue of Donaldson/Seiberg-Witten theory. Due to its discrete group-theoretic conceptual simplicity and universality, the AP Langlands program should also be considered to be a new type of intrinsic model-free symmetry, a universal Erlanger Program for Modern Physics.