A characterization of the Z^n lattice

TL;DR

This paper characterizes the Z^n lattice as the unique unimodular lattice of rank n lacking vectors with certain properties, using theta series and modular forms, and connects this to 4-manifold geometry via Seiberg-Witten theory.

Contribution

It provides a novel proof that Z^n is the only unimodular lattice without specific characteristic vectors, linking lattice theory with 4-manifold topology.

Findings

Z^n is uniquely characterized among unimodular lattices by the absence of certain vectors.

The proof employs theta series and modular forms techniques.

Connects lattice properties to results in 4-manifold geometry via Seiberg-Witten theory.

Abstract

We use theta series and modular forms to prove that Z^n is the only integral unimodular lattice of rank n without characteristic vectors of norm <n, i.e. the only integral unimodular lattice not containing a vector w such that (w,w)<n and 2|(v,v+w) for all lattice vectors v. By the work of Kronheimer and others on the Seiberg-Witten equation this yields an alternative proof of a theorem of Donaldson on the geometry of 4-manifolds.