Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory

TL;DR

This paper develops a geometric framework for Wilson surfaces in higher gauge theory using derived coadjoint orbits, extending classical orbit theory to a higher categorical setting, and lays groundwork for quantization.

Contribution

It introduces a novel derived coadjoint orbit theory and constructs associated line bundles and Poisson structures for higher gauge theories.

Findings

Formulated a higher version of Kirillov-Kostant-Souriau theory

Constructed derived unitary line bundles and Poisson structures

Identified a derived Bohr–Sommerfeld quantization condition

Abstract

This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4--dimensional higher Chern--Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr--Sommerfeld quantization condition is then identified. A version of derived prequantization is proposed. The difficulties hindering a full quantization, shared with other approaches to higher quantization, are pinpointed and a possible way--out is suggested. The theory we elaborate provide the geometric underpinning for the field theoretic constructions of the companion paper.