Loop Approach to Lattice Gauge Theories

TL;DR

This paper develops a loop-based formulation for SU(2) lattice gauge theories in arbitrary dimensions, solving key constraints and expressing dynamics through symmetric 3nj symbols, with extensions to matter fields and brief SU(N) discussion.

Contribution

It introduces a complete orthonormal loop basis using prepotential operators, enabling direct transcription of gauge dynamics without redundancies in any dimension.

Findings

Explicit construction of a loop basis in terms of prepotential intertwining operators

Expression of gauge dynamics via symmetric 3nj symbols

Extension of techniques to include matter fields and brief SU(N) discussion

Abstract

We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space ${\cal H}^{L}$ using the prepotential formulation of $(d+1)$ dimensional SU(2) lattice gauge theory. The resulting orthonormal and complete loop basis, explicitly constructed in terms of the $d(2d-1)$ prepotential intertwining operators, is used to transcribe the gauge dynamics directly in ${\cal H}^{L}$ without any redundant gauge and loop degrees of freedom. Using generalized Wigner-Eckart theorem and Biedenharn -Elliot identity in ${\cal H}^L$, we show that the loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by the real symmetric $3nj$ symbols of first kind (e.g., n=6, 10 for d=2, 3 respectively). The corresponding "ribbon diagrams" representing SU(2) loop dynamics are constructed. The prepotential techniques are trivially extended to include fundamental matter fields leading to a description in terms of loops and strings. The SU(N) gauge group is briefly discussed.