Loop Equations for + and - Loops in c = 1/2 Non-Critical String Theory

TL;DR

This paper develops new loop equations for + and - loops in c=1/2 non-critical string theory, extending algebraic structures and connecting to matrix model constraints, with applications to string field theory Hamiltonians.

Contribution

It constructs comprehensive loop equations for all genera in c=1/2 non-critical string theory, revealing an algebraic extension of W_3 and linking to matrix model constraints.

Findings

Loop equations include + and - loops for all genera.

Generated algebra extends W_3 algebra.

Connections established to matrix-model W_3 constraints.

Abstract

New loop equations for all genera in $c = \frac{1}{2}$ non-critical string theory are constructed. Our loop equations include two types of loops, loops with all Ising spins up (+ loops) and those with all spins down ( $-$ loops). The loop equations generate an algebra which is a certain extension of $W_3$ algebra and are equivalent to the $W_3$ constraints derived before in the matrix-model formulation of 2d gravity. Application of these loop equations to construction of Hamiltonian for $c = \frac{1}{2}$ string field theory is considered.