Construction of the two-dimensional O(3) nonlinear sigma model as an interacting continuum quantum field theory with a mass gap

Theorem 1.2: along bare couplings $\beta_N=H+\frac{\log L}{2\pi}N+O(\log N)$ with spacing $L^{-N}$, suitably normalized smeared lattice spins converge in law and in all moments; the limiting Schwinger distributions are Euclidean and $O(3)$ invariant, reflection positive and clustering with factorial bounds, reconstruct a local relativistic theory in $1+1$ dimensions with a unique vacuum and a Hamiltonian gap $m>0$ on the whole vacuum complement, and have a nonzero connected four-point function, so the theory is not Gaussian. Theorem 1.1: normalized by susceptibility and second-moment length, the limit exists along every real $\beta\to\infty$ without subsequences. The companion proves an isolated one-particle mass atom in the spin two-point spectral measure. Not shown: the value of the mass, the S-matrix of the constructed theory, or a continuum construction for other $O(N)$.

Theorem 1.2: along bare couplings $\beta_N=H+\frac{\log L}{2\pi}N+O(\log N)$ with spacing $L^{-N}$, suitably normalized smeared lattice spins converge in law and in all moments; the limiting Schwinger distributions are Euclidean and $O(3)$ invariant, reflection positive and clustering with factorial bounds, reconstruct a local relativistic theory in $1+1$ dimensions with a unique vacuum and a Hamiltonian gap $m>0$ on the whole vacuum complement, and have a nonzero connected four-point function, so the theory is not Gaussian. Theorem 1.1: normalized by susceptibility and second-moment length, the limit exists along every real $\beta\to\infty$ without subsequences. The companion proves an isolated one-particle mass atom in the spin two-point spectral measure. Not shown: the value of the mass, the S-matrix of the constructed theory, or a continuum construction for other $O(N)$.