Perturbative quantization of the two-dimensional supersymmetric sigma model

Abstract

The two-dimensional nonlinear sigma model is a classical field theory whose fields are maps from a Riemann surface Σ to a Riemannian manifold X, and whose classical solutions are minimal surfaces. In this thesis, we study a supersymmetric extension, which has an additional fermionic field. Using a mathematical formulation of the Batalin–Vilkovisky formalism developed by Costello and Gwilliam, we show that a perturbative quantization of this sigma model on flat surfaces exists if and only if the first Pontryagin class p₁(TX) ∈ H⁴(X; ℂ) vanishes. If X is in addition closed and oriented, we rigorously define the partition function of the resulting quantum field theory and show that it defines a weak modular form of weight ½dim X. We calculate it exactly as the Witten genus. <br /> The partition function is determined from local data on Σ through the factorization algebra structure on quantum observables, and we show that it is a deformation of a family of free quantum field theories. We prove existence of a quantization and calculate the partition function using a generalization of Gelfand–Kazhdan formal geometry to Riemannian manifolds, which reduces them to algebraic statements and Feynman diagram calculations. Our results are a first step in the Stolz–Teichner program for constructing geometric cocycles for elliptic cohomology.