983 - d nonlinear sigma model

Project Year: 2017
REU Student (s): Arthur Wang | Rutgers University
Student 1 Institution: Rutgers University-New Brunswick
Project Mentor: Anders Buch
Project Mentor Area: Mathematics
Project Abstract: The holonomy group of the tangent bundle of the two dimensional unit sphere is precisely the special orthogonal group. Let V denote the tangent space at some fixed point p on the sphere. We seek to consider the invariant space of $(V^{\otimes k})^{SO(2)} = \{x \in V^{\otimes k} \mid gx = x, \forall g \in SO(2)\}$ . Every tensor $(V^{\otimes k})^{SO(2)}$ defines globally a differential operator on $C^{\infty}(S^2)$ . This is of interest since $\coprod\limits_{k \in \mathbb{N}}(V^{\otimes k})^{SO(2)}$ forms a noncommutative vertex operator algebra. The Beltrami-Laplace operator Δ sits inside some $(V^{\otimes k})^{SO(2)}$ . The focus of this paper is to consider the eigenfunctions of Δ and characterize the set of functions in the orbit of differential operators in $\coprod\limits_{k \in \mathbb{N}}(V^{\otimes k})^{SO(2)}$ acting on an eigenfunction. Such a result will allow us to mathematically construct a 2-d nonlinear σ-model.

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