1-Dimensional Sigma Models

Question

I'm currently interested in 1-dimensional (linear) Sigma Models.

In the theory of 2-Dimensional GLSM, the fields can be viewed as an embedding of the worldsheet in some target Manifold of higher dimension.

Is there a similar, geometrical, interpretation of the 1-dimensional case?

So far i could only find references to superconformal quantum mechanics, where the $N$-dimensional target space is taken to be $N$ particles. I'm however interested in geometrical insight

See for example Witten's famous "Phases of N=2 Theories in 2 Dimensions", is there an equivalent in 1-Dimension?

Answer 1

Is there a similar, geometrical, interpretation of the 1-dimensional case?

Of course there is.

A 1d sigma model describes the embedding of a worldline into a manifold. In a linear sigma model, the manifold is isomorphic to $\mathbb{R}^n$. There's not much in the way of interesting geometry here. You can get curved target space geometry by putting a potential on the linear target space and looking at the low-energy effective field theory; the low-lying modes will be described by a nonlinear sigma model to the minima of the potential.

Whether this is trivial or an open research problem depends on whether you're a physicist or a mathematician.

For basic constructions of 1d nonlinear sigma models, look for papers by D. Fine & S. Sawin or B. Driver.

Fun puzzle: The real point of Witten's Phases paper is that the low energy dynamics of different phases of the gauged linear sigma model are described by different nonlinear sigma models, with different target spaces. What happens to this story when one compactifies the worldsheet to get 1d sigma model?