1 \todo{beginning of scott's attempt to write down what fields are...}

     3 \todo{beginning of scott's attempt to write down what fields are...}

     2 

     4 

     3 \newcommand{\manifolds}[1]{\cM_{#1}}

     5 \newcommand{\manifolds}[1]{\cM_{#1}}

     4 \newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}}

     6 \newcommand{\closedManifolds}[1]{\cM_{#1}^{\text{closed}}}

     5 \newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}}

     7 \newcommand{\boundaryConditions}[1]{\cM_{#1}^{\bdy}}