\part{Introduction and overview}
\input{sections/01-introduction}
\input{sections/02-overview}

\part{Supersymmetry for random band matrices}

\noindent Part II contains the calculus of Grassmann variables and Berezin integrals, the exact representation of the source ratios as the integral of a chain of supermatrices, and the sigma model on $[0, 1]$ in \zcref{s:sigma}, with its reduction at rank one in \zcref{s:sigma-radial}. \zcref[S]{s:ov-organization} describes what each section of Part II does, and \zcref{s:grassmann} is written for readers who have not used anticommuting variables.

Part III uses the dictionary \zcref{l:rep-dictionary} between resolvent entries and insertions in the chain, and the saddle manifold of \zcref{s:rep-heuristics}. A reader of Part III also needs the source supermatrices and local sources of \zcref{s:rep-sources}, the Sch\"afer--Wegner cycle of \zcref{s:rep-cycle}, and the onsite weights and the chain integral of \zcref{d:rep-chain}, in terms of which the dictionary is stated. Part IV starts from the exact representation \zcref{p:rep-exact} and from the anchored identity \zcref{p:rep-anchored} for partial chains. Both parts use the target space of \zcref{s:sigma}, the partition function with its semigroups, and the localization identity. The sigma model of rank one of \zcref{s:sigma-radial} is used throughout Part III and nowhere in Part IV.

A reader who takes the definitions on trust may skip the proofs in \zcref{s:grassmann,s:sigma-target}, and the proof of the regular realization in \zcref{s:regularity}. The start of \zcref{s:sigma} says which of its definitions a reader of Part III needs.

\input{sections/03-grassmann}
\input{sections/04-representation}
\input{sections/05-sigma-model}
\input{sections/05b-rank-one}