Grassmann-Berezin Calculus and Theorems of the Matrix-Tree Type

Abstract

We prove two generalizations of the matrix-tree theorem. The first one, a result essentially due to Moon for which we provide a new proof, extends the ``all minors'' matrix-tree theorem to the ``massive'' case where no condition on row or column sums is imposed. The second generalization, which is new, extends the recently discovered Pfaffian-tree theorem of Masbaum and Vaintrob into a ``Hyperpfaffian-cactus'' theorem. Our methods are noninductive, explicit and make critical use of Grassmann-Berezin calculus that was developed for the needs of modern theoretical physics.