On Generalized Pfaffians

TL;DR

This paper proves a conjecture proposing a polynomial generalization of Pfaffians for anti-symmetric matrices, with implications for super conformal field theories and the local Hitchin image in type D.

Contribution

It provides a complete proof of a conjecture extending Pfaffians, linking algebraic polynomials to super conformal field theories and Hitchin systems.

Findings

Confirmed the polynomial square root conjecture for generalized Pfaffians

Connected algebraic structures to super conformal field theories

Characterized the local Hitchin image for type D

Abstract

The determinant of an anti-symmetric matrix $g$ is the square of its Pfaffian, which like the determinant is a polynomial in the entries of $g$. Studies of certain super conformal field theories (of class S) suggested a conjectural generalization of this, predicting that each of a series of other polynomials in the entries of $g$ also admit polynomial square roots. Among other consequences, this conjecture led to a characterization of the local Hitchin image for type D. Several important special cases had been established previously. In this paper we prove the conjecture in full.