The Affine Connection of Supersymmetric SO(N)/Sp(N) Theories

TL;DR

This paper explores the geometric and covariance properties of generating functions in supersymmetric SO(N)/Sp(N) theories, revealing that one function acts as an affine connection and connecting these findings to Riemann surface theory.

Contribution

It demonstrates that the T generating function is an affine connection and constructs a modified connection leading to integrable generating functions, providing a geometric interpretation of recent theoretical maps.

Findings

T is an affine connection that cannot be integrated along cycles.

A modified affine connection allows for consistent integration of generating functions.

Provides a geometric explanation for the map in Sp(N) theories with antisymmetric tensors.

Abstract

We study the covariance properties of the equations satisfied by the generating functions of the chiral operators R and T of supersymmetric SO(N)/Sp(N) theories with symmetric/antisymmetric tensors. It turns out that T is an affine connection. As such it cannot be integrated along cycles on Riemann surfaces. This explains the discrepancies observed by Kraus and Shigemori. Furthermore, by means of the polynomial defining the Riemann surface, seen as quadratic-differential, one can construct an affine connection that added to T leads to a new generating function which can be consistently integrated. Remarkably, thanks to an identity, the original equations are equivalent to equations involving only one-differentials. This provides a geometrical explanation of the map recently derived by Cachazo in the case of Sp(N) with antisymmetric tensor. Finally, we suggest a relation between the Riemann surfaces with rational periods which arise in studying the Laplacian on special Riemann surfaces and the integrality condition for the periods of T.