On the relation between effective supersymmetric actions in different dimensions

TL;DR

This paper explores how supersymmetric effective actions in different dimensions are interconnected, revealing that renormalization properties in 4D gauge theories relate to lower-dimensional models, and establishing links between (0+1)D and (1+1)D theories.

Contribution

It demonstrates the connection between renormalization in 4D supersymmetric gauge theories and the metric on moduli space, and relates effective actions across different dimensions.

Findings

Renormalization of the effective charge is linked to the metric on the moduli space.

Supersymmetry constrains modifications of the metric, supporting nonrenormalization theorems.

A relationship between (0+1)-dimensional and (1+1)-dimensional Lagrangians is established.

Abstract

We make two remarks: (i) Renormalization of the effective charge in a 4--dimensional (supersymmetric) gauge theory is determined by the same graphs and is rigidly connected to the renormalization of the metric on the moduli space of the classical vacua of the corresponding reduced quantum mechanical system. Supersymmetry provides constraints for possible modifications of the metric, and this gives us a simple proof of nonrenormalization theorems for the original 4-dimensional theory. (ii) We establish a nontrivial relationship between the effective (0+1)-dimensional and (1+1)-dimensional Lagrangia (the latter represent conventional Kahlerian sigma models).