Euclidean SYM Theories by Time Reduction and Special Holonomy Manifolds

TL;DR

This paper introduces a method to derive Euclidean supersymmetric theories from Minkowskian ones via time reduction, revealing new insights into special holonomy manifolds and twisted SYM theories.

Contribution

It provides a systematic approach to obtain hermitian Euclidean SYM theories and reanalyzes twists of 4d N=2 and N=4 models using this framework.

Findings

Constructed twisted SYM theory on Kaehler 3-folds.

Clarified structure of SYM on hyper-Kaehler 4-folds.

Revealed hermitian Euclidean counterparts of non-mimimal SYM theories.

Abstract

Euclidean supersymmetric theories are obtained from Minkowskian theories by performing a reduction in the time direction. This procedure elucidates certain mysterious features of Zumino's N=2 model in four dimensions, provides manifestly hermitian Euclidean counterparts of all non-mimimal SYM theories, and is also applicable to supergravity theories. We reanalyse the twists of the 4d N=2 and N=4 models from this point of view. Other applications include SYM theories on special holonomy manifolds. In particular, we construct a twisted SYM theory on Kaehler 3-folds and clarify the structure of SYM theory on hyper-Kaehler 4-folds.