Euclidean SYM Theories by Time Reduction and Special Holonomy Manifolds

TL;DR

This work shows that Euclidean supersymmetric Yang–Mills theories can be systematically constructed by time-direction reduction from Minkowski theories, yielding hermitian Euclidean actions and natural non-compact R-symmetries. By reinterpreting known twists directly in the Euclidean setting, the authors clarify reality conditions for twisted theories and derive three distinct twists for $d=4$, $N=4$ SYM without complexification. They extend the framework to SYM on special-holonomy manifolds, explaining how Kähler and Calabi–Yau geometries support scalar supercharges via holonomy and spinor–form identifications, and they survey broader structures on Spin(7), $G_2$, and hyper-Kähler manifolds, including DUY-type equations on Kähler three-folds and holomorphic self-duality on Calabi–Yau four-folds. Overall, the paper links Euclidean SYM, topological twists, and special holonomy, providing a geometric and field-theoretic bridge between gauge theory, cohomological field theories, and D-brane/string theory contexts. The results offer a unified approach to constructing consistent Euclidean supersymmetric theories and to exploring their topological and geometric content.

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.