Partition functions of Supersymmetric Gauge Theories in Noncommutative R^{2D} and their Unified Perspective

TL;DR

This paper explores the invariance of partition functions in noncommutative supersymmetric gauge theories across dimensions, revealing their equivalence via dimensional reduction and calculating specific cases like the N=4 U(1) theory.

Contribution

It demonstrates the equivalence of partition functions in noncommutative gauge theories across different dimensions through dimensional reduction techniques.

Findings

Partition functions are invariant under noncommutative parameters.

Partition functions in different dimensions are equivalent via dimensional reduction.

Calculated the partition function of N=4 U(1) gauge theory in noncommutative R^4.

Abstract

We investigate cohomological gauge theories in noncommutative R^{2D}. We show that vacuum expectation values of the theories do not depend on noncommutative parameters, and the large noncommutative parameter limit is equivalent to the dimensional reduction. As a result of these facts, we show that a partition function of a cohomological theory defined in noncommutative R^{2D} and a partition function of a cohomological field theory in R^{2D+2} are equivalent if they are connected through dimensional reduction. Therefore, we find several partition functions of supersymmetric gauge theories in various dimensions are equivalent. Using this technique, we determine the partition function of the N=4 U(1) gauge theory in noncommutative R^4, where its action does not include a topological term. The result is common among (8-dim, N=2), (6-dim, N=2), (2-dim, N=8) and the IKKT matrix model given by their dimensional reduction to 0-dim.