Marginal and Relevant Deformations of N=4 Field Theories and Non-Commutative Moduli Spaces of Vacua

TL;DR

This paper explores how marginal and relevant deformations of N=4 super-Yang-Mills theories lead to non-commutative moduli spaces of vacua, revealing new geometric and string duality insights.

Contribution

It introduces a novel interpretation of moduli spaces as non-commutative spaces using quantum algebra representations, linking field theory deformations to string theory geometries.

Findings

Moduli spaces can be viewed as non-commutative spaces with novel features.

Mirror symmetry is explained via T-duality in deformed geometries.

Progress in understanding K-theory and discrete anomalies in non-commutative backgrounds.

Abstract

We study marginal and relevant supersymmetric deformations of the N=4 super-Yang-Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from F-term constraints. These field theories are dual to superstring theories propagating on deformations of the AdS_5xS^5 geometry. We study D-branes propagating in these vacua and introduce the appropriate notion of algebraic geometry for non-commutative spaces. The resulting moduli spaces of D-branes have several novel features. In particular, they may be interpreted as symmetric products of non-commutative spaces. We show how mirror symmetry between these deformed geometries and orbifold theories follows from T-duality. Many features of the dual closed string theory may be identified within the non-commutative algebra. In particular, we make progress towards understanding the K-theory necessary for backgrounds where the Neveu-Schwarz antisymmetric tensor of the string is turned on, and we shed light on some aspects of discrete anomalies based on the non-commutative geometry.