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

TL;DR

We address marginal and relevant deformations of 4D $N=4$ SYM and develop a non-commutative algebraic framework to describe their moduli spaces via F-term constraints. The approach treats vacua as finite-dimensional irreducible representations of deformed algebras, organizing multi-brane states through a symmetric-product moduli space and capturing D-brane fractionation. By connecting field theory deformations to dual string backgrounds, we reveal mirror symmetry via T-duality on a $T^2$-fibration and derive an algebraic K-theory description that encodes discrete anomalies and brane charges. The results provide a stringy, open-closed duality perspective on deformed AdS/CFT backgrounds and lay groundwork for non-commutative algebraic geometry in D-brane physics.

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.