Finite Theories and Marginal Operators on the Brane

TL;DR

The paper develops a Type IIB brane construction that yields broad classes of four-dimensional N=1 field theories with exactly marginal couplings, including many finite models. It establishes a concrete geometric link between brane bending and gauge beta functions, providing simple finiteness criteria (unbent branes) and a framework for analyzing elliptic and cylindrical quiver theories with multiple marginal parameters. It further explores duality structures, embedding the theories in M-theory and mapping parameter spaces to moduli spaces of punctured tori, while outlining partial duality groups for N=1 cases and highlighting open questions about complete SDR dualities. Together, these results connect brane geometry to conformal and marginal dynamics, expanding the landscape of controllable four-dimensional conformal field theories and their dual descriptions.

Abstract

We show how to use D and NS fivebranes in Type IIB superstring theory to construct large classes of finite N=1 supersymmetric four dimensional field theories. In this construction, the beta functions of the theories are directly related to the bending of branes; in finite theories the branes are not bent, and vice versa. Many of these theories have multiple dimensionless couplings. A group of duality transformations acts on the space of dimensionless couplings; for a large subclass of models, this group always includes an overall $SL(2,\ZZ)$ invariance. In addition, we find even larger classes of theories which, although not finite, also have one or more marginal operators.

Figures (13)

• Figure 1: The NS and NS$'$ branes form a grid; in each box $(i,j)$ of the grid lie $n_{i,j}$ D5 branes. The arrows denote the chiral multiplets $H_{i,j},V_{i,j},D_{i,j}$ which are in the fundamental of the group $SU(n_{i,j})$ and in the antifundamental of an adjacent group. The arrows at the far left indicate the $x^4$ and $x^6$ coordinates.
• Figure 2: The two superpotential interactions at each corner, with couplings $h^+_{i,j}$ and $h^-_{i,j}$, are represented by an oriented triangle of arrows.
• Figure 3: Construction of a model on a torus, expressed as a unit cell of a two-dimensional lattice. The unit cell of four boxes is highlighted. Note the twelve matter multiplets, indicated by the solid parts of the arrows; the dotted parts of the arrows correspond to their images under shifts of the unit cell by lattice vectors.
• Figure 4: The ${\cal N}=4$ gauge theory; the torus consists of the one highlighted box, of which the other boxes are images. The three arrows correspond to the three chiral multiplets in the adjoint representation.
• Figure 5: Construction of the ${\cal N}=2$ elliptic models. The theory has gauge group $SU(N)_1\times SU(N)_2\times \cdots \times SU(N)_k$; each box corresponds to one of these $SU(N)$ factors, as indicated by the indices in the upper left corners.