On Renormalization Group Flows and Exactly Marginal Operators in Three Dimensions

Abstract

As in two and four dimensions, supersymmetric conformal field theories in three dimensions can have exactly marginal operators. These are illustrated in a number of examples with N=4 and N=2 supersymmetry. The N=2 theory of three chiral multiplets X,Y,Z and superpotential W=XYZ has an exactly marginal operator; N=2 U(1) with one electron, which is mirror to this theory, has one also. Many N=4 fixed points with superpotentials W \sim Phi Q_i \tilde Q^i have exactly marginal deformations consisting of a combination of Phi^2 and (Q_i \tilde Q^i)^2. However, N=4 U(1) with one electron does not; in fact the operator Phi^2 is marginally irrelevant. The situation in non-abelian theories is similar. The relation of the marginal operators to brane rotations is briefly discussed; this is particularly simple for self-dual examples where the precise form of the marginal operator may be guessed using mirror symmetry.

Figures (6)

• Figure 1: The theory $W=\lambda(X^3+Y^3+Z^3)+hXYZ$ has one (a) or more (b,c) one-complex-dimensional spaces of SCFTs, which may (a,c) or may not (b) connect the SCFTs with $\lambda=0$ and $h=0$. The SCFTs separate regions where $\gamma>1/3$ from those with $\gamma<1/3.$ Renormalization group flow toward the infrared is indicated by arrows. The dotted line in (b) indicates a line of infrared unstable SCFTs.
• Figure 2: Renormalization group flow connecting the ${\cal N}=4$ and ${\cal N}=2$ theories of $U(1)$ with $N_f=1$. The horizontal direction represents the gauge coupling; the vertical represents the coupling of $\Phi^2$ (which grows as one moves up the diagram) or equivalently the coupling of $(Q\tilde{Q})^2$ (which grows as one moves down the diagram.)
• Figure 3: For $N_f>1$, the $U(1)$${\cal N}=4$ fixed point, at $y_0=\infty, y_k=0$, has exactly marginal deformations which preserve ${\cal N}=2$ supersymmetry. In the same space, the ${\cal N}=2$ theory with $W=0$ is an isolated SCFT. There could be more SCFTs than shown.
• Figure 4: In self-dual theories the line of SCFTs lies at $h=k$.
• Figure 5: Brane construction of $U(1)$ with $N_f=2$ out of a D3 brane stretched between two NS branes with two D5 branes placed along it. Strings with both ends on the D3 are in a vector multiplet; strings with ends on D3 and D5 are in a hypermultiplet. Rotations of the D5 and NS5 branes break ${\cal N}=4$ supersymmetry to ${\cal N}=2$ and generate the $\Phi^2$ and $(Q\tilde{Q})^2$ terms required for the exactly marginal operators. In this self-dual case, the exactly marginal operator preserves self-duality and lies at $\phi_{NS}=\phi_D$, as shown.