Grassmannian heterotic sigma model

We study the nonminimal supersymmetric heterotically deformed $\mathcal{N}=(0,2)$ sigma model with the Grassmannian target space ${\mathcal{G}}_{M,N}$. To develop the appropriate superfield formalism, we begin with a simplified model with flat target space, find its beta function up to two loops, and prove a nonrenormalization theorem. Then we generalize the results to the full model with the Grassmannian target space. Using the geometric formulation, we calculate the beta functions and discuss the ’t Hooft and Veneziano limits.

Figure 1

One-loop corrections. The wavy line is $A$, the solid line is $B$, and the composition of the wavy and solid lines is $\mathcal{B}$.

Figure 2

Two-loop corrections for $A$. The diagrams (a) and (b) are common for both the linearized and the full models, while the remaining ones involve vertices with more than three lines that appear only in the geometric model.

Figure 3

Two-loop corrections for $B$. In the same way as above the first two diagrams are applicable for both the linearized and the full models, while the other diagrams appear only in the latter one.

Figure 4

Two-loop corrections for $\mathcal{B}$. Once again, the diagrams (a) and (b) are common for both models while the diagrams (c)–(f) are applicable only in the full model.

Figure 5

Fixed point ${\rho }_{*}$ for the ratio $\rho \equiv {\gamma }^2/g^2$ as a function of $M$ and $N$. The value of ${\rho }_{*}$ is denoted by brightness (the scale is on the right) while $M$ and $N$ run along both axes. We see that if one of the parameters scales with the other one, ${\rho }_{*}$ approaches 0 no matter what that ratio is.

Figure 6

The renormalization group flow of the coupling constants $g^2$ and ${\gamma }^2$, based on the two-loop expressions (32) and (33). Here $M=10$, $N=30$. The dashed line has the slope ${\rho }_{*}=\frac{M+N}{2(MN+1)}$, so the curves above and below correspond to $\rho$ being greater and smaller than ${\rho }_{*}$, respectively. We see that the coupling constants flow to that ratio at the IR.