^{1,2}

^{2}

^{2,3}

^{3}.

We study the nonminimal supersymmetric heterotically deformed

Sigma models have a long history dating to 1960

The most widely considered target spaces of the sigma models studied in the literature are

Since the Grassmannian is a Kähler manifold, the Lagrangian of the (2,2) model can be written as

In Sec.

The nonminimal heterotic (0,2) model combines the original (2,2) model and its deformation that partly breaks supersymmetry. Before exploring their interplay, let us take a separate look at the two parts—the interaction imposed by geometry and the interaction generated by deformation. The case of the undeformed (2,2) model is well studied in the literature

The complex dimension of the Grassmannian manifold is

In the (0,2) formalism, the Lagrangian of the linearized model acquires the form of

In this section, due to the flatness of the target space, the position of the fields’ indices is not of importance. For the same reason, here we do not use the barred indices either.

Here we have already introduced the

After eliminating the auxilary fields

The diagrams for one-loop wave function renormalization are shown in Fig.

One-loop corrections. The wavy line is

Two-loop corrections for

Two-loop corrections for

Two-loop corrections for

As was shown in

The positivity of the beta function

We now prove a version of the nonrenormalization theorem for the interaction term and the kinetic term of the

Let us take a look at the

Assume that we have a function of

The renormalized kinetic term for

The complex Grassmannian manifold is a manifold consisting of all

The complex dimension of the Grassmannian manifold is

Since interactions in the undeformed model are caused by geometry, we now review some details about geometric structure of the target space. The Kähler potential of the Grassmannian manifold is given by

The Kähler metric is obtained as follows:

The small-

The Ricci tensor is proportional to the metric,

We are now in a position to apply these results to the full model. which combines the geometric structure with the partial supersymmetry breaking. To break supersymmetry, we add another term to the Lagrangian, which is similar to the one in the previous section.

Since

Next, we proceed to finding the beta functions of this model.

In components, the Lagrangian of the full model

Here

In the

Note that the Lagrangian

Extending the analysis to the two-loop level (see Figs.

As expected, Eqs.

We now briefly discuss what we can learn from the results above regarding the large-

From

There are a number of limits one can study:

The second case is somewhat analogous to the Veneziano limit it QCD

In this case, the beta functions can be defined purely in terms of the t’ Hooft parameters,

What are the other consequences of considering such a limit? Let us take a look at the fixed-point behavior of the theory. The ratio

Fixed point

The renormalization group flow of the coupling constants

From

In this paper, we have discussed the perturbative aspects of the heterotically deformed (0,2) Grassmannian

A similar kind of behavior takes place in the full model—however, only for a certain range of parameters. There, in contradistinction with the linearized model, for a different set of parameters one may also end up with a well-defined asymptotically free theory, reaching its conformal point at the UV.

Lastly, we have discussed the large-

The authors are grateful to J. Chen both for the stimulating exchange that gave a start to this work and for the advise provided at its later stages. The authors also thank X. Cui for useful discussions on the topics raised in this paper. The work of M. S. was supported in part by the DOE Grant No. DE-SC0011842. E. K. was supported in part by the Robert E Greiling Jr. Scholarship for Particle Physics and Cosmic Rays. M. K. was supported by DOE HEP Award No. 0000240346.

Gamma matrices, metric,

Following the lines for

We begin with splitting the quantum field

In Eqs.

Here

Strictly speaking, the presented approach is noncovariant from the target-space point of view. Under the assumption that both

We then can calculate the renormalized coupling constant by integrating out the quantum corrections to the field configuration

For definiteness, we present the calculation of the one-loop beta function of the bosonic Grassmannian model. Following the steps from Sec. 28 of

Comparing the equation above with the

Next, we follow the steps of

To the lowest order, the diagram for the wave function renormalization takes the form

This way, for the

To the first order, the bosonic beta function

To proceed further, we take into account the diagrams contributing to the renormalization of