OU-HET 382

hep-th/0104184

April 2001

Supersymmetric Nonlinear Sigma Models

on Ricci-flat Kähler Manifolds with Symmetry

Kiyoshi Higashijima^{*}^{*}*
E-mail: ,
Tetsuji Kimura^{†}^{†}†
E-mail: and
Muneto Nitta^{‡}^{‡}‡
E-mail:

Department of Physics, Graduate School of Science, Osaka University,

Toyonaka, Osaka 560-0043, Japan

We propose a class of supersymmetric nonlinear sigma models on the Ricci-flat Kähler manifolds with symmetry.

String theory propagating in a curved spacetime is described by a conformally invariant nonlinear sigma model in two dimensions. Spacetime supersymmetry of the string theory requires worldsheet supersymmetry. The conformal invariance is realized in finite field theories with vanishing -functions. In this letter, we propose a class of nonlinear sigma models on Ricci-flat Kähler manifolds with symmetry.

Two-dimensional supersymmetric nonlinear sigma models are described by (anti-)chiral superfields: () where , and are complex scalar, Dirac fermion, auxiliary scalar fields, respectively. To define supersymmetric nonlinear sigma models on the Ricci-flat Kähler manifolds with symmetry, we prepare dynamical chiral superfields (; ), belonging to the vector representation of , and an auxiliary chiral superfield , belonging to an singlet. The most general Lagrangian with symmetry, composed of these chiral superfields, is given by

(1) |

where is the invariant defined by

(2) |

and is an arbitrary function of . In the Lagrangian (1), we can assume that is a positive real constant, using the field redefinition. By the integration over the auxiliary field , we obtain the constraint among the superfields , , whose bosonic part is

(3) |

The manifold defined by this constraint with the Kähler potential is a non-compact Kähler manifold with the complex dimension , where the symmetry acts as a holomorphic isometry. One of the component, say , can be expressed in terms of the independent fields ()

(4) |

To obtain the Ricci-flat Kähler manifold with symmetry, we calculate the Ricci form and solve the Ricci-flat condition. To do this, we first calculate the Kähler metric of the manifold:

(5) |

where the prime denotes differentiation with respect to , and is given by (4). Here we have used the same letter for its lowest component: . Using this metric the nonlinear sigma model Lagrangian is written as

(6) |

where is the Riemann curvature tensor and [1]. Without loss of generality, an arbitrary point on the manifold can be transformed by the symmetry to

(7) |

where is complex. The Kähler metric on this point is

(8) |

and its determinant is given by

(9) |

The Ricci form is given by . Hence, the Ricci-flat condition is equivalent to the condition that the determinant is a constant up to products of holomorphic and anti-holomorphic functions: . From this condition, we obtain a differential equation

(10) |

where is a constant. Using this equation, the metric (5) can be rewritten as

(11) |

where is given by (4). Therefore, we only need the solution of but not itself, to calculate the Ricci-flat Kähler metric.

To solve the nonlinear differential equation (10), we transform it to a linear differential equation:

(12) |

The general solution of is immediately obtained as

(13) |

where is an integration constant and . In order to obtain the finite solution at , we must set the parameter the solution of

(14) |

Using this equation, we obtain an integral representation of :

(15) |

By performing the integration we can express using the hypergeometric function :

(16) |

We can also obtain the Ricci-flat metric by substituting to (11). We thus have obtained -dimensional Ricci-flat Kähler manifolds with symmetry.

When is odd ( and ), the hypergeometric function reduces to a polynomial:

(17) |

If is even ( and ) except for [see (19a), below, for the solution], the solution can be written as

(18) |

where . Explicit expressions of for to are

(19a) | ||||

(19b) | ||||

(19c) | ||||

(19d) | ||||

(19e) | ||||

(19f) |

For definiteness, we give the explicit solution of for and . For , can be obtained as

(20) |

By the field redefinition ( C), this Kähler potential becomes

(21) |

Here, and are holomorphic and anti-holomorphic functions, respectively, which can be eliminated by the Kähler transformation. Thus we obtain the free field theory with the flat metric as we expected, since the Ricci-flatness implies the vanishing Riemann curvature in real two-dimensional manifolds. The solution of for can be calculated, to yield

(22) |

The Ricci-flat Kähler metric for is

(23) |

where is given in (4). This defines a (real) four-dimensional hyper-Kähler manifold with symmetry, the Eguchi-Hanson space [2]. The metric for calculated from (19c) coincides with the metric of the deformed conifold, obtained earlier in [3].

We discuss the limit of vanishing . In this limit, the manifold becomes a conifold, in which the point represented by in (7) is singular. In this limit, we can obtain the explicit solution of for any , given by

(24) |

In the limit, the Kähler potential (24) becomes the simplest form:

(25) |

Therefore, when we discuss non-perturbative effects of the sigma model on the conifold defined by (24), using the expansion method, it is sufficient to consider the simplest Kähler potential instead of (24).

If we prepare invariants of other groups in the superpotential of (1), we would obtain Ricci-flat Kähler manifolds with other symmetry.

We have studied the nonlinear sigma models on Ricci-flat Kähler manifolds. These models have vanishing -function up to the fourth order in the perturbation theory [4, 5]. Despite the appearance of non-zero -function at the four-loop order, we will be able to obtain the conformally invariant field theory for the background metric related to the Ricci-flat manifolds through non-local field redefinition [6].

After the completion of this work we came to know that our metric was also discussed in other context [7, 8]. Let us give a comment on the relation to their work. Defining the new variable by the relation in Eq. (15), the integral can be rewritten as , where has been defined by . Here and are the same ones introduced in [8]. We obtain the Kähler potential differentiated by , given by

(26) |

## Acknowledgements

We would like to thank Gary Gibbons for pointing out references [8]. We are also grateful to Takashi Yokono for useful comments.

## References

- [1] B. Zumino, Phys. Lett. 87B (1979) 203.
- [2] T. Eguchi and A. J. Hanson, Phys. Lett. 74B (1978) 249.
- [3] P. Candelas and X. C. de la Ossa, Nucl. Phys. B342 (1990) 246. K. Ohta and T. Yokono, JHEP 0002 (2000) 023, hep-th/9912266.
- [4] L. Alvarez-Gaumé, Nucl. Phys. B184 (1981) 180. C. M. Hull, Nucl. Phys. B260 (1985) 182. L. Alvarez-Gaumé and P. Ginsparg, Comm. Math. Phys. 102 (1985) 311.
- [5] M. T. Grisaru, A. E. M. van de Ven and D. Zanon, Phys. Lett. 173B (1986) 423; Nucl. Phys. B277 (1986) 388; Nucl. Phys. B277 (1986) 409.
- [6] D. Nemeschansky and A. Sen, Phys. Lett. 178B (1986) 365.
- [7] M. B. Stenzel, Manuscripta Math. 80 (1993) 151.
- [8] M. Cvetič, G. W. Gibbons, H. Lü and C. N. Pope, hep-th/0012011; hep-th/0102185.