I have not found a clear definition of this. A teacher told me that it was a field having some constrains but that is not very convincing for me. He told me also that some examples could be skyrme model, sigma model and sine-gordon model. Is this right? Could you give me a complete definition of chiral fields and some examples? Please I also would very grateful if you could refer me some book or paper about this topic.
There are several inequivalent definitions, used in different contexts, which is the reason for your confusion.
The word "Chiral" originally refered to chirality, or handedness of spin along the direction of motion. This is still the most often used definition. The spinor representations of the Lorentz group in even dimensions have components with a definite eigenvalue of gamma-5 (or the higher diemnsional equivalent). The spinors two different eigenvalues are the chiralities.
In 4d, there is a two-index formalism for the Lorentz group representations, because the Lorentz group is SU(2)xSU(2). The index for one SU(2) is undotted, and the other is dotted. The dotted and undotted indices are different chiralities.
Light quarks have a "chiral symmetry" which rotates the left and right handed chiralities of the up and down quarks into each other separately, making an SU(2)xSU(2) symmetry group which includes isospin. Isospin is when you rotate both chiralities the same way, and chiral isospin is when you rotate the two chiralities oppositely.
Isospin is a symmetry of the observed particles, but the chiral isospin symmetry is broken by the vaccuum. The reason is that the quarks form a "chiral condensate", so called because it breaks the chiral symmetry. The chiral condensate is not chiral by itself, but it is called "chiral" anyway. A better name would be "chiral symmetry breaking condensate", but that is too much of a mouthful. Models of the chiral condensate are called "chiral models", even though they are usually bosonic scalars. They should better be called "chiral symmetry breaking models", but that is also too long.
The term "chiral model" is then used to describe effective scalar or scalar-vector theories of mesons, which describe the effective oscillations of the chiral condensate. It has been extended to describe many different nonlinear sigma models which resemble the original Gell-Mann Levy sigma model. A model is then called a chiral model if it can be thought of as the effective oscillations of a condensate of fermions.
The Sine-Gordon model can be thought of as a chiral model, where the scalar field takes values in a circle, which is the group U(1). So in some sense, it is the simplest chiral model. In the fermionic point of view, the bosonic field is a two-fermion composite. So the bosonic excitation can be thought of as made up of two fermions. This is a chiral model in this sense.
here is no book or paper about this topic--- the word chiral is just overloaded by convention of different authors. Once they write down a Lagrangian, however, there is no confusion.