# What is a chiral field?

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.

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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 term "chiral model" is used by Polyakov as an alternate name for Wess-Zumino-Witten models and variations. These are not chiral in the sense of handedness, but in the sense of resembling the sigma model. The scalar fields take value in a lie group.
• Chiral models in 4d are nonlinear sigma models of the chiral condensate and associated effective gauge fields. The skyrme model is in this category. They are also called "constrained fields" because the fields take values in a group, which is a curved space, which can be thought of as embedded in a larger Euclidean field space with a constraint that the fields have to lie on a certain surface. This is also called a "nonlinear sigma model". But the chiral models are specific nonlinear sigma models where the field values are in a lie group.
• Chiral fermions are Weyl fermions, which have a definite handedness (definite gamma-5 eigenvalue). The handedness of the antiparticle is sometimes opposite, sometimes the same, depending on the dimension. In dimensions 2,6,10 you have chiral majorana fermions, in dimensions 4,8 chiral fermions come with antifermions of opposite chirality. In odd dimensions you don't have chiral fermions, because you don't have a gamma-5.
• Chiral supermultiplets are constrained superfields which combine a complex scalar and a chiral fermion in four dimensions. They are called chiral because they only have one chirality of fermion. The scalar is not chiral, of course, but the supersymmetry mixes up the nonchiral scalar with the chiral fermion.
• The superspace for a chiral field superpotential is called a chiral superspace, because it uses only half the superspace coordinates. These superspace coordinates have a definite type of Lorentz index --- undotted or dotted--- so they are chiral.

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.

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Thank you, this answer was very useful. But I have a doubt. How is that the scalar field takes values of U(1) in sine-gordon? I believed that in this model the field was real and not complex. Im not clear about what is the constrain. –  Anthonny Sep 20 '11 at 16:42
@Anthony: U(1) is just a circle, so you can parametrize it by the angle $\theta$. The Sine-Gordon model has a periodic potential, so the real field can be thought of as an angle. If you exponentiate $e^{i\phi}$ you get a complex U(1) group element, if you like. –  Ron Maimon Sep 20 '11 at 16:46
The model of n-field is also a chiral model. In this case the field takes value on the 2-dimensional sphere and this is not a lie group. I think you should say that the fields take value in a manifold not necessarily in a lie group. –  Anthonny Sep 23 '11 at 1:50
@Anthonny-- I thought of that, but then it is no different from the definition of a nonlinear sigma model. I think the term "chiral model" is most often used in a different sense, of a lie group valued field, but it might just be a synonym for nonlinear sigma model. –  Ron Maimon Sep 23 '11 at 1:57