# Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation

## Abstract

We continue the study of hidden Z{sub 2} symmetries of the four-point sl(2){sub k} Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector {omega}=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS{sub 3} has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it ismore »

- Authors:

- Departamento de Fisica, Universidad de Buenos Aires, FCEN UBA, Ciudad Universitaria, Pabellon I, 1428 Buenos Aires (Argentina)

- Publication Date:

- OSTI Identifier:
- 20929615

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2424789; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANTI DE SITTER SPACE; CONFORMAL INVARIANCE; CORRELATION FUNCTIONS; EQUATIONS; MATHEMATICAL SOLUTIONS; QUANTUM FIELD THEORY; STRING MODELS; SYMMETRY

### Citation Formats

```
Giribet, Gaston E.
```*Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation*. United States: N. p., 2007.
Web. doi:10.1063/1.2424789.

```
Giribet, Gaston E.
```*Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation*. United States. doi:10.1063/1.2424789.

```
Giribet, Gaston E. Mon .
"Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation". United States.
doi:10.1063/1.2424789.
```

```
@article{osti_20929615,
```

title = {Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation},

author = {Giribet, Gaston E.},

abstractNote = {We continue the study of hidden Z{sub 2} symmetries of the four-point sl(2){sub k} Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector {omega}=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS{sub 3} has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories.},

doi = {10.1063/1.2424789},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 48,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}