Correlation functions in conformal Toda field theory I

Abstract

Two-dimensional sl(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.

Comments on Rényi entropy in AdS3/CFT2

“…However, unlike in Liouville theory, for generic external states the correlator does not obey an ordinary differential equation that could be used to fix f Toda . (This is related to our previous point about W conformal blocks [79].) In the case that the external operators have some degeneracy, the correlator does obey a differential equation, but it is of higher than second order [79,80].…”

“…(This is related to our previous point about W conformal blocks [79].) In the case that the external operators have some degeneracy, the correlator does obey a differential equation, but it is of higher than second order [79,80]. It would be interesting to understand this work's conclusions from the behavior of f T oda for the specific case of external twist operators Φ ± .…”

“…Despite this, we are entitled to ask how the twist field correlators specifically behave in the semiclassical limit, c → ∞. When the higher spin algebra is W N , for example, the where the semiclassical block f Toda is determined by a regularized on-shell Toda action evaluated on some solution of the Toda equation [79]. However, unlike in Liouville theory, for generic external states the correlator does not obey an ordinary differential equation that could be used to fix f Toda .…”

Null-polygonal minimal surfaces in AdS4 from perturbed W minimal models

“…To compute this, we first note that the normalized three-point function is generally given by [71,72] 14) where O α := B(α)V α is the normalized operator so that O α (z)O α * (0) normalized = |z| −4h(α) , and α * for α is defined through (α, e j ) = (α * , e k−j ). The normalization constant B(α) is given by with the product being over the positive roots.…”

“…In particular, a class of the structure constants has been evaluated in eq. (1.53) of [71]: We now concentrate on the case of k = 4. Since α 0 = −bω 2 for p = 5, q = 7 of our interest, we first evaluate C −bω 2 −be 0 ,−bω 2 with b being generic.…”