Knizhnik-Zamolodchikov equations and spectral flow inAdS₃string theory

Abstract

I generalize the Knizhnik-Zamolodchikov equations to correlators of spectral flowed fields in AdS 3 string theory. If spectral flow is preserved or violated by one unit, the resulting equations are equivalent to the KZ equations. If spectral flow is violated by two units or more, only some linear combinations of the KZ equations hold, but extra equations appear. Then I explicitly show how these correlators and the associated conformal blocks are related to Liouville theory correlators and conformal blocks with degenerate field insertions, where each unit of spectral flow violation removes one degenerate field. A similar relation to Liouville theory holds for noncompact parafermions.

Correlation Functions of Winding Strings in AdS[sub 3]

“…As a successful check, the result in (12) was trasformed to the m-basis in [2] using (1), thus giving precisely the expression for the amplitude of three unflowed operators computed in [10], up to the powers of z i j . This is consistent with the claim that the coefficient of all winding conserving amplitudes is the same in the m-basis (for a given number of external states) [3,6].…”

“…(7) and (8), respectively generalize (5) and (6) It would be interesting to establish the connection between the results above and those in [6] regarding the KZ equation for amplitudes of spectral flowed operators, where the calculations were performed in a different basis than here.…”

“…In order to obtain the generalizations of (5) and (6) to the case of an N point function including one w = 1 operator and N − 1 unflowed operators, the proposal in [1] was to start from an amplitude of N + 1 unflowed operators, one of which was required to be a spectral flow operator. This should satisfy (5) with the replacement of N by N + 1.…”

Winding strings in AdS3

“…It is convenient to rename x 2 , z 2 → x 3 , z 3 and set x 5 ≡ y 3 = x 3 + s and z 5 ≡ ζ 3 = z 3 + ξ. By definition we have Namely, taking s = ex 13 u+e it can be shown that the exponents of ξ,ξ cancel and the integral can be rewritten as…”

“…The coefficient Q can be determined multiplying both sides of Eq. (8.12) by Φ j 3 , namely 13) and taking the expectation values as…”