Analytic Continuation of Liouville Theory

Correlation functions in Liouville theory are meromorphic functions of the Liouville momenta, as is shown explicitly by the DOZZ formula for the three-point function on the sphere. In a certain physical region, where a real classical solution exists, the semiclassical limit of the DOZZ formula is known to agree with what one would expect from the action of the classical solution. In this paper, we ask what happens outside of this physical region. Perhaps surprisingly we find that, while in some range of the Liouville momenta the semiclassical limit is associated to complex saddle points, in general Liouville's equations do not have enough complex-valued solutions to account for the semiclassical behavior. For a full picture, we either must include "solutions" of Liouville's equations in which the Liouville field is multivalued (as well as being complex-valued), or else we can reformulate Liouville theory as a Chern-Simons theory in three dimensions, in which the requisite solutions exist in a more conventional sense. We also study the case of "timelike" Liouville theory, where we show that a proposal of Al. B. Zamolodchikov for the exact three-point function on the sphere can be computed by the original Liouville path integral evaluated on a new integration cycle.Comment: 86 pages plus appendices, 9 figures, minor typos fixed, references added, more discussion of the literature adde

String Theory on AdS Orbifolds

We consider worldsheet string theory on $Z_N$ orbifolds of $AdS_3$ associated with conical singularities. If the orbifold action includes a similar twist of $S^3$, supersymmetry is preserved, and there is a moduli space of vacua arising from blowup modes of the orbifold singularity. We exhibit the spectrum, including the properties of twisted sectors and states obtained by fractional spectral flow. A subalgebra of the spacetime superconformal symmetry remains intact after the $Z_N$ quotient, and serves as the spacetime symmetry algebra of the orbifold.Comment: 37 pages, 3 eps figures. v2: Substantial revision to section 7, on spacetime CFT interpretatio

Liouville's Imaginary Shadow

N=1 super Liouville field theory is one of the simplest non-rational conformal field theories. It possesses various important extensions and interesting applications, e.g. to the AGT relation with 4D gauge theory or the construction of the OSP(1|2) WZW model. In both setups, the N=1 Liouville field is accompanied by an additional free fermion. Recently, Belavin et al. suggested a bosonization of the product theory in terms of two bosonic Liouville fields. While one of these Liouville fields is standard, the second turns out to be imaginary (or time-like). We extend the proposal to the R sector and perform extensive checks based on detailed comparison of 3-point functions involving several super-conformal primaries and descendants. On the basis of such strong evidence we sketch a number of interesting potential applications of this intriguing bosonization.Comment: 31 page