48,321 research outputs found
H(3)+ correlators from Liouville theory
We prove that arbitrary correlation functions of the H(3)+ model on a sphere
have a simple expression in terms of Liouville theory correlation functions.
This is based on the correspondence between the KZ and BPZ equations, and on
relations between the structure constants of Liouville theory and the H(3)+
model. In the critical level limit, these results imply a direct link between
eigenvectors of the Gaudin Hamiltonians and the problem of uniformization of
Riemann surfaces. We also present an expression for correlation functions of
the SL(2)/U(1) coset model in terms of correlation functions in Liouville
theory.Comment: 24 pages, v3: minor changes, references adde
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
The Stoyanovsky-Ribault-Teschner Map and String Scattering Amplitudes
Recently, Ribault and Teschner pointed out the existence of a one-to-one
correspondence between N-point correlation functions for the SL(2,C)_k/SU(2)
WZNW model on the sphere and certain set of 2N-2-point correlation functions in
Liouville field theory. This result is based on a seminal work by Stoyanovsky.
Here, we discuss the implications of this correspondence focusing on its
application to string theory on curved backgrounds. For instance, we analyze
how the divergences corresponding to worldsheet instantons in AdS_3 can be
understood as arising from the insertion of the dual screening operator in the
Liouville theory side. We also study the pole structure of N-point functions in
the 2D Euclidean black hole and its holographic meaning in terms of the Little
String Theory. This enables us to interpret the correspondence between CFTs as
encoding a LSZ-type reduction procedure. Furthermore, we discuss the scattering
amplitudes violating the winding number conservation in those backgrounds and
provide a formula connecting such amplitudes with observables in Liouville
field theory. Finally, we study the WZNW correlation functions in the limit k
-> 0 and show that, at the point k=0, the Stoyanovsky-Ribault-Teschner
dictionary turns out to be in agreement with the FZZ conjecture at a particular
point of the space of parameters where the Liouville central charge becomes
c=-2. This result makes contact with recent studies on the dynamical tachyon
condensation in closed string theory.Comment: 30 pages; no figure
A Note on Quantum Liouville Theory via Quantum Group; an Approach to Strong Coupling Liouville Theory
Quantum Liouville theory is analyzed in terms of the infinite dimensional
representations of with q a root of unity. Making full use of
characteristic features of the representations, we show that vertex operators
in this Liouville theory are factorized into `classical' vertex operators and
those which are constructed from the finite dimensional representations of
. We further show explicitly that fusion rules in this model also
enjoys such a factorization. Upon the conjecture that the Liouville action
effectively decouples into the classical Liouville action and that of a quantum
theory, correlation functions and transition amplitudes are discussed,
especially an intimate relation between our model and geometric quantization of
the moduli space of Riemann surfaces is suggested. The most important result is
that our Liouville theory is in the strong coupling region, i.e., the central
charge c_L satisfies . An interpretation of quantum space-time is
also given within this formulation.Comment: 25 pages, Latex file, no figure
Two and three-point functions in Liouville theory
Based on our generalization of the Goulian-Li continuation in the power of
the 2D cosmological term we construct the two and three-point correlation
functions for Liouville exponentials with generic real coefficients. As a
strong argument in favour of the procedure we prove the Liouville equation of
motion on the level of three-point functions. The analytical structure of the
correlation functions as well as some of its consequences for string theory are
discussed. This includes a conjecture on the mass shell condition for
excitations of noncritical strings. We also make a comment concerning the
correlation functions of the Liouville field itself.Comment: 15 pages, Latex, Revised version: A sign error in formula (50) is
correcte
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