642 research outputs found
R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
A certain class of unitary representations of U_q(sl(2,R)) has the property
of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a
particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum
groups U_q(sl(2,R)) and U_{tilde{q}}(sl(2,R)) into some enlarged object for
which he has coined the name ``modular double''. We study the R-operator, the
co-product and the Haar-measure for the modular double of U_q(sl(2,R)) and
establish their main properties. In particular it is shown that the
Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator.Comment: 27 pages, LaTex (smfart.sty
Semiclassical and quantum Liouville theory
We develop a functional integral approach to quantum Liouville field theory
completely independent of the hamiltonian approach. To this end on the sphere
topology we solve the Riemann-Hilbert problem for three singularities of finite
strength and a fourth one infinitesimal, by determining perturbatively the
Poincare' accessory parameters. This provides the semiclassical four point
vertex function with three finite charges and a fourth infinitesimal. Some of
the results are extended to the case of n finite charges and m infinitesimal.
With the same technique we compute the exact Green function on the sphere on
the background of three finite singularities. Turning to the full quantum
problem we address the calculation of the quantum determinant on the background
of three finite charges and of the further perturbative corrections. The zeta
function regularization provides a theory which is not invariant under local
conformal transformations. Instead by employing a regularization suggested in
the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the
correct quantum conformal dimensions from the one loop calculation and we show
explicitly that the two loop corrections do not change such dimensions. We then
apply the method to the case of the pseudosphere with one finite singularity
and compute the exact value for the quantum determinant. Such results are
compared to those of the conformal bootstrap approach finding complete
agreement.Comment: 12 pages, 1 figure, Contributed to 5th Meeting on Constrained
Dynamics and Quantum Gravity (QG05), Cala Gonone, Sardinia, Italy, 12-16 Sep
200
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
Liouville field theory with heavy charges. II. The conformal boundary case
We develop a general technique for computing functional integrals with fixed
area and boundary length constraints. The correct quantum dimensions for the
vertex functions are recovered by properly regularizing the Green function.
Explicit computation is given for the one point function providing the first
one loop check of the bootstrap formula.Comment: LaTeX 26 page
Irregular singularities in Liouville theory
Motivated by problems arising in the study of N=2 supersymmetric gauge
theories we introduce and study irregular singularities in two-dimensional
conformal field theory, here Liouville theory. Irregular singularities are
associated to representations of the Virasoro algebra in which a subset of the
annihilation part of the algebra act diagonally. In this paper we define
natural bases for the space of conformal blocks in the presence of irregular
singularities, describe how to calculate their series expansions, and how such
conformal blocks can be constructed by some delicate limiting procedure from
ordinary conformal blocks. This leads us to a proposal for the structure
functions appearing in the decomposition of physical correlation functions with
irregular singularities into conformal blocks. Taken together, we get a precise
prediction for the partition functions of some Argyres-Douglas type theories on
the four-sphere.Comment: 84 pages, 6 figure
Loop and surface operators in N=2 gauge theory and Liouville modular geometry
Recently, a duality between Liouville theory and four dimensional N=2 gauge
theory has been uncovered by some of the authors. We consider the role of
extended objects in gauge theory, surface operators and line operators, under
this correspondence. We map such objects to specific operators in Liouville
theory. We employ this connection to compute the expectation value of general
supersymmetric 't Hooft-Wilson line operators in a variety of N=2 gauge
theories.Comment: 60 pages, 11 figures; v3: further minor corrections, published
versio
Strings in AdS_3 and the SL(2,R) WZW Model. Part 3: Correlation Functions
We consider correlation functions for string theory on AdS_3. We analyze
their singularities and we provide a physical interpretation for them. We
explain which worldsheet correlation functions have a sensible physical
interpretation in terms of the boundary theory. We consider the operator
product expansion of the four point function and we find that it factorizes
only if a certain condition is obeyed. We explain that this is the correct
physical result. We compute correlation functions involving spectral flowed
operators and we derive a constraint on the amount of winding violation.Comment: 87 pages, 7 figures; minor change
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