Moduli integrals and ground ring in minimal Liouville gravity

Straightforward evaluation of the correlation functions in 2D minimal gravity requires integration over the moduli space. For degenerate fields, the Liouville higher equations of motion allow one to turn the integrand to a derivative and, thus, to reduce it to the boundary terms plus a so-called curvature contribution. The last is directly related to the expectation value of the corresponding ground ring element. We use the operator product expansion technique to reproduce the ground ring construction explicitly in terms of the (generalized) minimal matter and Liouville degenerate fields. The action of the ground ring on the generic primary fields is evaluated explicitly. This permits us to directly construct the ground ring algebra. Detailed analysis of the ground ring mechanism is helpful in the understanding of the boundary terms and their evaluation.     

Parafermionic polynomials,Selberg integrals and three-point correlation function in parafermionic Liouville field theory

In this paper we consider parafermionic Liouville field theory. We study integral representations of three-point correlation functions and develop a method allowing us to compute them exactly. In particular, we evaluate the generalization of Selberg integral obtained by insertion of parafermionic polynomial. Our result is justified by different approach based on dual representation of parafermionic Liouville field theory described by three-exponential model.     

Three-point function in perturbed Liouville gravity

Three-point correlation function in perturbed conformal field theory coupled to two-dimensional quantum gravity (perturbed Liouville gravity) is explicitly computed by using the free field approach. The representation considered here is the one recently proposed in [G. Giribet, Nucl. Phys. B 737 (2006) 209] to describe the string theory in AdS₃${\mathit{AdS}}_3$ space. Consequently, this computation extends previous results which presented free field calculations of particular cases of string amplitudes, and confirms that the free field approach leads to the exact result. Remarkably, this representation allows to compute winding violating three-point functions without making use of the spectral flow operator.     

Schramm-Loewner evolution and Liouville quantum gravity

We show that when two boundary arcs of a Liouville quantum gravity random surface are conformally welded to each other (in a boundary length-preserving way) the resulting interface is a random curve called the Schramm-Loewner evolution. We also develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under conformal welding maps related to Schramm-Loewner evolution. As an application, we construct quantum length and boundary intersection measures on the Schramm-Loewner evolution curve itself.     

Nonperturbative model of Liouville gravity

We formulate nonperturbative 2D gravity in the framework of Liouville theory. In particular, we express the specific heat of pure gravity in terms of an expansion of integrals on moduli spaces of punctured Riemann spheres. We recognize the relevant divisors on moduli spaces and write the integrands in terms of the Liouville action. We evaluate the integrals (rational intersections) and show that satisfies the Painlevé I.     

Moduli integrals in liouville gravity

We consider moduli integrals appearing in four-point correlation functions of the (p, q) minimal models coupled to Liouville gravity on a sphere, which is sometimes called 2D minimal gravity or minimal string theory on a sphere. Liouville gravity on a sphere is the quantized metric of the spherical topology in the conformal gauge. Reviewing the previous results on such four-point functions (Y. Ishimoto and Sh. Yamaguchi: Phys. Lett. B607 (2005) 172), we show logarithmic correlation functions of ‘tachyons’ in the Liouville sector, and its moduli integrals of the full correlation functions, in particular in the Majorana fermion model coupled to 2D gravity. Further discussions and related results are given in the final section and in Y. Ishimoto and Al. Zamolodchikov: Theor. Math. Phys.147 (2006) 755.     

Invariant characterization of Liouville metrics and polynomial integrals

In this paper a criterion for a metric on a surface to be Liouville is established, and it is given in terms of differential invariants of the metric. Moreover, here we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals the geodesic flow of a given metric possesses? The method is also applied to recognition of higher degree polynomial integrals of geodesic flows.     

Liouville mode in gauge/gravity duality

We establish solutions corresponding to AdS\(_4\) static charged black holes with inhomogeneous two-dimensional horizon surfaces of constant curvature. Depending on the choice of the 2D constant curvature space, the metric potential of the internal geometry of the horizon satisfies the elliptic wave/elliptic Liouville equations. We calculate the charge diffusion and transport coefficients in the hydrodynamic limit of gauge/gravity duality and observe the exponential suppression in the diffusion coefficient and in the shear viscosity-per-entropy density ratio in the presence of an inhomogeneity on black hole horizons with planar, spherical, and hyperbolic geometry. We discuss the subtleties of the approach developed for a planar black hole with inhomogeneity distribution on the horizon surface in more detail and find, among others, a trial distribution function, which generates values of the shear viscosity-per-entropy density ratio falling within the experimentally relevant range. The solutions obtained are also extended to higher-dimensional AdS space. We observe two different DC conductivities in 4D and higher-dimensional effective strongly coupled dual media and formulate conditions under which the appropriate ratio of different conductivities is qualitatively the same as that observed in an anisotropic strongly coupled fluid. We briefly discuss ways of how the Liouville field could appear in condensed matter physics and outline prospects of further employing the gauge/gravity duality in CMP problems.     

Torus amplitudes in minimal Liouville gravity and matrix models

We evaluate one-point correlation numbers on the torus in the Liouville theory coupled to the conformal matter M(2,2p+1)$M(2,2p+1)$. We find agreement with the recent results obtained in the matrix model approach.     

Discretization of Liouville type nonautonomous equations preserving integrals

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.     

Negative screenings in Liouville theory

We demonstrate how negative powers of screenings arise as a non-perturbative effect within the operator approach to Liouville theory. This explains the origin of the corresponding poles in the exact Liouville three-point function proposed by Dorn/Otto and (Zamolodchikov)² (DOZZ) and leads to a consistent extension of the operator approach to arbitrary integer numbers of screenings of both types. The general Liouville three-point function in this setting is computed without any analytic continuation procedure, and found to support the DOZZ conjecture. We point out the importance of the concept of free-field expansions with adjustable monodromies - recently advocated by Petersen, Rasmussen and Yu - in the present context, and show that it provides a unifying interpretation for two types of previously constructed local observables.     

Convergence of Feynman integrals in Coulomb gauge QCD

At 2-loop order, Feynman integrals in the Coulomb gauge are divergent over the internal energy variables. Nevertheless, it is known how to calculate the effective action, provided that the external gluon fields are all transverse. We show that, for the two-gluon Greens function as an example, the method can be extended to include longitudinal external fields. The longitudinal Greens functions appear in the BRST identities. As an intermediate step, we use a flow gauge, which interpolates between the Feynman and Coulomb gauges.     

Conformal bootstrap in Liouville field theory

We consider the recently proposed analytic expression for the three-point function in the Liouville field theory on a sphere. It is verified that in the classical limit this expression reduces to what the classical Liouville theory predicts. Using the suggested three-point function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and check numerically that it satisfies the conformal bootstrap equations. The Liouville reflection amplitude which follows explicitly from the structure constants is also considered and compared with the results of the Bethe ansatz technique.