Liouville Theory Research Articles

Holographic JT gravity with quartic couplings

We construct the most general theory of 2D Einstein-dilaton gravity coupled with U(1) gauge fields that contains all the 2-derivative and the 4-derivative interactions allowed by the diffeomorphism invariance. We renormalise the 2D action and obtain the vacuum solution as well as the black hole solution. The vacuum solution in the UV is dominated by Lifshitz2 with dynamical exponent (z = frac{7}{3} ) while on the other hand, the spacetime curvature diverges as we move towards the deep IR limit. We calculate the holographic stress tensor and the central charge for the boundary theory. Our analysis shows that the central charge goes as the inverse power of the coupling associated to 4-derivative interactions. We also compute the Wald entropy for 2D black holes and interpret its near horizon divergence in terms of the density of states. We compare the Wald entropy with the Cardy formula and obtain the eigen value of Virasoro operator (L0) for our model. Finally, we explore the near horizon structure of 2D black holes and calculate the central charge corresponding to the CFT near horizon. We further show that the near horizon CFT may be recast as a 2D Liouville theory with higher derivative corrections. We study the Weyl invariance of this generalised Liouville theory and identify the Weyl anomaly associated to it. We also comment on the classical vacuum structure of the theory.

Holographic boundary actions in AdS3/CFT2 revisited

The generating functional of stress tensor correlation functions in two-dimensional conformal field theory is the nonlocal Polyakov action, or equivalently, the Liouville or Alekseev-Shatashvili action. I review its holographic derivation within the AdS3/CFT2 correspondence, both in metric and Chern-Simons formulations. I also provide a detailed comparison with the well-known Hamiltonian reduction of three-dimensional gravity to a flat Liouville theory, and conclude that the two results are unrelated. In particular, the flat Liouville action is still off-shell with respect to bulk equations of motion, and simply vanishes in case the latter are imposed. The present study also suggests an interesting re-interpretation of the computation of black hole spectral statistics recently performed by Cotler and Jensen as that of an explicit averaging of the partition function over the boundary source geometry, thereby providing potential justification for its agreement with the predictions of a random matrix ensemble.

The two-sphere partition function in two-dimensional quantum gravity

We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere.

Liouville perturbation theory for Laughlin state and Coulomb gas

We consider the generating functional (logarithm of the normalization factor) of the Laughlin state on a sphere, in the limit of a large number of particles N. The problem is reformulated in terms of a perturbative expansion of a 2d QFT, resembling the Liouville field theory. We develop an analog of the Liouville loop perturbation theory, which allows us to quantitatively study the generating functional for an arbitrary smooth metric and an inhomogeneous magnetic field beyond the leading orders in large N.

Correlation between Spin and Orbital Dynamics during Laser-Induced Femtosecond Demagnetization

Spin and orbital angular momenta are two intrinsic properties of an electron and are responsible for the physics of a solid. How the spin and orbital evolve with respect to each other on several hundred femtoseconds is largely unknown, but it is at the center of laser-induced ultrafast demagnetization. In this paper, we introduce a concept of the spin-orbital correlation diagram, where spin angular momentum is plotted against orbital angular momentum, much like the position-velocity phase diagram in classical mechanics. We use four sets of highly accurate time-resolved x-ray magnetic circular dichroism (TR-XMCD) data to construct four correlation diagrams for iron and cobalt. To our surprise, a pattern emerges. The trace on the correlation diagram for iron is an arc, and at the end of demagnetization, it has a pronounced cusp. The correlation diagram for cobalt is different and appears more linear, but with kinks. We carry out first-principles calculations with two different methods: time-dependent density functional theory (TDDFT) and time-dependent Liouville density functional theory (TDLDFT). These two methods agree that the experimental findings for both Fe and Co are not due t experimental errors. It is the spin-orbit coupling that correlates the spin dynamics to the orbital dynamics.Microscopically, Fe and Co have different orbital occupations, which leads to distinctive correlation diagrams. We believe that this correlation diagram presents a useful tool to better understand spin and orbital dynamics on an ultrafast time scale. A brief discussion on the magnetic anisotropy energy is also provided.

N = 3-extended supersymmetric Schwarzian and Liouville theories

N = 3 super-Schwarzian and N = (3, 0) super-Liouville theories are formulated by the coadjoint orbit method. We study the coadjoint orbit dependence of the respective theories, represented by a superfield b. We show that it is renormalized into the N = 3 super-Schwarzian derivative when the b field takes an appropriate configuration at the initial point of the orbit. Then the renormalized actions of the respective theories are invariant under OSp(2|3) transformations. If the configuration gets further specified, the initial point of the orbit turns out to be stable under one other kind of OSp(2|3) transformations as well.

Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation

In this paper, we consider a non-homogeneous time–space-fractional telegraph equation in n-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm–Liouville operator defined in terms of right and left fractional Riemann–Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.

The large-c Virasoro identity block is a semi-classical Liouville correlator

It will be shown analytically that the light sector of the identity block of a mixed heavy-light correlator in the large central charge limit is given by a correlation function of light operators on an effective background geometry. This geometry is generated by the presence of the heavy operators. It is shown that this background geometry is a solution to the Liouville equation of motion sourced by corresponding heavy vertex operators and subsequently that the light sector of the identity block matches the Liouville correlation function in the semi-classical limit. This method effectively captures the spirit of Einstein gravity as a theory of dynamical geometry in AdS/CFT. The reason being that Liouville theory is closely related to semi-classical asymptotically AdS3 gravity.

Private life of the Liouville field that causes new anomalies in the Nambu-Goto string

I consider higher-order terms of the Seeley expansion of the heat kernel, which for smooth metrics are suppressed as inverse powers of the UV cutoff Λ, and demonstrate how they result in an anomalous contribution to the string effective action after doing uncertainties Λ−2×Λ2. For the Polyakov string these anomalies precisely reproduce at one loop the result of KPZ-DDK obtained for the Liouville theory by the conformal field theory technique. For the Nambu-Goto string I find a deviation from this result which shows that the two string formulations may differ.

Degenerate operators in JT and Liouville (super)gravity

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight h ∉ −ℕ/2. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the mathcal{N} = 1 super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound λL = 2π/β at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.

Comments on the quantum field theory of the Coulomb gas formalism

The holomorphic Coulomb gas formalism, as developed by Feigin-Fuchs, Dotsenko-Fateev and Felder, is a set of rules for computing minimal model observables using free field techniques. We attempt to derive and clarify these rules using standard techniques of quantum field theory. We begin with a careful examination of the timelike linear dilaton. Although the background charge of the model breaks the scalar field’s continuous shift symmetry, the exponential of the action remains invariant under a discrete shift because the background charge is imaginary. Gauging this symmetry makes the dilaton compact and introduces winding modes into the spectrum. One of these winding operators corresponds to the anti-holomorphic completion of the BRST current first introduced by Felder, and the full left/right cohomology of this BRST charge isolates the irreducible representations of the Virasoro algebra within the degenerate Fock space of the linear dilaton. The “supertrace” in the BRST complex reproduces the minimal model partition function and exhibits delicate cancellations between states with both momentum and winding. The model at the radius R=sqrt{pp^{prime }} has two marginal operators corresponding to the Dotsenko-Fateev “screening charges”. Deforming by them, we obtain a model that might be called a “BRST quotiented compact timelike Liouville theory”. The Hamiltonian of the zero-mode quantum mechanics of this model is not Hermitian, but it is PT-symmetric and exactly solvable. Its eigenfunctions have support on an infinite number of plane waves, suggesting an infinite reduction in the number of independent states in the full quantum field theory. Applying conformal perturbation theory to the exponential interactions reproduces the Coulomb gas calculations of minimal model correlation functions. In contrast to spacelike Liouville, these “resonance correlators” are finite because the zero mode is compact. We comment on subtleties regarding the reflection operator identification, as well as naive violations of truncation in correlators with multiple reflection operators inserted. This work is part of an attempt to understand the relationship between the JT model of two dimen- sional gravity and the worldsheet description of the (2, p) minimal string as suggested by Seiberg and Stanford.

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Previous works in this series have shown that an instance of a sqrt{8/3}-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the sqrt{8/3}-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and sqrt{8/3}-LQG surfaces with other topologies.

Ensemble averages, Poisson processes, and microstates

Ensemble averages of random field theories have recently become promising candidates of the holographic dual of classical gravity. Since quantum theories are discrete, we initiate the study of consistent averages over theories with discrete random variables---which admit a mathematically rigorous description as Poisson processes---and get results that mirror Liouville gravity. We show that this is equivalent to averaging over an ensemble of states in a single microscopic theory, which is an early top-down example trying to answer the crucial question of whether the gravitational path integral computes an ensemble average over true randomness or over pseudorandomness coming from a large number of microstates.

Uniform full stability of recovering convolutional perturbation of the Sturm–Liouville operator from the spectrum

Recently, there appeared new stability results in the inverse Sturm–Liouville theory that, unlike the usual local stability, involve uniform estimates. Besides more advanced justification of numerical simulations, the uniform stability can reveal the nature of an inverse problem to a significantly greater extent. For example, it allows obtaining global solvability of an inverse problem as a corollary of its solvability for any dense subclass of input data. In this paper, we obtain uniform stability of an inverse problem for one important class of integro-differential operators, which is a natural generalization of the classical Sturm–Liouville operator. For this purpose, we develop a new method, which is different from that used for the classical problem. Moreover, the considered situation arose relevance of the so-called full stability, i.e. the stability with respect to the full set of the input data including all a priori known components of the operator. The obtained results illustrate some essential difference from the inverse Sturm–Liouville problem.

Higher rank FZZ-dualities

We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the mathfrak{sl}(2)/mathfrak{u}(1) coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from mathfrak{sl}(2) Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type mathfrak{sl}left(N+1right)/left(mathfrak{sl}(N)times mathfrak{u}(1)right) and investigate the equivalence to a theory with an mathfrak{sl}left(N+left.1right|Nright) structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for mathfrak{sl}(N) and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

Liouville quantum gravity surfaces with boundary as matings of trees

For γ∈(0,2), the quantum disk and γ-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE16/γ2 curve on a quantum disk or on a γ-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation −cos(πγ2/4). This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for γ∈(2,2) was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.

On the S-matrix of Liouville theory

The S-matrix for each chiral sector of Liouville theory on a cylinder is computed from the loop expansion of correlation functions of a one-dimensional field theory on a circle with a non-local kinetic energy and an exponential potential. This action is the Legendre transform of the generating function of semiclassical scattering amplitudes. It is derived from the relation between asymptotic in- and out-fields. Its relevance for the quantum scattering process is demonstrated by comparing explicit loop diagrams computed from this action with other methods of computing the S-matrix, which are also developed.

Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model

Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension 2 or 3. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions.As an application, we describe the logarithmic structures of the critical two-dimensional O(n) and Q-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the Q-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino-Viti conjecture for the three-point connectivity. Our results hold for generic values of Q in the complex plane and beyond.

Theory of oblique-field magnetoresistance from spin centers in three-terminal spintronic devices

We present a general stochastic Liouville theory of electrical transport across a barrier between two conductors that occurs via sequential hopping through a single defect's spin-0 to spin-1/2 transition. We find magneto-conductances similar to Hanle features (pseudo-Hanle features) that originate from Pauli blocking without spin accumulation, and also predict that evolution of the defect's spin modifies the conventional Hanle response, producing an inverted Hanle signal from spin center evolution. We propose studies in oblique magnetic fields that would unambiguously determine if a magnetoconductance results from spin-center assisted transport.

A probabilistic approach of Liouville field theory

In this article, we present the Liouville field theory, which was introduced in the eighties in physics by Polyakov as a model for fluctuating metrics in 2D quantum gravity, and outline recent mathematical progress in its study. In particular, we explain the probabilistic construction of this theory carried out by David–Kupiainen–Rhodes–Vargas in [] and how this construction connects to the modern and general approach of Conformal Field Theories in physics, called conformal bootstrap and based on representation theory.