We construct the four-point correlation functions containing the top component of the supermultiplet in the Neveu-Schwarz sector of the N=1 SUSY Liouville field theory. The construction is based on the recursive representation for the NS conformal blocks. We test our results in the case where one of the fields is degenerate with a singular vector on the level 3/2. In this case, the correlation function satisfies a third-order ordinary differential equation, which we derive. We numerically verify the crossing symmetry relations for the constructed correlation functions in the nondegenerate case.
Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.
Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function o…
We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0,1.
We construct a four supercharges Liouville superconformal field theory in four dimensions. The Liouville superfield is chiral and its lowest component is a log-correlated complex scalar whose real part carries a background charge. The action consists of a supersymmetric Paneitz operator, a background supersymmetric $\mathcal{Q}$-curvature charge and an exponential potential. It localizes semiclassically on solutions that describe curved superspaces with a constant complex supersymmetric $\mathcal{Q}$-curvature. The theory is non-unitary with a continuous spectrum of scaling dimensions. We stud…
We consider the 2D super Liouville gravity coupled to the minimal superconformal theory. We analyze the physical states in the theory and give the general form of the n-point correlation numbers on the sphere in terms of integrals over the moduli space. The three-point correlation numbers are presented explicitly. For the four-point correlators, we show that the integral over the moduli space reduces to the boundary terms if one of the fields is degenerate. It turns out that special logarithmic fields are relevant for evaluating these boundary terms. We discuss the construction of these fields…
The four-point integral of the minimal super Liouville gravity on the sphere is evaluated numerically. The integration procedure is based on the effective elliptic parameterization of the moduli space. The analysis is performed for a few different gravitational four-point amplitudes. The results agree with the analytic results recently obtained using the Higher super Liouville equations of motion.
In this dissertation we present some basic features about Liouville and $\mathcal{N}=1$ Super Liouville Theory, and focus in the computation of their three point functions. Additionally, we include an introduction to Conformal Field Theories (CFT) and Supersymmetry, which are the basic tools of the present research.
We construct a Liouville superconformal field theory with eight real supercharges in four dimensions. The Liouville superfield is an $\mathcal{N}=2$ chiral superfield with sixteen bosonic and sixteen fermionic component fields. Its lowest component is a log-correlated complex scalar field whose real part carries a background charge. The theory is non-unitary with a continuous spectrum of scaling dimensions. We study its quantum dynamics on the supersymmetric 4-sphere and show that the classical background charge is not corrected quantum mechanically. We calculate the super-Weyl anomaly coeffic…
We study the bosonic super Liouville system which is a statistical transmutation of super Liouville system. Lax pair for the bosonic super Liouville system is constructed using prolongation method, ensuring the Lax integrability, and the solution to the equations of motion is also considered via Leznov-Saveliev analysis.
The AGT motivated relation between the tensor product of the N = 1 super-Liouville field theory with the imaginary free fermion (SL) and a certain projected tensor product of the real and the imaginary Liouville field theories (LL) is analyzed. Using conformal field theory techniques we give a complete proof of the equivalence in the NS sector. It is shown that the SL-LL correspondence is based on the equivalence of chiral objects including suitably chosen chiral structure constants of all the three Liouville theories involved.
A four point function of basic Neveu-Schwarz exponential fields is constructed in the N = 1 supersymmetric Liouville field theory. Although the basic NS structure constants were known previously, we present a new derivation, based on a singular vector decoupling in the NS sector. This allows to stay completely inside the NS sector of the space of states, without referencing to the Ramond fields. The four-point construction involves also the NS blocks, for which we suggest a new recursion representation, the so-called elliptic one. The bootstrap conditions for this four point correlation functi…
We present a new supersymmetric integrable model: the $N=2$ superconformal affine Liouville theory. It interpolates between the $N=2$ super Liouville and $N=2$ super sine-Gordon theories and possesses a Lax representation on the complex affine Kac-Moody superalgebra ${\hat {sl(2| 2)^{(1)}}}$. We show that the higher spin $W_{1+\infty}$-type symmetry algebra of ordinary conformal affine Liouville theory extends to a $N=2\; W_{1/2 + \infty}$-type superalgebra.
The two-dimensional manifestly locally supersymmetric actions describing the N=2 and N=4 extended super-Liouville theory coupled to the N=2 and N=4 conformal supergravity, respectively, are constructed in superspace. It is shown that the N=4 super-Liouville multiplet is described by the improved twisted-II scalar multiplet TM-II, whose kinetic terms are given by the SU(2)xU(1) WZNW model.
We show that the crossing symmetry of the four-point function in the Liouville conformal field theory on the sphere contains more information than what was hitherto considered. Under certain assumptions, it provides the special structure constants that were previously computed perturbatively and allows to solve the theory without using the Liouville interaction.
We derive a boundary action of N=2 super-Liouville theory which preserves both N=2 supersymmetry and conformal symmetry by imposing explicitly $T={\bar T}$ and $G={\bar G}$. The resulting boundary action shows a new duality symmetry.
As an existence proof of the (0,2) heterotic supercurrent supermultiplets in (1+1) dimensional quantum field theories which are consistent with the warped superconformal algebra, we construct the (0,2) chiral Liouville field theories. The two distinct possibilities of the heterotic supercurrent supermultiplets are both realized.
We study conformal boundary conditions and corresponding one-point functions of the N=2 super-Liouville theory using both conformal and modular bootstrap methods. We have found both continuous (`FZZT-branes') and discrete (`ZZ-branes') boundary conditions. In particular, we identify two different types of the discrete ZZ-brane solutions, which are associated with degenerate fields of the N=2 super-Liouville theory.
An N=(4,0) supersymmetric Liouville theory is formulated by the coadjoint orbit method. It is discovered that the action has symmetry under PSU(1,1|2).
The relation between super-Virasoro anomaly and super-Weyl anomaly in $N=1$ NSR superstring coupled with 2D supergravity is investigated from canonical theoretical view point. The WZW action canceling the super-Virasoro anomaly is explicitly constructed. It is super-Weyl invariant but nonlocal functional of 2D supergravity. The nonlocality can be remedied by the super-Liouvlle action, which in turn recovers the super-Weyl anomaly. The final gravitational effective action turns out to be local but noncovariant super-Liouville action, describing the dynamical behavior of the super-Liouville fiel…
A recursive formula for an infinity of integrals of motion for the super-Liouville theory is derived. The integrable boundary interactions for this theory and the super-Toda theory based on the affine superalgebra $B^{(1)} (0,1)$ are computed. In the first case the boundary interactions are unambiguously determined by supersymmetry, whilst in the latter case there are free parameters.
We consider a ${\hat c}=1$ model in the fermionic black hole background. For this purpose we consider a model which contains both the N=1 and the N=2 super-Liouville interactions. We propose that this model is dual to a recently proposed type 0A matrix quantum mechanics model with vortex deformations. We support our conjecture by showing that non-perturbative corrections to the free energy computed by both the matrix model and the super-Liouville theories agree exactly by treating the N=2 interaction as a small perturbation. We also show that a two-point function on sphere calculated from the …
So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are performed with the help of Neveu--Schwarz superalgebra. We consider another version of these superLiouville equations based on the Ramond superalgebra, their explicit solutions are given by Ivanov--Krivonos' scheme. Open problems are offered.
We describe in superspace a classical theory of of two dimensional $(1,1)$ dilaton supergravity coupled to a super-Liouville field, and find exact super black hole solutions to the field equations that have non-constant curvature. We consider the possibility that a gravitini condensate forms and look at the implications for the resultant spacetime structure. We find that all such condensate solutions have a condensate and/or naked curvature singularity.
We present a superconformally invariant and integrable model based on the twisted affine Kac-Moody superalgebra $\hat{osp(2|2)}^{(2)}$ which is the supersymmetrization of the purely bosonic conformal affine Liouville theory recently proposed by Babelon and Bonora. Our model reduces to the super-Liouville or to the super sinh-Gordon theories under certain limit conditions and can be obtained, via hamiltonian reduction, from a superspace WZNW model with values in the corresponding affine KM supergroup. The reconstruction formulae for classical solutions are given. The classical $r$-matrices in t…
In a series of recent papers, a special kind of AdS$_2$/CFT$_1$ duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS$_2$ background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS$_2$ is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the $\mathcal W$-algebr…
We present an infinite set of higher equations of motion in N=2 supersymmetric Liouville field theory. They are in one to one correspondence with the degenerate representations and are enumerated in addition to the U(1) charge ωby the positive integers m or (m,n) respectively. We check that in the classical limit these equations hold as relations among the classical fields.
We reformulate the Omega-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Omega-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.
Noncritical Conformal Gravity and Four-Dimensional Liouville Theory
National Taiwan University（台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南阳一郎理论与实验物理研究所（NITEP））；Institute of Fundamental Physics and Quantum Technology, Department of Physics, School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院物理学与量子技术基础物理研究所）
We consider a four-dimensional theory in the z=3 Lifshitz context, with an exponential (Liouville) potential. We determine the exact renormalized potential of the theory and derive the non-perturbative relation between the renormalized and bare couplings. In addition, we show that Lorentz symmetry is naturally generated by quantum fluctuations in the infrared regime, and conclude that the model can be relevant to High Energy Physics.
We study an integrable deformation of the super-Liouville theory which generates a RG flows to the critical Ising model as the IR fixed point. This model turns out to be a supersymmetric sinh-Gordon model with spontaneously broken N=1 supersymmetry. The resulting massless Goldstino is the only stable on-shell particle which controls the IR behaviours. We propose the exact $S$-matrix of the Goldstino and compare associated thermodynamic Bethe ansatz equations with the quantization conditions derived from the reflection amplitudes of the the super-Liouville theory to provide nonperturbative chec…
In this paper we consider a strong-weak coupling duality of the N=2 super-Liouville field theory (SLFT). Without the self-duality found in other Liouville theories, the N=2 SLFT, we claim, is associated with a `dual' action by a transformation $b\to 1/b$ where $b$ is the coupling constant. To justify our conjecture, we compute the reflection amplitudes (or two-point functions) of the (NS) and the (R) operators of the N=2 SLFT based on the conjectured dual action and show that the results are consistent with known results.
We calculate three- and four-point functions in super Liouville theory coupled to super Coulomb gas on world sheets with spherical topology. We first integrate over the zero mode and assume that a parameter takes an integer value. After calculating the amplitudes, we formally continue the parameter to an arbitrary real number. Remarkably the result is completely parallel to the bosonic case, the amplitudes being of the same form as those of the bosonic case.
We derive one-point functions of the N=2 super-Liouville theory on a half line using the modular transformations of the characters in terms of the bulk and boundary cosmological constants. We also show that these results are consistent with conformal bootstrap equations which are based on the bulk and boundary actions. We provide various independent checks for our results.
Based on a path integral prescription for anomaly calculation, we analyze an effective theory of the two-dimensional $N=2$ supergravity, i.e., $N=2$ super-Liouville theory. We calculate the anomalies associated with the BRST supercurrent and the ghost number supercurrent. From those expressions of anomalies, we construct covariant BRST and ghost number supercurrents in the effective theory. We then show that the (super-)coordinate BRST current algebra forms a superfield extension of the topological conformal algebra for an {\it arbitrary\/} type of conformal matter or, in terms of the string t…
We derive a relation between correlation functions of supergroup WZNW models and conformal field theories with extended superconformal symmetry. The supergroups considered have a bosonic subgroup of the form SL(2) x A for some Lie group A. The corresponding conformal field theory is a super Liouville field theory coupled with the WZNW model on A. An example is a correspondence between the PSU(1,1|2) WZNW model and small N=4 super Liouville field theory. The OSP(n|2) WZNW model is related to a superconformal field theory with SO(n) extended superconformal symmetry of the Knizhnik-Bershadsky typ…
We propose that N=2 U(2) gauge theories on the A_1 ALE space, with asymptotic holonomies not in SU(2), correspond to the Ramond sector of super Liouville theory. As evidence, we show that the instanton partition functions for the theories with and without a fundamental hypermultiplet, computed with such holonomies, coincide with the norms or the inner products of the Whittaker vectors in the Ramond sector. This correspondence suggests that S-duality of U(2) gauge theories interchanges the sectors with different boundary conditions.
We show that N=1 supersymmetric Liouville theory can be continued to central charge c=3/2, and that the limiting non-rational superconformal field theory can also be obtained as a limit of supersymmetric minimal models. This generalises a result known for the non-supersymmetric case. We present explicit expressions for the three-point functions of bulk fields, as well as a set of superconformal boundary states. The main technical ingredient to take the limit of minimal models consists in determining analytic expressions for the structure constants. In the appendix we show in detail how the str…
The finite form of the $N=2$ super-Weyl transformations in the chiral and twisted-chiral irreducible formulations of the two-dimensional $N=2$ superfield supergravity are found in $N=2$ superspace. The super-Weyl anomaly of the $N=2$ extended fermionic string theory is computed in terms of the $N=2$ superfields, by using a short time expansion of the $N=2$ chiral heat kernel. The super-Weyl invariant $N=2$ superconformal structure is introduced, and a new definition of the $N=2$ super-Riemannian surfaces is proposed.
The symmetry algebra of $N=1$ Super-Liouville field theory in two dimensions is the infinite dimensional $N=1$ superconformal algebra, which allows one to prove, that correlation functions, containing degenerated fields obey some partial linear differential equations. In the special case of four point function, including a primary field degenerated at the first level, this differential equations can be solved via hypergeometric functions. Taking into account mutual locality properties of fields and investigating s- and t- channel singularities we obtain some functional relations for three- poi…
The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions, which may be expressed in terms of hypergeometric functions, a set of functional equations is found for the general three point function. It is shown that the expression proposed by the Zamolodchikovs solves these functional equations and that under certain assumptions the solution is unique.
We point out that the non-critical version of the k-fractional superstring theory can be described by k-cut critical points of the matrix models. In particular, in comparison with the spectrum structure of fractional super-Liouville theory, we show that (p,q) minimal fractional superstring theories appear in the Z_k-symmetry breaking critical points of the k-cut two-matrix models and the operator contents and string susceptibility coincide on both sides. By using this correspondence, we also propose a set of primary operators of the fractional superconformal ghost system which consistently pro…
In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space-time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular $S$-matrices. Using this result, we conjecture the dependence of the bo…
Analytic expressions for the two dimensional $\mathcal{N}=1$ SLFT blocks in the light semi-classical limit are found for both Neveu-Schwarz and Ramond sectors. The calculations are done by using the duality between $SU(2)$ $\mathcal{N}=2$ super-symmetric gauge theories living on $R^4/Z_2$ space and two dimensional $\mathcal{N}=1$ super Liouville field theory. It is shown that in the light asymptotic limit only a restricted set of Young diagrams contribute to the partition function. This enables us to sum up the instanton series explicitly and find closed expressions for the corresponding $\mat…
We study four-point correlation functions with logarithmic behaviour in Liouville field theory on a sphere, which consist of one kind of the local operators. We study them as non-integrated correlation functions of the gravitational sector of two-dimensional quantum gravity coupled to an ordinary conformal field theory in the conformal gauge. We also examine, in the (p,q) minimal conformal field theories, a condition of the appearance of logarithmic correlation functions of gravitationally dressed operators.
We study N=1 super Liouville theory on worldsheets with and without boundary. Some basic correlation functions on a sphere or a disc are obtained using the properties of degenerate representations of superconformal algebra. Boundary states are classified by using the modular transformation property of annulus partition functions, but there are some of those whose wave functions cannot be obtained from the analysis of modular property. There are two ways of putting boundary condition on supercurrent, and it turns out that the two choices lead to different boundary states in quality. Some proper…
The conjecture about the correspondence between instanton partition functions in the N=2 SUSY Yang-Mills theory and conformal blocks of two-dimensional conformal field theories is extended to the case of the N=1 supersymmetric conformal blocks. We find that the necessary modification of the moduli space of instantons requires additional restriction of Z(2)-symmetry. This leads to an explicit form of the N=1 superconformal blocks in terms of Young diagrams with two sorts of cells.
An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere. In the classical limit it coincides with what the classical Liouville theory predicts. Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative. We consider also the Liouville reflection amplitude which follows explicitly from the structure constants.
In this paper we compute the N-point correlation functions of the tachyon operator from the Neveu Schwarz sector of super Liouville theory coupled to matter fields (with $\hat c\le 1$) in the super Coulomb gas formulation, on world sheets with spherical topology. We first integrate over the zero mode assuming that the $s$ parameter takes an integer value, subsequently we continue the parameter to an arbitrary real number. We included an arbitrary number of screening charges (s.c.) and as a result, after renormalizing the s.c., the external legs and the cosmological constant, the form of the fi…
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…
An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group U_q(sl(2,R)) is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.
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…
