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Conformal field theory in two dimensions has, over the years, been an extremely useful tool for a variety of physical and mathematical problems. In this paper, perhaps one of the most profound and foundational aspects of the subject is reviewed in detail, namely that of an operator algebra of the theory. This aspect is then demonstrated in some modern applications, first stochastic Loewner evolution and then logarithmic conformal field theory.

In a recent paper, a variational principle for Liouville dynamics (vector fields preserving a volume form) was introduced. In the present paper, we discuss the relation between symmetries and conservation laws in Liouville dynamics. We recall a result by Crampin relating symmetries and conserved quantities in Liouville dynamics, and discuss how symmetry reduction is implemented in the frame of the variational principle for Liouville dynamics.

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

Using the kinetic equation of subdynamics and the suitable time ordering rule we present a type of irreversible Liouville equation and nonlinear Liouville equation for open quantum systems. We also found a spectral relation for obtaining the non-equilibrium statistical distribution based on the boundary conditions which consistent with H theorem. Furthermore, a formalism of non-equilibrium project Green functions is given, and a non-perturbative method to solve the irreversible Liouville equation is studied. Finally, we give a model of spin-based quantum computing to show that the decoherence free states can be constructed using the non-perturbative approach.

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of three parts. In part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also presented. In part II, we review the supersymmetric extension of the Liouville theory. We first discuss the bulk structure constants and the branes as in the bosonic Liouville theory, and then we present the matrix dual descriptions with some applications. In part III, the Liouville theory on unoriented surfaces is reviewed. After introducing the crosscap state, we discuss the matrix model dual description and the tadpole cancellation condition. This review also includes some original material such as the derivation of the conjectured dual action for the N = 2 Liouville theory from other known dualities and the comparison of the Liouville crosscap state with the c = 0 unoriented matrix model. This is based on my master's thesis submitted to Department of Physics, Faculty of Science, University of Tokyo on January 2004., 264 pages, 31 figures, based on master's thesis defended 31/01/2004, v2,3,4,5,6,7: references added, typos corrected, abridged version published in IJMPA

An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group [Formula: see text] 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.

Three explicit and equivalent representations for the monodromy of the conformal blocks in the non compact SL(2)k WZNW model are proposed in terms of the same quantity computed in Liouville field theory.

A brief summary is given of the currently available low dimensional fuzzy spaces and some of the simplest field theories permitted on these spaces.

We study SU(2)[Formula: see text]×[Formula: see text]U(1) gauge theory with Chern–Simons term, coupled to scalar field in adjoint, in a probe approximation by ignoring back reaction on metric. Considering a simple ansatz for non-Abelian gauge field with helical structure, we find it admits s-wave and p-wave phases along with their coexistence. We study free energies for different phases along with those for p-wave phases for different values of pitch and frequency dependence of optical conductivities below critical temperature.

It has been found empirically that the Virasoro centre and 3-point functions of quantum Liouville field theory with potential e2bΦ(x) and external primary fields exp(αΦ(x)), are invariant with respect to the duality transformations ℏα→q-α where q=b-1+b. The steps leading to this result (via the Virasoro algebra and 3-point functions) are reviewed in the path-integral formalism. The duality stems from the fact that the quantum relationship between the α and the conformal weights Δα is two-to-one. As a result the quantum Liouville potential may actually contain two exponentials (with related parameters). It is shown that in the two-exponential theory the duality appears in a natural way and that an important extrapolation which was previously conjectured can be proved.

The dual relationship between two n-1 parameter families of quantum field theories based on extended complex numbers is investigated in two dimensions. The nonlocal conserved charges approach is used. The lowest rank affine Toda field theories are generated and identified as integrability submanifolds in parameter space. A truncation of the model leads to a conformal field theory in extended complex space. Depending on the projection over the usual complex space chosen, a parametrized central charge is calculated.

We calculate the normalization factors and reflection amplitudes in the W-invariant conformal quantum field theories. Using these CFT data we derive vacuum expectation values of exponential fields in affine Toda theories and related perturbed conformal field theories. We also calculate the asymptotics of cylindrically symmetric solutions of the classical Toda equations.

The essentials of quantum mechanics are derived from Liouville's theorem in statistical mechanics. An elementary solution, g, of Liouville's equation helps to construct a differentiable N-particle distribution function (DF), F(g), satisfying the same equation. Reality and additivity of F(g): (i) quantize the time variable; (ii) quantize the energy variable; (iii) quantize the Maxwell–Boltzmann distribution; (iv) make F(g) observable through time-elimination; (v) produce the Planck constant; (vi) yield the black-body radiation spectrum; (vii) support chronotopology introduced axiomatically; (viii) the Schrödinger and the Klein–Gordon equations follow. Hence, quantum theory appears as a corollary of Liouville's theorem. An unknown connection is found allowing the better understanding of space-times and of these theories.

We consider a class of conformal field theories on Riemann surfaces represented as a Zk invariant covering of the sphere. The introduction of exchange interactions among couples of sheets generate effective parafermions. The outgoing theory can be seen as a fractional supersymmetry conformal field theory.

We extend the induced-matter interpretation of Kaluza–Klein field theory to obtain the Klein–Gordon equation for a scalar particle. The motion of the particle is correctly recovered, and we give an exact plane-wave solution whose induced energy-momentum tensor depends only on first derivatives of the scalar field, as in flat-space quantum field theory. However, in general the energy and momenta of the particle also include terms in second derivatives, which should repay investigation.

It is known that the Vilkovisky–DeWitt effective potential for Einstein gravity depends on the choice of the metric of configuration space. We will show that this metric dependence occurs not only in gauge field theories, but also in scalar fields as well as other field theories with "unnatural" initial choices of metric. This means that the problem of metric dependence relates to initial field parametrization but not gauge symmetry.

We use the connection between the Frobenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search for a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensures the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has an explicit and simple form in the flat coordinates on the Frobenius manifold in the general case of (p,q) Minimal Liouville gravity.

The quantum stochastic Liouville equation of Ito type is derived, for the first time, within Nonequilibrium Thermo Field Dynamics (NETFD), a unified canonical formalism for dissipative and/or stochastic fields. With the stochastic time-evolution generator of Ito type, the whole framework is inspected. Since most of the mathematical formulations of noncommutative stochastic variables are based on the equations of Ito type, the construction of the quantum stochastic Liouville equation has been highly desired. It is expected that the unified framework may provide us with a deeper insight for the mathematical foundation of quantum stochastic variables as well as for the physical one.

The quaternion solutions of four-dimensional Liouville and sine-Gordon equations are explicity obtained by using the quaternion basis.

In this work we investigate the five-dimensional Kaluza-Klein theory with a scalar field contained in the metric, where a Dirac-field is coupled to the metric field. We find that in the four-dimensional theory a nontrivial ground state for the scalar field exists and therefore the mass term in the Dirac equation can be interpreted, for example, as the electron mass.

A new geometric formulation of Liouville gravity based on the area preserving diffeomorphism is given and a possible alternative to reinterpret Liouville gravity is suggested, namely, a scalar field coupled to two-dimensional gravity with a curvature constraint.

We review the physical properties of random surfaces from the view point of the continuum Liouville theory. We shall summarize physical motivations of this subject and the field-theoretic treatment of the random surfaces. In the case of subcritical strings propagating in less than two dimensions, a complete understanding has been obtained recently through matrix models. We discuss the Liouville theory understanding of these results by studying the correlation functions. We also discuss promising avenues of further investigations such as black holes in 2-dimensional gravity and 3- and 4-dimensional string theory which may be relevant to Ising 3 and QCD 4.

The Bethe–Salpeter equations for the n-particle irreducible Green functions in quantum field theory are derived. Their kernels are expressed in terms of the functional Legendre transforms, and n-particle irreducibility of the perturbation theory graphs for the kernel of the n-particle equation is proved. The method for obtaining the Faddeev–Yakubovski equations in quantum field theory is demonstrated for the three-particle case.

We discuss a target space field theory of macroscopic loops W(ℓ,…) in two-dimensional gravity. The propagator 1)W(ℓ2)> and topology-changing amplitudes 1)W(ℓ2)W(ℓ3)> (string interactions) are considered as off-shell Euclidean Green's functions in this field theory. In the course of the analysis, we identify a new set of operators in the c = 1 system and interpret them in two-dimensional gravity. We also identify an infinite set of new conserved charges in the c = 1 system which are associated with the special states in the theory. The analysis also shows that the eigenvalue coordinate of the matrix model and a zero mode of the Liouville field are not functionally related but are conjugate variables in an integral transform.

This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics.

We discuss the relation between topological conformal field theories and singularity theory. We argue that the problem of the explicit solution of these theories is equivalent to that of finding the flat-coordinate basis in the space of couplings. The latter problem is the well-known problem of singularity theory. It has especially simple solutions in the case of minimal models.

String field theory for 2D gravity coupled to a c=1 matter field is described. We use the free field representation and take into account the semi-bounded of the Liouville mode in the last step of calculations. We call such representation semi-free field representation. It is shown that the Witten string field theory gives a tachyon action which up to some, factor in the interaction term coincides with the Das-Jevicki-Polchinski action. Starting from string field theory with semi-free field representation we calculate string amplitudes. These amplitudes instead of energy conservation δ-functions contain energy denominators which residues coincide with the Veneziano (Shapiro-Virasoro) amplitudes for open (closed) string and exhibit very interesting property: the s, t and u poles become the poles in respect to individual external momentum. These poles are related with higher levels excitations living only with fixed momentum.

The general three-point correlation functions of the minimal conformal field theories coupled to gravity are calculated using the specific Coulomb gas type quantization technique. The gravity interaction modifies in an essential way the operator algebra of the corresponding minimal model, in particular in canceling the decoupling of a finite number of primary fields, in case of rational theories. This is suggested to be related to the appearance of new physical states, under the BRST analysis of Lian and Zuckerman. Within the analytic technique, this effect is due to the corresponding singularities of the gravity sector operator algebra structure constants.

We compute a class of four-point correlation functions of physical operators on a sphere in the unitary minimal conformal field theories coupled to 2-dimensional gravity. We use the continuum Liouville field theory approach and they are obtained as integrals over the moduli (positions of the operators). We examine the integrands near the boundaries of the moduli space and compare their singular behaviors with the operator product expansion.

[Formula: see text]-invariant conformal field theories admit perturbations preserving integrability and leading to massive quantum field theories. The unitary and crossing-invariant S matrices of such QFTs are explicitly constructed by restricting the SL (n, q)-symmetric solutions of the Yang-Baxter equations when qn+k=1. These scattering theories possess level-rank duality (n↔k).

The factorized scattering theory, corresponding to integrable deformations of parafermionic ZN-symmetrical models of the conformal quantum field theory, is studied.

Two-point Green’s function is measured on the manifolds of a 2-dimensional quantum gravity. The recursive sampling technique is used to generate the triangulations, lattice sizes being up to hundred thousand triangles. The grid Laplacian was inverted by means of the algebraic multi-grid solver. The free field model of the Quantum Gravity assumes the Gaussian behavior of Liouville field and curvature. We measured histograms as well as six momenta of these fields. Our results support the Gaussian assumption.

The S-matrices for ϕ1,2-perturbated minimal models are shown to be connected with RSOS projections of the [Formula: see text]R-matrix. The agreement with the known particular cases (E6, E7, E8, D4) is established.

The equivalence of 2-dimensional quantum gravity and Liouville theory quantized with the standard translation invariant measure, is shown to hold to all orders in perturbation theory. Also, an explicit equation for the Liouville effective action as a function of the effective potential is derived from Weyl Ward identities. We speculate on some of the non-perturbative aspects of the theory.

All genus characters define a complete solution of a two-dimensional rational conformal field theory. An arbitrary point correlator can be obtained by an appropriate combination of the pinchings of zero-homology and non-zero-homology cycles of the characters on the higher genus Riemann surface.

The Gaussian effective potential for the supersymmetric Liouville model is computed both at zero temperature and at a finite temperature. It is noted that the supersymmetric Liouville theory, just like the ordinary Liouville model, does not possess a translationally invariant ground state. The broken translational symmetry is not restored by temperature effects. The supersymmetric Liouville theory is also non-trivial.

A Lorentz invariant and Hermitian statistical field theory is presented. Its derivation is based on the random field theory, the evolution operator and the path integration. It yields the density matrix, the Gibbs states and the temperatures for equilibrium, nonequilibrium and variable particles numbers states. The chemical potential follows from a spontaneous energy renormalization of the Hamiltonian in the evolution operator. The generalization to any field other than scalar fields is straightforward.

A numerical approach to 2-D relativistic field theories is suggested. Considering a field theory model as an ultraviolet conformal field theory perturbed by a suitable relevant scalar operator one studies it in finite volume (on a circle). The perturbed Hamiltonian acts in the conformal field theory space of states and its matrix elements can be extracted from the conformal field theory. Truncation of the space at a reasonable level results in a finite dimensional problem for numerical analyses. The nonunitary field theory with the ultraviolet region controlled by the minimal conformal theory [Formula: see text] is studied in detail.

We quantize the Liouville theory, or 2-D quantum gravity, and quantum supergravity in the conformal gauge. We explicitly calculate the Jacobian accompanying the change from the Weyl invariant measure to the translation invariant one. We show that it is of the same form as the original Liouville action, thus establishing a conjecture of David and Distler and Kawai. This calculation yields dressed gravitational central charges and anomalous dimensions from first principles.

Theory of l-reduction of KP hierarchy is investigated from a view point of conformal field theory. Twisted conformal field theory emerges naturally in the discussion. Possible application of this kind of connection between hierarchy and CFT is suggested.

This paper gives a heuristic derivation of the skein relation for the Homfly polynomial in an integral formalism. The derivation is formally correct but highly simplified. In the light of Witten's proposal for invariants of links via functional integrals, it is useful to have a formal pattern to compare with the complexities of the full approach. The formalism is a heuristic. However, it is closely related to the actual structure of the Witten functional integral.

Particular perturbations of a 2D Conformal Field Theory leading to Integrable massive Quantum Field Theories are examined. The mass spectra and S-matrices for some models, including the field theory of the Ising Model with magnetic field and “thermal” deformations of the tricritical Ising and 3-state Potts models, are proposed. The hidden Lie-algebraic structures of these spectra and their relation to the Toda systems are discussed.

In this paper we extend our previous results on the bosonic Liouville theory, to the supersymmetric case. As in the bosonic case, we find that the quantization of the N=1 theory is limited to the region D≤1. We compute the exact critical exponents and the analogue of the Hausdorff dimension of super random surfaces. Our procedure is manifestly covariant and our results hold for the surface of arbitrary topology. We also examine the N=2, O(2) string theory and find that it appears to be well-defined for all D.

The general class of scalar-tensor theories is discussed in which, in contrast to Brans-Dicke theory, the coupling parameter ω is not a constant but a function of the scalar field ϕ. Some other generalizations of the Brans-Dicke theory have also been discussed in brief. The physical aspects of these theories have been dealt. The cosmological models in scalar-tensor theories have been discussed in detail. Finally some open problems in these theories have been pointed out.

It is shown that the contact term discovered by Wendt is sufficient to ensure finiteness of all tree-level scattering amplitudes in Witten’s field theory of open superstrings. Its inclusion in the action also leads to a gauge-invariant theory. Thus, no additional higher-order counterterms in the action are needed.

It is shown that all two dimensional world-sheet reparametrization invariant and renormalizable theories of string field Xμ, metric gab and nonlinearly coupled scalar field ϕ can be reduced to the Liouville theory provided there is no direct ϕ−Xμ coupling and the theory is constrained to be ghost-free.

We present a survey of two-dimensional conformal field theory and show how the mathematical structures underlying conformal field theory can be used to construct invariants of links imbedded in a general class of three-dimensional manifolds. After a general introduction, we discuss chiral algebras and their representation theory. Chiral vertices are introduced as analogues of Clebsch-Gordan operators in group theory. Braiding and fusing of chiral vertices is analyzed, and it is sketched how to define conformal field theory on arbitrary Riemann surfaces by a sewing procedure. We then show how to construct link invariants from the data provided by a conformal field theory and sketch connections with quantum group theory.

The unique effective action recently suggested by Vilkovisky should be considered as an alternative to the standard effective action in field theory.

Modern development of conformal field theory in two dimensions and its applications to critical phenomena are briefly reviewed. The specific properties of the renormalization group in two dimensions and the fundamentals of 2-dimensional conformal field theory are presented. The properties of degenerate representations of the Virasoro algebra and other infinite dimensional algebras, "minimal" models of conformal and superconformal field theory, "parafermionic" and other symmetries are discussed. We also investigate a perturbation theory around conformal solutions of field theory.