Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Conformally invariant field theories in two dimensions and corresponding statistical mechanical models

Belavin, Zamlodochikov and Polyakov have recently proposed a class of conformally invariant field theories in two dimension with exactly determined rational critical indices. We establish a tentative identification of a subset of these theories in terms of the O(n) model and theq-state Potts model in 2-dimensions for appropriaten andq. The results of this work were reported in the conference on “Structural Similarities in Exactly Solved Models” at I.T.P. Santa Barbara, August 1984.     

Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

On the twistor space of a quaternionic contact manifold

In this note, we prove that the CR manifold induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.     

Tau functions and the twistor theory of integrable systems

We first define τ-functions as generalized cross-ratios of four points on a finite- or infinite-dimensional Grassmannian. We show how this definition can be used to construct a natural flat connection on a determinant line bundle associated with two equivariant holomorphic vector bundles over a twistor space, provided that the action of the symmetries on the bundles has the same normal form at the fixed points for the two bundles. The determinant line bundle has a natural meromorphic section of which the logarithmic covariant derivative is the logarithmic derivative of the τ-function. We establish a natural product formula for this τ-function; we show that it vanishes at the jumping lines of one bundle and has poles at the jumping lines of the other. We also show that this definition leads to standard expressions for the τ-functions of the KdV equation, the Ernst equation, and the isomonodromic deformation equations. We describe a new twistor treatment of the isomonodromic deformation equations.     

Boundary conditions changing operators in non-conformal theories

Boundary conditions changing operators have played an important role in conformal field theory. Here, we study their equivalent in the case where a mass scale is introduced, in an integrable way, either in the bulk or at the boundary. More precisely, we propose an axiomatic approach to determine the general scalar products _bθ₁, … ,θ_mθ₁′, … ,θ_n′_a, between asymptotic states in the Hilbert spaces with a and b boundary conditions respectively, and compute these scalar products explicitly in the case of the Ising and sinh-Gordon models with a mass and a boundary interaction. These quantities can be used to study statistical systems with inhomogeneous boundary conditions, and, more interestingly maybe, dynamical problems in quantum impurity problems. As an example, we obtain a series of new exact results for the transition probability in the double-well problem of dissipative quantum mechanics.     

Stability of twistor lifts for surfaces in four-dimensional manifolds as harmonic sections

We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts.     

Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

New observables in topological instantonic field theories

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type for the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on the linking of points.     

Types of two-dimensionalN = 4 superconformal field theories

Various types ofN = 4 superconformal symmetries in two dimensions are considered. It is proposed that apart from the well-known cases ofSU (2)and SU(2) × SU(2) ×U (1), their Kac-Moody symmetry can also be SU(2) × (U (1))⁴. Operator product expansions for the last case are derived. A complete free field realization for the same is obtained     

On explicit free field realization of current algebras

We construct the explicit free field representations of the current algebras so(2n)_k, so(2n+1)_k and sp(2n)_k for a generic positive integer n and an arbitrary level k. The corresponding energy–momentum tensors and screening currents of the first kind are also given in terms of free fields.     

Disordered loop model and coupled conformal field theories

A family of models for fluctuating loops in a two-dimensional random background is analyzed. The models are formulated as O(n) spin models with quenched inhomogeneous interactions. Using the replica method, the models are mapped to the M→0 limits of M-layered O(n) models coupled each other via _1,3 primary fields. The renormalization group flow is calculated in the vicinity of the decoupled critical point, by an epsilon expansion around the Ising point (n=1), varying n as a continuous parameter. The one-loop beta function suggests the existence of a strongly coupled phase (0<n<n_*) near the self-avoiding walk point (n=0) and a line of infrared fixed points (n_*<n<1) near the Ising point. For the fixed points, the effective central charges are calculated. The scaling dimensions of the energy operator and the spin operator are obtained up to two-loop order. The relation to the random-bond q-state Potts model is briefly discussed.     

The bases of effective field theories

With reference to the equivalence theorem, we discuss the selection of basis operators for effective field theories in general. The equivalence relation can be used to partition operators into equivalence classes, from which inequivalent basis operators are selected. These classes can also be identified as containing Potential-Tree-Generated (PTG) operators, Loop-Generated (LG) operators, or both, independently of the specific dynamics of the underlying extended models, so long as it is perturbatively decoupling. For an equivalence class containing both, we argue that the basis operator should be chosen from among the PTG operators, because they may have the largest coefficients. We apply this classification scheme to dimension-six operators in an illustrative Yukawa model as well in the Standard Model (SM). We show that the basis chosen by Grzadkowski et al. [5] for the SM satisfies this criterion. In this light, we also revisit and verify our earlier result [6] that the dimension-six corrections to the triple-gauge-boson couplings only arise from LG operators, so the magnitude of the coefficients should only be a few parts per thousand of the SM gauge coupling if BSM dynamics respects decoupling. The same is true of the quartic-gauge-boson couplings.     

A discrete approach to topological quantum field theories

First, we describe a rather general scheme for constructing three-dimensional euclidean topological quantum field theories, whose basic building blocks are provided by the representation theory of a certain class of (bi-)algebras. Secondly, we discuss in some detail examples, where the algebra is either the function algebra of a finite group, the group algebra of a finite group or a deformation of the enveloping algebra of a classical simple Lie group.     

Bayesian methods for parameter estimation in effective field theories

We demonstrate and explicate Bayesian methods for fitting the parameters that encode the impact of short-distance physics on observables in effective field theories (EFTs). We use Bayes’ theorem together with the principle of maximum entropy to account for the prior information that these parameters should be natural, i.e., O(1) in appropriate units. Marginalization can then be employed to integrate the resulting probability density function (pdf) over the EFT parameters that are not of specific interest in the fit. We also explore marginalization over the order of the EFT calculation, M, and over the variable, R, that encodes the inherent ambiguity in the notion that these parameters are O(1). This results in a very general formula for the pdf of the EFT parameters of interest given a data set, D. We use this formula and the simpler “augmented χ²” in a toy problem for which we generate pseudo-data. These Bayesian methods, when used in combination with the “naturalness prior”, facilitate reliable extractions of EFT parameters in cases where χ² methods are ambiguous at best. We also examine the problem of extracting the nucleon mass in the chiral limit, M₀, and the nucleon sigma term, from pseudo-data on the nucleon mass as a function of the pion mass. We find that Bayesian techniques can provide reliable information on M₀, even if some of the data points used for the extraction lie outside the region of applicability of the EFT.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.