Two-dimensional Quantum Field Theory Research Articles

The composite operatorTbar T in sinh-Gordon and a series of massive minimal models

The composite operator T\bar{T}, obtained from the components of the energy-momentum tensor, enjoys a quite general characterization in two-dimensional quantum field theory also away from criticality. We use the form factor bootstrap supplemented by asymptotic conditions to determine its matrix elements in the sinh-Gordon model. The results extend to the breather sector of the sine-Gordon model and to the minimal models M_{2/(2N+3)} perturbed by the operator phi_{1,3}.

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.

Spacetime in string theory

We give a brief overview of the nature of spacetime emerging from string theory. This is radically different from the familiar spacetime of Einstein's relativity. At a perturbative level, the spacetime metric appears as ‘coupling constants’ in a two-dimensional quantum field theory. Nonperturbatively (with certain boundary conditions), spacetime is not fundamental but must be reconstructed from a holographic, dual theory.

Matrix elements of the operator [formula omitted] in integrable quantum field theory

Recently A. Zamolodchikov obtained a series of identities for the expectation values of the composite operator T T ¯ constructed from the components of the energy–momentum tensor in two-dimensional quantum field theory. We show that if the theory is integrable the addition of a requirement of factorization at high energies can lead to the exact determination of the generic matrix element of this operator on the asymptotic states. The construction is performed explicitly in the Lee–Yang model.

G-FUNCTION IN PERTURBATION THEORY

We present some explicit computations checking a particular form of gradient formula for a boundary beta function in two-dimensional quantum field theory on a disk. The form of the potential function and metric that we consider were introduced in Refs. 16 and 18 in the context of background independent open string field theory. We check the gradient formula to the third order in perturbation theory around a fixed point. Special consideration is given to situations when resonant terms are present exhibiting logarithmic divergences and universal nonlinearities in beta functions. The gradient formula is found to work to the given order.

Virasoro algebra with central charge c=1 on the horizon of a two-dimensional-Rindler space–time

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler space–time admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra defined in the Fock representation. That representation has central charge c=1. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated quantum field theory on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy h≔μ2/2 vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for h=1/2, the ground state of that generator seems to correspond to a thermal state when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. Finally, under Wick rotation in Rindler time, the pair of quantum field theories which are built up on the future and past horizon defines a proper two-dimensional conformal quantum field theory on a cylinder.

Polarization-Free Quantum Fields and Interaction

A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Field algebra, Hilbert space and observables in two-dimensional higher-derivative field theories

We discuss structural aspects related to the field algebra of two-dimensional higher-derivative quantum field theories. We present general selection criteria for the proper field subalgebra that generates Wightman functions satisfying the asymptotic factorization property and which define a semi-definite inner product Hilbert space. The positive definite inner product Hilbert space, which contains as a subspace of states the general Wightman functions of the corresponding standard canonical models, is a quotient space obtained by equivalence classes. For higher-derivative local gauge theories, besides the Lowenstein–Swieca condition, an additional condition must be imposed on the field algebra in order to obtain a physical subspace of states satisfying a reasonable set of physically meaningful axioms.

An Exact Holographic RG Flow Between 2d Conformal Fixed Points

We describe a supersymmetric RG flow between conformal fixed points of a two-dimensional quantum field theory as an analytic domain wall solution of the three-dimensional SO(4) × SO(4) gauged supergravity. Its ultraviolet fixed point is an N = (4,4) superconformal field theory related, through the double D1-D5 system, to theories modeling the statistical mechanics of black holes. The flow is driven by a relevant operator of conformal dimension Δ = 3/2 which breaks conformal symmetry, breaks supersymmetry down to N = (1,1), and sends the theory to an infrared conformal fixed point with central charge cIR = cUV/2. Using the supergravity description, we compute counterterms, one-point functions and fluctuation equations for inert scalars and vector fields, providing the complete framework to compute two-point correlation functions of the corresponding operators throughout the flow in the two-dimensional quantum field theory. This produces a toy model for flows of N = 4 super Yang-Mills theory in 3+1 dimensions, where conformal-to-conformal flows have resisted analytical solution.

Field-theoretical interpretation of the holographic renormalization group

A quantum-field theoretical interpretation is given to the holographic RG equation by relating it to a field-theoretical local RG equation which determines how Weyl invariance is broken in a quantized field theory. Using this approach we determine the relation between the holographic C theorem and the C theorem in two-dimensional quantum field theory which relies on the Zamolodchikov metric. Similarly we discuss how in four dimensions the holographic C function is related to a conjectured field-theoretical C function. The scheme dependence of the holographic RG due to the possible presence of finite local counterterms is discussed in detail, as well as its implications for the holographic C function. We also discuss issues special to the situation when mass deformations are present. Furthermore we suggest that the holographic RG equation may also be obtained from a bulk diffeomorphism which reduces to a Weyl transformation on the boundary.

Quantum field theory of treasury bonds.

The Heath-Jarrow-Morton (HJM) formulation of treasury bonds in terms of forward rates is recast as a problem in path integration. The HJM model is generalized to the case where all the forward rates are allowed to fluctuate independently. The resulting theory is shown to be a two-dimensional Gaussian quantum field theory. The no arbitrage condition is obtained and a functional integral derivation is given for the price of a futures and an options contract.

Boundary bootstrap principle in two-dimensional integrable quantum field theories

We study the reflection amplitudes of affine Toda field theories with boundary, following the ideas developed by Fring and Koberle and focusing our attention on the $E_{n}$ series elements, because of their interesting structure of higher order poles. We also investigate the corresponding minimal reflection matrices, finding, with respect to the bulk case, a more complicated relation between the spectra of bound states associated to the minimal and to the ''dressed'' amplitudes.

Polarization-Free Generators and the S-Matrix

Polarization-free generators, i.e. ``interacting'' Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.

Renormalization group flow with unstable particles

The renormalization group flow of an integrable two-dimensional quantum field theory which contains unstable particles is investigated. The analysis is carried out for the Virasoro central charge and the conformal dimensions as a function of the renormalization group flow parameter. This allows us to identify the corresponding conformal field theories together with their operator content when the unstable particles vanish from the particle spectrum. The specific model considered is the ${\mathrm{SU}(3)}_{2}$-homogeneous sine-Gordon model.

On second quantization of quantum groups

We construct a deformation of the function algebra on the quantum group SLq(2) into a trialgebra in the sense of Crane and Frenkel. We show that this naturally acts on the trialgebraic deformation of the Manin plane, previously introduced by the authors. Alternatively, one can view it as acting on the trialgebraic deformation of the fermionic Manin plane. We prove that the trialgebraic deformation of SLq(2) defines a 2−C*-category, a structure as needed for the superselection structure of massive two-dimensional quantum field theories. Besides this, we investigate another approach to trialgebra deformations of a bialgebra as a deformation of a Fock space construction over the bialgebra.

Canonical Tensor Product Subfactors

Canonical tensor product subfactors (CTPS's) describe, among other things, the embedding of chiral observables in two-dimensional conformal quantum field theories. A new class of CTPS's is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding two-dimensional theories.

Yang-Mills theory as a deformation of topological field theory, dimensional reduction, and quark confinement

We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group Hof G invariant. Then we find topological field theory $(D>2)$ has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the $(D\ensuremath{-}2)$-dimensional coset $G/H$ nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories.

Construction of Kink Sectors for Two-Dimensional Quantum Field Theory Models – An Algebraic Approach

Several two-dimensional quantum field theory models have more than one vacuum state. Familiar examples are the Sine-Gordon and the [Formula: see text]-model. It is known that in these models there are also states, called kink states, which interpolate different vacua. A general construction scheme for kink states in the framework of algebraic quantum field theory is developed in a previous paper. However, for the application of this method, the crucial condition is the split property for wedge algebras in the vacuum representations of the considered models. It is believed that the vacuum representations of P(ϕ)2-models fulfill this condition, but a rigorous proof is only known for the massive free scalar field. Therefore, we investigate in a construction of kink states which can directly be applied to a large class of quantum field theory models, by making use of the properties of the dynamics of a P(ϕ)2 and Yukawa2 models.

The reduced covariant phase space quantization of the three-dimensional Nambu-Goto string

The reduced covariant phase space associated with the three-dimensional Euclidean Nambu-Goto action can be identified, via the Enneper-Weierstrass representation of minimal surfaces, with the space of complex analytic functions plus three translational zero-modes. The symplectic structure induced through the Enneper-Weierstrass map can be explicitly computed. Quantization is then straightforward, yielding as a result a target-space Euclidean-invariant, positive-definite, two-dimensional quantum field theory. The physical states are shown to correspond with particles states of integer spin and arbitrary mass.

Non-integrable aspects of the multi-frequency sine-Gordon model

We consider the two-dimensional quantum field theory of a scalar field self-interacting via two periodic terms of frequencies α and β. Looking at the theory as a perturbed sine-Gordon model, we use form factor perturbation theory to analyse the evolution of the spectrum of particle excitations. We show how, within this formalism, the non-locality of the perturbation with respect to the solitons is responsible for their confinement in the perturbed theory. The effects of the frequency ratio α/β being a rational or irrational number and the occurrence of massless flows from the gaussian to the Ising fixed point are also discussed. A generalisation of the Ashkin-Teller model and the massive Schwinger model are presented as examples of application of the formalism.