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Register a new userVertex Operators and Soliton Time Delays in Affine Toda Field Theory
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In a space-time of two dimensions the overall effect of the collision of two solitons is a time delay (or advance) of their final trajectories relative to their initial trajectories. For the solitons of affine Toda field theories, the space-time displacement of the trajectories is proportional to the logarithm of a number $X$ depending only on the species of the colliding solitons and their rapidity difference. $X$ is the factor arising in the normal ordering of the product of the two vertex operators associated with the solitons. $X$ is shown to take real values between $0$ and $1$. This means that, whenever the solitons are distinguishable, so that transmission rather than reflection is the only possible interpretation of the classical scattering process, the time delay is negative and so an indication of attractive forces between the solitons.
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A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.
Exact solutions to the quantum S-matrices for solitons in simply-laced affine Toda field theories are obtained, except for certain factors of simple type which remain undetermined in some cases. These are found by postulating solutions which are consistent with the semi-classical limit, $hbarrightarrow 0$, and the known time delays for a classical two soliton interaction. This is done by a `$q$-deformation procedure, to move from the classical time delay to the exact S-matrix, by inserting a special function called the `regularised quantum dilogarithm, which only holds when $|q|=1$. It is then checked that the solutions satisfy the crossing, unitarity and bootstrap constraints of S-matrix theory. These properties essentially follow from analogous properties satisfied by the classical time delay. Furthermore, the lowest mass breather S-matrices are computed by the bootstrap, and it is shown that these agree with the particle S-matrices known already in the affine Toda field theories, in all simply-laced cases.
Soliton theory and the theory of Hankel (and Toeplitz) operators have stayed essentially hermetic to each other. This paper is concerned with linking together these two very active and extremely large theories. On the prototypical example of the Cauchy problem for the Korteweg-de Vries (KdV) equation we demonstrate the power of the language of Hankel operators in which symbols are conveniently represented in terms of the scattering data for the Schrodinger operator associated with the initial data for the KdV equation. This approach yields short-cuts to already known results as well as to a variety of new ones (e.g. wellposedness beyond standard assumptions on the initial data) which are achieved by employing some subtle results for Hankel operators.
For $mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $text{Aut}, mathcal O$-equivariant identification between $text{Fun},text{Op}_{mathfrak g}(D)$, the algebra of functions on the space of ${mathfrak g}$-opers on the disc, and $Wsubset pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $delta_{-1}left|0right>$. We show that the latter endows $pi_0$ with a canonical notion of translation $T^{text{(aff)}}$, and use it to define the densities in $pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $text{Aut},mathcal O$-action defines a bundle $Pi$ over $mathbb P^1$ with fibre $pi_0$. We show that the product bundles $Pi otimes Omega^j$, where $Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $ abla^{text{(aff)}} - alpha T^{text{(aff)}}$, $alphain mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[mathbf v_j dt^{j+1} ] in H^1(mathbb P^1, Piotimes Omega^j, abla^{text{(aff)}})$ of the de Rham cohomology of $ abla^{mathrm{(aff)}}$. Any choice of ${mathfrak g}$-Miura oper $chi$ gives a connection $ abla^{mathrm{(aff)}}_chi$ on $Omega^j$. Using coinvariants, we define a map $mathsf F_chi$ from sections of $Pi otimes Omega^j$ to sections of $Omega^j$. We show that $mathsf F_chi abla^{text{(aff)}} = abla^{text{(aff)}}_chi mathsf F_chi$, so that $mathsf F_chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(mathbb P^1, Omega^j, abla^{text{(aff)}}_chi)$ defined by the ${mathfrak g}$-oper underlying $chi$.
We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.
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