Abstract
We discuss the quantization of non-ultralocal integrable models directly in the continuous case, using the example of the Alday–Arutyunov–Frolov model. We show that by treating fields as distributions and regularizing the operator product, it is possible to avoid all the singularities, and allow to obtain results consistent with perturbative calculations. We illustrate these results by considering the reduction to the massive free fermion model and extracting the quantum Hamiltonian as well as other conserved charges directly from the regularized trace identities. Moreover, we show that our regularization recovers Maillet's prescription in the classical limit.
Highlights
The Alday-Arutyunov-Frolov (AAF) model is a purely fermionic classical integrable model arising from the reduction of the AdS5 × S5 superstring theory to the su(1|1) subsector in the uniform gauge [1, 2]
One that recovers the symmetrization prescription pertaining the definition of the Maillet brackets in the classical limit
For non-ultralocal models, such as the AAF model, there does not exist such quantum relation, from which one can, for example, extract the quantum Hamiltonian. That such quantum Hamiltonian should be written, according to our main result in the previous section, in terms of the regularized fields (3.1), where the operator products are written in terms of Sklyanin’s product (3.12), in order to avoid the problems associated with singularities and to implement the principal value prescription from the beginning
Summary
The Alday-Arutyunov-Frolov (AAF) model is a purely fermionic classical integrable model arising from the reduction of the AdS5 × S5 superstring theory to the su(1|1) subsector in the uniform gauge [1, 2]. It reduces essentially to the following four steps: (i) ultralocalize the Kac-Moody type algebra satisfied by the classical continuous theory; (ii) regularize the ultralocalized current algebra to get rid of the singularities at coinciding points by invoking a lattice discretization; (iii) quantize the lattice current algebra by means of the quantum inverse scattering method; (iv) check that in the scaling limit the quantized discrete algebra reproduces the classical Kac-Moody algebra This recipe breaks down for the AAF model already in step (i), as all the ultralocalization procedures so far developed work only for models plagued by non-ultralocalities up to the first derivative of the delta function, while the algebra of Lax operators for the AAF model is even more non-ultralocal, including terms proportional to the second derivative of the delta function [17, 18]. In appendices we collect various computational details used in the text
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