We consider some generalizations of a ((2+1))-dimensional Davey–Stewartson-type equation. In particular, we propose a dynamical system that does not admit an explicit formulation in terms of differential equations, but needs an additional independent variable.
We present exactly solvable modifications of the two-matrix Zinn-Justin–Zuber model and write it as a tau function. The grand partition function of these matrix integrals is written as the fermion expectation value. The perturbation theory series is written explicitly in terms of a series in strict partitions. The related string equations are presented.
We summarize some (mostly geometric) facts underlying the relation between (2)D integrable sigma models and generalized Gross–Neveu models, emphasizing connections to the theory of nilpotent orbits, Springer resolutions, and quiver varieties. This is meant to shed light on the general setup when this correspondence holds.
We develop a technique based on the boost automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces. We initiate the method by implementing it in two-dimensional models and resolve a classification problem, which not only confirms the known vertex model solution space but also extends to the new (mathfrak{sl}_2) deformed sector. A generalization of the approach to integrable string backgrounds is provided and allows finding new integrable deformations and associated (R)-matrices. The new integrable solutions appear to be of a nondifference or pseudo-difference form admitting (AdS_2) and (AdS_3) (S)-matrices as special cases (embeddings), which also includes a map of the double-deformed sigma model (R)-matrix. The corresponding braiding and conjugation operators of the novel models are derived. We also demonstrate implications of the obtained free-fermion analogue for (AdS) deformations.
We review some recent results of the studies of quantum corrections in supersymmetric theories derived using Slavnov’s higher covariant derivative regularization. In particular, we demonstrate that the (beta)-function of (mathcal{N}=1) supersymmetric theories is related to the anomalous dimensions of matter superfields by the NSVZ relation if the theory is regularized by higher covariant derivatives and the renormalization group functions are defined in terms of the bare couplings, because the corresponding loop corrections are given by integrals of double total derivatives in the momentum space. For the standard renormalization-group functions, we show that an all-loop NSVZ renormalization scheme is given by the HD(,+,)MSL renormalization prescription when the higher covariant derivative regularization is supplemented by minimal subtractions of logarithms. Applications of these results to the precise calculations in various supersymmetric theories are briefly described.
We report on a recent progress in constructing off-shell (4) D, (mathcal{N}=2) supersymmetric integer higher-superspin theory in terms of unconstrained harmonic analytic gauge superfields and their cubic interaction with matter hypermultiplets. For even superspins, a new equivalent representation of the hypermultiplet couplings in terms of an analytic (omega) superfield is presented. It involves both cubic and quartic vertices.
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