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We study the deformation describing a mapping any dynamical system with irreducible first-class constraints in the phase space into another dynamical system with first-class constraints. It is shown that such a deformation problem can be effectively explored in the framework of the Batalin-Fradkin-Vilkovisky (BFV) formalism. The basic objects of this formalism are the BRST-BFV charge and a generalized Hamiltonian that satisfy the defining equations in the extended phase space in terms of (super)Poisson brackets. Solution to the deformation problem is found in terms of a (super)canonical transformation with a special generating function which is explicitly established. It is proved that this generating function is defined by a single arbitrary function which depends only on coordinates of initial dynamical system. To illustrate the developed approach, we have constructed a non-local deformation of the Abelian gauge theory into a non-local non-Abelian gauge theory whose local sector coincides with the Yang-Mills theory.

We study the off-shell structure of the two-loop effective action in 6D,$$ \mathcal{N} $$ N = (1, 1) supersymmetric gauge theories formulated in $$ \mathcal{N} $$ N = (1, 0) harmonic superspace. The off-shell effective action involving all fields of 6D,$$ \mathcal{N} $$ N = (1, 1) supermultiplet is constructed by the harmonic superfield background field method, which ensures both manifest gauge covariance and manifest $$ \mathcal{N} $$ N = (1, 0) supersymmetry. We analyze the off-shell divergences dependent on both gauge and hypermultiplet superfields and argue that the gauge invariance of the divergences is consistent with the non-locality in harmonics. The two-loop contributions to the effective action are given by harmonic supergraphs with the background gauge and hypermultiplet superfields. The procedure is developed to operate with the harmonic-dependent superpropagators in the two-loop supergraphs within the superfield dimensional regularization. We explicitly calculate the gauge and the hypermultiplet-mixed divergences as the coefficients of $$ \frac{1}{\varepsilon^2} $$ 1 ε 2 and demonstrate that the corresponding expressions are non-local in harmonics.

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