Том 199, № 1 (2019)

We review properties of generalized Macdonald functions arising from the AGT correspondence. In particular, we explain a coincidence between generalized Macdonald functions and singular vectors of a certain algebra \({\cal A}(N)\) obtained using the level-(N, 0) representation (horizontal representation) of the Ding-Iohara-Miki algebra. Moreover, we give a factored formula for the Kac determinant of \({\cal A}(N)\), which proves the conjecture that the Poincaré-Birkhoff-Witt-type vectors of the algebra \({\cal A}(N)\) form a basis in its representation space.

We describe the real forms of classical elliptic integrable systems such as the elliptic Calogero-Moser system and the elliptic Euler-Arnold top in the framework of a general scheme for constructing real reductions for a Hitchin system.

For a planar Dirac-Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge Z > Z_cr,1and radius R₀, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density \({\rho _{{\rm{VP}}}}(\vec r)\), we thoroughly consider the calculation of the vacuum energy ε_VPbased on the renormalization, the convergence of the partial expansion for \({\rho _{{\rm{VP}}}}(\vec r)\), and the behavior of the integral induced charge Q_VPin the overcritical region. In particular, we show that the renormalization using the fermionic loop with two external lines turns out to be a universal technique, which eliminates the divergence of the theory in the purely perturbative and essentially nonperturbative modes for \({\rho _{{\rm{VP}}}}(\vec r)\) and ε_VP. The most significant result is that for Z ≫ Z_cr,1in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z reaching large negative values; its behavior is estimated from below (in absolute value) as ∼ −|η_effZ³|/R₀. We also study the dependence of the effect of the decrease in ε_VPon the cutoff of the Coulomb asymptotics of the external field at different scales R₁ gt; R₀and R₁ ≫ R₀.

We consider the influence of N long-lived states characterized by resonance energies E_i and widths Γ_i(E) on the elastic scattering process and obtain an expression for the partial S-matrix S_l(E) in the form of a sum over the resonance levels (poles) at which the residues have the form \({{\rm{\Gamma}}_i}\prod\nolimits_{\matrix{\hfill {k = 1,} \cr \hfill {k \ne i} \cr}}^N {{\gamma _{ik}}} \), where \({\gamma _{ik}} = ({z_i} - z_k^*)\imath /2({z_i} - {z_k})\) and z_i = E_i−lΓ_i/2. We show that a necessary condition for the unitarity of the partial S-matrix in the presence of N resonance levels can be written as \(\sum\nolimits_{i = 1}^N {{{\rm{\Gamma}}_i}(E)\prod\nolimits_{\matrix{\hfill {k = 1,} \cr \hfill {k \ne i} \cr}}^N {{\gamma _{ik}} = \sum\nolimits_{i = 1}^N {{{\rm{\Gamma}}_i}(E)}}} \).

In the framework of general relativity theory, we consider a space-time whose metric depends on only one coordinate and time. We choose a gauge class such that all the constraint conditions of the theory of gravity as a gauge theory are explicitly solved and construct a Hamiltonian depending only on dynamical physical variables (gravitons). We show that such a Hamiltonian can be obtained from the Polyakov action for a string in an anti-de Sitter background space with a “string constant” depending on time.