On Dynamics of 5D superconformal theories

5D superconformal theories involve vacuum valleys characterized in the simplest case by the vacuum expectation value of the real scalar field σ. If 〈σ〉 ≠ 0, conformal invariance is spontaneously broken and the theory is not renormalizable. In the conformally invariant sector 〈σ〉 = 0, the theory is intrinsically nonperturbative. We study classical and quantum dynamics of this theory in the limit when field dependence of the spatial coordinates is disregarded. The classical trajectories “fall” on the singularity at σ = 0. The quantum spectrum involves ghost states with negative energies unbounded from below, but such states fail to form complete 16-plets as is dictated by the presence of four complex supercharges and should be rejected for that reason. Physical excited states come in supermultiplets and have all positive energies.We conjecture that the spectrum of the complete field theory Hamiltonian also becomes positive definite (ghost-free) when invoking supersymmetry considerations.We speculate that the ghosts in higher derivative supersymmetric field theories are exterminated by a similar mechanism. The text was submitted by the author in English. On leave of absence from ITEP, Moscow, Russia     

Unitarity of rationalN = 2 superconformal theories

We demonstrate that all rational models of theN = 2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu’s algebraA(H₀) (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms ofsu(2) and two free fermions. Some of our arguments generalise to the Kazama-Suzuki models indicating that all rationalN = 2 supersymmetric models might be unitary.     

Modular invariance in N=2 superconformal field theories

The modular invariance properties of two-dimensional N=2 superconformal field theories are studied. It is shown that the character formulae of the central charge c<3 unitary highest weight representation for the untwisted algebras can be written in terms of the string functions and the theta functions of the affine su(2) Kac-Moody algebra. Deriving the modular transformation of the characters we construct the modular invariant partition functions on a torus. The character relation corresponding to the coset space construction of the unitary discrete series in the N=2 algebra is also obtained.     

On the classification ofN=2 superconformal coset theories

We show that two dimensionalN=2 superconformal field theories cannot be constructed by applying the supersymmetric extension of the GKO construction to the so-called special subalgebras, i.e. subalgebras for which at least one generator associated to a root of the subalgebra does not correspond to a root of the algebra itself. We thus prove the completeness of the classification ofN=2 supersymmetric coset models obtained by Kazama and Suzuki. Furthermore we point out that compared to their papers an additional criterion has to be added in theN=2 conditions.     

Simple superconformal field theories in two dimensions

Superconformal field theories with first-order lagrangians, the super BC-systems, are considered on an arbitrary Riemann surface with an arbitrary gravitino field. The corresponding functional integrals are carefully defined, and the Weyl, supersymmetry, and holomorphic anomalies are computed. All the anomalies have a common coefficient. The holomorphic anomaly involves terms quadratic in the gravitino field.     

Study of N = 2 superconformal field theories in 4 dimensions

Making use of the exact solutions of the N = 2 supersymmetric gauge theories, we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional N = 2 SCFTs.