N = 4 Yang-Mills theory in terms of N = 3 and N = 2 light-cone superfields

The N = 4 Yang-Mills theory is truncated to an N = 3 Yang-Mills theory and to an N = 2 Yang-Mills theory coupled to an N = 2 Wess-Zumino field. The whole procedure is performed in the light-cone gauge. It is then shown that these theories are unique even if we only insist on N = 3 or N = 2 supersymmetry respectively. Finally we show in detail how the introduction of the fermionic Wess-Zumino field renders the one-loop self-energy finite.     

The relaxed hypermultiplet: An unconstrained N = 2 superfield theory

We present an off-shell version of the hypermultiplet, without a central charge or gauge antisymmetric tensors. We give the unconstrained superfield form of the action and its coupling to N = 2 super-Yang-Mills theory. This allows an N = 2 superspace formulation of the N = 4 super-Yang-Mills theory, whose ultraviolet finiteness is then ensured by N = 2 non-renormalization theorems.     

Seiberg-Witten geometry with various matter contents

We obtain the Seiberg-Witten geometry for four-dimensional N = 2 gauge theory with gauge group SO(2N_c) (N_c ⩽ 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N = 2 E₆ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N = 2 SU(N_c) (N_c ⩽ 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable agreement. We also show that these results are reproduced by using N = 1 confining phase superpotentials.     

Quartic mass matrix and renormalization constants in supersymmetric Yang-Mills theories

We derive the general formula for the supertrace of the quartic mass matrix in a general supersymmetric gauge theory, with arbitrary representations for the chiral multiplets. This formula clarifies the non-renormalization theorems in presence of gauge interactions and gives “extended renormalization theorems” for N = 2 and N = 4 supersymmetric Yang-Mills theories. In particular we find the known result that g_ren = g_bare for the N = 4 theory and the new result m_ren = m_bare for the N = 2 gauge interactions of massive hypermultiplets. We give arguments to the extent that the latter non-renormalization theorem persists to all orders in perturbation theory.     

On the unitary representations of N = 2 superconformal theory

The Kac formula for superconformal dimensions (generalized to N = 2) is further developed (compared to a previous article). A list of discrete values of the central charge for which unitary representations are expected to exist is proposed. For several of these, unitarity is checked by computer. For two values, unitarity is proven by providing explicit fermionic representations. For one of those values, the N = 2 theory coincides with a sub theory of one of the known unitary N = 1 theories, thus extending a similar situation between N = 0 and N = 1.     

Counterterms in extended supergravities

Geometrical invariants respecting all necessary symmetries of the theory are shown to exist, starting from the 8th (4th) loop approximation in N = 8 (N = 4) on-shell supergravity. 3-loop counterterms are presented on a linearized level for N = 4 and N = 8 theories. The corresponding 3-loop non-linear invariants are discussed.     

Renormalization character and quantum S-matrix for a classically integrable theory

The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sine-Gordon model) and N = 2 (reduced O(4) σ-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the one-loop order. Its effect is like a local quartic coupling.     

Dual quark-gluon theory with dynamical color

A modification of a previously proposed dual resonance theory of quarks and gluons is presented. It consists of incorporating new oscillator modes carrying color indices. The specific properties of these operators and the way they are included into the theory are completely determined by various consistency requirements. This modification of the theory has two important consequences. First, quark statistics are properly taken into account. Second, the critical dimension of space-time is reduced to d = 10?2N, where N is the number of colors. Thus, the physically preferred choices N = 3 and d = 4 are compatible.     

Extended supergravity on the supergroup manifold: N = 3 and N = 2 theories

In this paper I construct the group-manifold first-order formulation of N = 2 and N = 3 supergravity based on the $\text{Osp}\text{(}\text{4}\text{2}\text{)}\text{and Osp}\text{(}\text{4}\text{3}\text{)}$ supergroups, respectively. In the case N = 2, a group manifold version of the theory was already presented in a previous paper. Here a simpler formulation is given which shows exact factorization in the SO(2) subgroup absent in the previous one. Particular attention is paid to the algebraic role played by the $\text{spin}\text{-}\text{1}\text{2}$ field which is the novel feature of the N = 3 case with respect to N = 2. It is shown how the “non-geometrical” term in the gravitino transformation law in the N = 2 theory arises from the rheonomic symmetry mechanism.     

Integrability and Seiberg-Witten theory curves and periods

The interpretation of exact results on the low-energy limit of 4D N = 2 supersymmetric Yang-Mills theory in the language of 1D integrable system of particles is discussed. The Riemann surfaces of the Seiberg-Witten theory are explicitly described as spectral curves of Lax operators. The case of the elliptic Calogero system, associated with the flow between N = 4 and N = 2 supersymmetric in 4D, is considered in some detail. Equations for the corresponding Riemann surfaces are written down rather explicitly for all the SU(n) groups.     

The action of the supersymmetric N = 2 gauge theory in harmonic superspace

We consider a solution of constraints for the analytic representation of the harmonic superfield connection in the N = 2 supersymmetric gauge theory. A simple geometric expression for the action of this theory is obtained in the form of the integral over the N = 2 superspace.     

Liquid 3He droplets: Energy systematics and magic numbers

Ground state properties of ³He clusters are investigated in the framework of the Hartree-Fock theory using an effective interaction of phenomenological type. Binding energies and density distributions are evaluated for clusters up to N = 168 atoms. The analysis yields magic number effects at N = 40 and 70. The method predicts the existence of a minimum number of particles forming a bound droplet around N = 20.     

Mean field and Monte Carlo studies of SU(N) chiral models in three dimensions

SU(N) chiral models defined on three-dimensional cubic lattices arestudied using mean field and Monte Carlo techniques. Mean field theory predicts first-order transitions for all finite N greater than two. The mean field estimates of the transition temperature and discontinuity of the order parameter are in good agreement with computer simulations for N = 3 and 4. The N → ∞ limit of mean field theory has a first-order phase transition.     

The off-shell infrared problem in N = 1 supersymmetric yang-mills theories (II). General massless models

We prove that in a general massless N = I SYM theory off-shell Green functions exist such that Green functions of gauge invariant operators are supersymmetrically covariant. The off-shell infrared problem present in the superfield treatment of these theories is thus shown to remain a gauge artefact. The N = 2, 4 pure SYM theories are covered by this result and thus exist as N = 1 SYM theories.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Auxiliary field-free supersymmetric gauges

We find that, in perturbation theory, non-light-cone axial gauges, N ° A^a (x) = 0, preserve the supersymmetry remaining in N = 1 supersymmetric YM theories, after imposing the Wess-Zumino gauge.     

N = 2, D = 10 non-chiral supergravity and its spontaneous compactification

The non-chiral N = 2, D = 10 supergravity theory is constructed using dimensional reduction from N = 1, D = 11 supergravity. It is shown that this theory may spontaneously compactify, yielding S⁴ × S², CP² × S² and S² × S² × S² spaces for the extra dimensions.