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.     

Ten-dimensional supersymmetric Yang-Mills theory in terms of four-dimensional superfields

We show that the usual formulation of the N = 4 supersymmetric Yang-Mills theory in terms of N = 1 superfields can be generalized to describe the full ten-dimensional theory.     

N = 4 Yang-Mills theory on the light cone

The 10-dimensional supersymmetric Yang-Mills theory is constructed in the light-cone gauge. When the theory is dimensionally reduced to four dimensions it is shown that the corresponding N = 4 theory is conveniently described in terms of a scalar superfield. This formalism avoids the problem of auxiliary fields but is Lorentz invariant only on the mass shell. Similar formalisms in terms of scalar superfields are also sketched for the other supersymmetric Yang-Mills as well as for N = 8 supergravity.     

A six-dimensional superspace approach to extended superfields

We present an improved formalism for treating six-dimensional superspace and apply it to writing linearized N = 2 Yang-Mills theory and the N = 2 relaxed hypermultiplet in SU(2) covariant forms that are simpler than the known four-dimensional forms.     

Quantum free Yang-Mills on the plane

We construct a free-probability quantum Yang-Mills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that this theory is the N=∞ limit of U(N) quantum Yang-Mills theory on the plane. Our model provides an example of a stochastic geometry, motivated by quantum field theory, based on free probability theory.     

The property of maximal transcendentality for the universal anomalous dimension in the N = 4 SYM

We show the results for the universal anomalous dimension γ _uni(j) of Wilson twist-2 operators in the N = 4 Supersymmetric Yang-Mills theory.     

Schwinger-Dyson equations for <Emphasis Type="Italic">N</Emphasis> = 1 supersymmetric Yang-Mills theory

The Schwinger-Dyson equations for the N = 1 supersymmetric Yang-Mills theory are constructed.     

N = 2 supersymmetric Yang-Mills theory and its supercurrent

Using the off-shell two-component formulation of N = 2 supersymmetric Yang-Mills theory and the gauge covariantized supersymmetry transformations for the fields, the non-abelian supercurrent and anomaly multiplets are developed and the transformation rules for their components are obtained. The superfields in N = 2 superspace whose components are the non-abelian supercurrent and anomaly multiplets are written down.     

Background field formalism and construction of effective action for N = 2, d = 3 supersymmetric gauge theories

We review the background field method for three-dimensional Yang-Mills and Chern-Simons models in N = 2 superspace. Superfield proper time (heat kernel) techniques are developed and exact expressions of heat kernels for constant backgrounds are presented. The background field method and heat kernel techniques are applied for evaluating the low-energy effective actions in N = 2 supersymmetric Yang-Mills and Chern-Simons models as well as in N = 4 and N = 8 SYM theories.     

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.     

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.     

Supergraphity: (II). Manifestly covariant rules and higher-loop finiteness

We continue our discussion of the background field formalism in supersymmetric theories, deriving new covariant Feynman rules for chiral superfields. As a result, we obtain improved power-counting rules for both simple and extended supersymmetry which can be used to make the following statements: If the corresponding extended superfield formalism exist, (a) N=2 supersymmetric Yang-Mills theory is finite beyond one loop, (b) N=4 Yang-Mills is finite at all loops, and (c) N=8 supergravity is finite through six loops. We also find that in simple super-Yang-Mills the radiative corrections to the Fayet-Iliopoulos (“D”) term, which are known to vanish for higher loops, also vanish automatically at one loop for arbitrary couplings.     

Flat coordinates,topological Landau-Ginzburg models and the Seiberg-Witten period integrals

We study the Picard-Fuchs differential equations for the Seiberg-Witten period integrals in N = 2 supersymmetric Yang-Mills theory. For A-D-E gauge groups we derive the Picard-Fuchs equations by using the flat coordinates in the A-D-E singularity theory. We then find that these are equivalent to the Gauss-Manin system for two-dimensional A-D-E topological Landau-Ginzburg models and the scaling relation for the Seiberg-Witten differential. This suggests an interesting relationship between four-dimensional N = 2 gauge theories in the Coulomb branch and two-dimensional topological field theories.     

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the N=4{\mathcal N=4} supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure N=2{\mathcal N=2} and the N=2^*{\mathcal N=2^*} supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional N=2{\mathcal N=2} superconformal gauge theory is treated similarly.     

Superspace gauge theories with central charges

The superspace constraints leading to off-shell formulations of extended supersymmetric Yang-Mills theories for N = 2 and 4 are generalised to N = 6 in the abelian case. This leads to an on-shell representation of linearized N = 6 supergravity.     

Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories

We study the existence of monopole bound states saturating the BPS bound in N = 2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an L² index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N = 2 Yang-Mills theories.     

The property of maximal transcendentality: Calculation of anomalous dimensions in the N = 4 SYM and master integrals

We review results for the universal anomalous dimension γ _uni (j) of Wilson twist-2 operators in the N = 4 Supersymmetric Yang-Mills theory having the property of maximal transcendentality. It is shown that the similar property is ibserved in the results for master integrals.     

Braves and the dynamics of QCD

A brane configuration is described that is relevant to understanding the dynamics of N = 1 supersymmetric Yang-Mills theory. Confinement and spontaneous breaking of a discrete chiral symmetry can be understood as consequences of the topology of the brane. Because of the symmetry breaking, there can be domain walls separating different vacua; the QCD string can end on such a domain wall. The model in which these properties can be understood semiclassically does not coincide with supersymmetric Yang-Mills theory but is evidently in the same universality class.     

A new derivation of the Picard-Fuchs equations for effective N = 2 super Yang-Mills theories

A new method to obtain the Picard-Fuchs equations of effective, N = 2 supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. It applies to all classical gauge groups, and directly produces a decoupled set of second-order, partial differential equations satisfied by the period integrals of the Seiberg-Witten differential along the 1-cycles of the algebraic curves describing the vacuum structure of the corresponding N = 2 theory.     

Operators with Large R-Charge in N=4 Yang-Mills Theory

It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong-weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant λ and have a well-defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions. In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two-loop level that the anomalous dimension has a finite limit when the R-charge J→∞ keeps λ/J² finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model.