Towards the construction of N = 2 supersymmetric integrable hierarchies

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super-W_n second Hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct Hamiltonians we are led to a new definition of super-residues for degenerate N = 2 supersymmetric pseudo-differential operators. We have found a new non-polynomials Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.     

Operator algebra of the N=1 super Wess-Zumino model

The “projective” Ward identities for the super Kac-Moody algebra are solved in order to determine the three- and four-point functions in the N=1 superspace. Linear super-equations are derived for the correlation functions of degenerate operators. These equations are solved for the three- and four-point functions, proving the closure of the operator algebra of the super Wess-Zumino model.     

N = 2 superconformal algebra and integrable O(2) fermionic extensions of the Korteweg-de Vries equation

The N = 2 superconformal algebra is shown to be related to the second hamiltonian structure of three integrable fermionic extensions of the Korteweg-de Vries equation. One of these systems is bi-hamiltonian but not supersymmetric while the reverse is true for the other two.     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.     

Determinant formula for the topological N = 2 superconformal algebra

The Kac determinant for the topological N = 2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing ‘no-label’ singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the Verma modules and singular vectors of the Ramond N = 2 algebra, which have been very poorly studied so far. This work clarifies several misconceptions and confusing claims appeared in the literature about the singular vectors, Verma modules and submodules of the topological N = 2 superconformal algebra.     

The integrable hierarchy constructed from a pair of KdV-type hierarchies and its associatedW algebra

For any two arbitrary positive integers n and m, using them ^th KdV hierarchy and the (n+m)^th KdV hierarchy as building blocks, we are able to construct another integrable hierarchy (referred to as the (n, m)^th KdV hierarchy). TheW-algebra associated to the second Hamiltonian structure of the (n, m)^th KdV hierarchy (calledW(n, m) algebra) is isomorphic via a Miura map to the direct sum of aW _m -algebra, aW _n+m -algebra and an additionalU(1) current algebra. In turn, from the latter, we can always construct a representation of aW -algebra.     

Quantizing the N=2 super sigma-model in two dimensions

A manifestly covariant background field formalism for the N=2 supersymmetric non-linear σ-model is presented. The formalism allows the symmetries of the model to be exploited to the full in the discussion of the ultraviolet divergences in the quantum theory. This proves the cohomological triviality of the metric counterterms at the l⩾2 loop orders. The formalism confirms the finiteness of models with Ricci-flat metrics through the three-loop order. However, it seems unlikely that these cancellations will persist to higher orders. This general analysis is borne out by a study of the supercurrent structure. It is shown that while there is a component axial U(1) current which obeys an Adler-Bardeen theorem, this current is not in the supercurrent multiplet and its existence cannot therefore be used to prove conformal invariance at the quantum level.     

N=2 super Yang-Mills theory in projective superspace

We constructN=2 Yang-Mills theory in projective superspace by exploiting the analogy to Ward's twistor construction of self-dual Yang-Mills fields.Work supported in part by NSF grant No. PHY 85-07627     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

Broken N=2 super Yang-Mills theories with a finite vacuum energy

It is shown that an explicit symmetry breaking of the N=2 ultra-violet finite super Yang-Mills theories through the addition of soft mass terms does not affect the finiteness of the vacuum energy above one loop. One-loop vacuum energy is finite provided the mass terms satisfy the sum rules Σ_J(2J+1)(−1)^2Jm_J^2K = 0 (K = 0, 1, 2).     

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra ?_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra ?_2 into loop algebra ?_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.     

Discrete integrable couplings associated with Toda-type lattice and two hierarchies of discrete soliton equations

With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations.     

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with sl(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.