Ramond sector of superconformal algebras via quantum reduction

Quantum hamiltonian reduction of affine superalgebras is studied in the twisted case. The Ramond sector of "minimal" superconformal W-algebras is described in detail, the determinant formula is obtained. Extensive list of examples includes all the simple Lie superalgebras of rank up to 2. The paper generalizes the results of Kac and Wakimoto (math-ph/0304011) to the twisted case.Comment: 50 pages, 8 figures; v2: examples added, determinant formula derivation modified, section order change

q,k-generalized gamma and beta functions

We introduce the q,k-generalized Pochhammer symbol. We construct $\Gamma_{q,k}$ and $B_{q,k}$, the q,k-generalized gamma and beta fuctions, and show that they satisfy properties that generalize those satisfied by the classical gamma and beta functions. Moreover, we provide integral representations for $\Gamma_{q,k}$ and $B_{q,k}.$Comment: 17 page

Vacuum structure in supersymmetric Yang-Mills theories with any gauge group

We consider the pure supersymmetric Yang--Mills theories placed on a small 3-dimensional spatial torus with higher orthogonal and exceptional gauge groups. The problem of constructing the quantum vacuum states is reduced to a pure mathematical problem of classifying the flat connections on 3-torus. The latter problem is equivalent to the problem of classification of commuting triples of elements in a connected simply connected compact Lie group which is solved in this paper. In particular, we show that for higher orthogonal SO(N), N > 6, and for all exceptional groups the moduli space of flat connections involves several distinct connected components. The total number of vacuumstates is given in all cases by the dual Coxeter number of the group which agrees with the result obtained earlier with the instanton technique.Comment: 41 pages, 9 figures, 9 tables. Final version to be published in the Yuri Golfand memorial volume. We added the Appendix D with classification of all non-trivial commuting n-tuples for arbitrary

News from the Virasoro algebra

It is shown that the local quantum field theory of the chiral energy- momentum tensor with central charge $c=1$ coincides with the gauge invariant subtheory of the chiral $SU(2)$ current algebra at level 1, where the gauge group is the global $SU(2)$ symmetry. At higher level, the same scheme gives rise to $W$-algebra extensions of the Virasoro algebra.Comment: 4 pages, Latex, DESY 93-11

Highest weight irreducible representations of the Lie superalgebra gl(1/∞)gl(1/\infty)

Two classes of irreducible highest weight modules of the general linear Lie superalgebra $gl(1/\infty)$ are constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down.Comment: 24 pages, TeX; Journ. Math. Phys. (to be published

Higher level twisted Zhu algebras

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper we consider the general set-up of a vertex algebra $V$, graded by \G/\Z for some subgroup \G of $\R$ containing $\Z$, and with a Hamiltonian operator $H$ having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level $p$ Zhu algebras \zhu_{p, \G}(V), and we prove the following theorems: For each $p$ there is a bijection between the irreducible \zhu_{p, \G}(V)-modules and the irreducible \G-twisted positive energy $V$-modules, and $V$ is (\G, H)-rational if and only if all its Zhu algebras \zhu_{p, \G}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for $H$. We provide an explicit description of the level $p$ Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \vir^c and the universal affine Kac-Moody vertex algebra V^k(\g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.Comment: 47 pages, no figure

Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras

We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.Comment: 24 page

Irreducible modules over finite simple Lie conformal superalgebras of type K

We construct all finite irreducible modules over Lie conformal superalgebras of type KComment: Accepted for publication in J. Math. Phys