Representations of affine superalgebras and mock theta functions III

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers type modifier is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $mathfrak{g}$ is non-degenerate, to that for the subalgebra $ mathfrak{g}^! $ of $ mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $ mathfrak{g}. $

Representations of affine superalgebras and mock theta functions II

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $widehat{sl}_{2|1}$ can be modified, using Zwegers real analytic corrections, to form an $SL_2(mathbf{Z})$-invariant family of functions. Using a variation of Zwegers correction, we obtain a similar result for $widehat{osp}_{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-frac{2m+2}{M})$, where $m in mathbf{Z}_{geq 0}, M in mathbf{Z}_{geq 2}$, gcd$(2m+2,M)=1$ if $m>0$ (resp. $c=-3frac{2m+1}{M}$, where $m in mathbf{Z}_{geq 0}, M in mathbf{Z}_{geq 2}$ gcd$(4m +2, M) =1)$, whose modified supercharacters form an $SL_2(mathbf{Z})$-invariant family of functions.

Representations of affine superalgebras and mock theta functions

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $hat{sell}_{2|1}$ (resp. $hat{psell}_{2|2}$) can be modified, using Zwegers real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-frac{2m+2}{M})$, where $m in mathbb{Z}_{geq 0}$, $Min mathbb{Z}_{geq 2}$, $gcd(2m+2,M)=1$ if $m>0$ (resp. $6(frac{m}{M}-1)$, where $m in mathbb{Z}_{geq 1}, Min mathbb{Z}_{geq 2}$, $gcd(2m,M)=1$ if $m>1$), whose modified characters and supercharacters form a modular invariant family.

Irreducible Modules over Finite Simple Lie Pseudoalgebras II. Primitive Pseudoalgebras of Type K

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[partial] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work. The present paper is the second in our series on representation theory of simple Lie pseudoalgebras. In the first paper we showed that any finite irreducible module over a simple Lie pseudoalgebra of type W or S is either an irreducible tensor module or the kernel of the differential in a member of the pseudo de Rham complex. In the present paper we establish a similar result for Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by Rumin.

On the Kernel of the affine Dirac operator

Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, sigma an elliptic automorphism of L leaving the form invariant, and A a sigma-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the sigma-twisted Clifford module over A^ associated to the orthocomplement of A in L. Under suitable hypotheses onsigma and A, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L^-module and F, into irreducible A^-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.

On rationality of W-algebras

We study the problem of classification of triples ($mathfrak{g}, f, k$), where $mathfrak{g}$ is a simple Lie algebra, $f$ its nilpotent element and $k in CC$, for which the simple $W$-algebra $W_k (mathfrak{g}, f)$ is rational.

Multiplets of representations, twisted Dirac operators and Vogans conjecture in affine setting

We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogans conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.