Integrable Highest Weight Modules over Affine Superalgebras and Number Theory

Abstract

The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an "irrefutable proof" of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.

Free fermion resolution of supergroup WZNW models

“…We expect that this observation extends to more general current superalgebras and that it will be helpful in the study of modular transformations. Representations of affine Lie superalgebras and their behaviour under modular transformations have also been studied in [64,65,13].…”

Some characters of Kac and Wakimoto and nonholomorphic modular functions

“…Their 1994 paper [3] shed new light on the representation of integers as sums of squares, and it also inspired subsequent works in the subject by Milne, the second author, and Zagier [5,6,7,10]. In the same paper, Kac and Wakimoto also noticed that a q-series related to Ramanujan's mock theta functions is the denominator identity for the affine superalgebra s (2, 1) ∧ .…”

“…In an important series of papers [3,4,9], Kac and Wakimoto discovered deep connections between the representation theory of affine Lie superalgebras and number theory. Their 1994 paper [3] shed new light on the representation of integers as sums of squares, and it also inspired subsequent works in the subject by Milne, the second author, and Zagier [5,6,7,10].…”

On a class of tensor product representations for the orthosymplectic superalgebra

“…The highest weight is different with respect to this choice of positive roots, µ, and we denote the representation also by K m|2n µ . Calculating µ from Λ and vice versa can be done using the technique of odd reflections from [13,20]. For the cases C(n) = osp(2|2n − 2) and B(0|n) = osp(1|2n) this is not necessary, as explained in Section 2.…”

“…We calculate the highest weight of the representations in the standard choice of positive roots, again using the method from [13,20] explained in Section 4. Since…”