level 2,3 characters of affine su(2)

Question

Does anyone know where I can find an explicit formula to compute the level 2 or level 3 characters of affine $su(2)$? I have found several sources that give a formula to compute the level 1 characters in terms of theta functions, but I cannot find anywhere in the literature nice formulas for level 2 or level 3. Thanks

Answer 1

The books Affine Lie algebras, weight multiplicities, and branching rules by Kass, Moody, Patera, and Slansky contain a lot of such explicit computations (I don't have them right now in front of me, so I don't know if this answers your specific question). Also, Volume 2 of that series of two books contains a lot of tables that might be useful for what you're doing.

At any rate, I highly recommend those books: they are very readable and contain a lot of information. That's where I learned that subject.

Answer 2

By "level 2 or level 3 characters of affine $su(2)$" I hope you meant "characters of level 2 or 3 integrable highest weight $A_1^{(1)}$-modules". Also, I assume you don't want principally specialized characters. Those can be easily computed by using the so-called "numerator formula". See this paper for more details in the $A_1^{(1)}$ case. For example, level three (principally specialized) characters are known to be Rogers-Ramanujan series multiplied with something like $\prod_{n=1}^ \infty (1-q^{2n-1})^{-1}$.

Level 2 modules can be realized via fermions, so not surprisingly their characters admit nice form as in the level one case. Here's what you get directly from Weyl-Kac:

Neveu-Schwarz sector:

$$\chi_{L(2 \Lambda_0)} \pm e^{-\delta/2} \chi_{L(2 \Lambda_1)}=e^{2 \Lambda_0} \left( \prod_{n=1}^\infty (1 \pm e^{-\delta(n-1/2)})(1-e^{-n \delta})^{-1} \right) \sum_{n \in \mathbb{Z}} (\pm 1)^n e^{n \alpha_1} e^{-\delta n^2/2}$$

Ramond sector:

$$\chi_{L(\Lambda_0+\Lambda_1)}=e^{\Lambda_0+\Lambda_1} \left( \prod_{n=1}^\infty (1 + e^{-\delta n})(1-e^{-n \delta})^{-1} \right) \sum_{n \in \mathbb{Z}} e^{n \alpha_1} e^{-\delta (n^2+n)/2}$$

I guess you can write down (more complicated) level 3 characters. But if I remember correctly level 3 modules have no known bosonic/fermionic constructions, so I doubt they will look as nice as those of level 1 and 2.

Answer 3

I would suggest "Fock Representations of the Affine Lie Algebra $A^{(1)}_1$ by Wakimoto.

Answer 4

Affine Lie algebra :

Let $\mathfrak{g} = \mathfrak{su}_{2}$, $L\mathfrak{g}$ its loop algebra and $\mathcal{L}\mathfrak{g} = L\mathfrak{g} \oplus \mathbb{C}\mathcal{L}$ its affine Lie algebra (the central extension) :
$$[X^{a}_{n},X^{b}_{m}] = [X^{a},X^{b}]_{m+n} + m\delta_{ab}\delta_{m+n}\mathcal{L}$$ with $(X^{a})$ the basis of $\mathfrak{g}$. The irreducible unitary highest weight representations of $\mathcal{L}\mathfrak{g}$ are $(H_{i}^{\ell})$ with :

• $\mathcal{L} \Omega = \ell \Omega$ with $\ell \in \mathbb{N}$ the level and $\Omega$ the vacuum vector.

• $i \in \frac{1}{2}\mathbb{N}$ and $i \le \frac{\ell}{2}$, the spin (related to the irreducible representation $V_{i}$ of $\mathfrak{g}$)

Character at level $\ell$ and spin $i$ :

Definition : Let $m \in \mathbb{N}^{\star} $, $n \in \mathbb{Z} $, $t, z \in \mathbb{C}$ with $\Vert t \Vert < 1$.
Let the theta functions: $$ \theta_{n,m}(t,z) = \sum_{k \in \frac{n}{2m} + \mathbb{Z} }t^{mk^{2}}z^{mk} $$ Theorem (see [K1], [K2] or [W] p 65) : $$ ch(H_{i}^{\ell}) = \frac{ \theta_{2j+1,\ell + 2}- \theta_{-2j-1,\ell + 2} }{ \theta_{1,2}- \theta_{-1,2}} $$

References :

• [K1] V. G. Kac, Infinite-dimensional Lie algebras, and the Dedekind $\eta$-function. (Russian) Funkcional. Anal. i Prilozen. 8 (1974), no. 1, 77--78 (English translation: Functional Anal. Appl. 8 (1974), 68--70).
• [K2] V. G. Kac, Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990.
• [W] A. J. Wassermann, Kac-Moody and Virasoro algebras, 1998, arXiv:1004.1287 (2010)