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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

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Have you looked into the papers by Jim Lepowsky and Robert Wilson in Invent. Math.? For references like this it's useful to consult MathSciNet if you have access, since some of the reviews there are also helpful. The literature seems to go about as far as levels 2 and 3, though you may not find all the "formulas" made explicit enough. –  Jim Humphreys Apr 17 '12 at 23:29
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4 Answers 4

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.

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Thank you for the suggestion. –  Kevin Wray Apr 17 '12 at 17:14
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I would suggest "Fock Representations of the Affine Lie Algebra $A^{(1)}_1$ by Wakimoto.

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Thank you for the reference, I'm going to check now. –  Kevin Wray Apr 17 '12 at 17:12
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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.

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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)
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