# Concise mathematical definition of the fusion product on the Verlinde ring?

The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the loop of $G$ with the 'positive energy' condition. I'm trying to find a concise mathematical definition of the product on this ring, the so-called fusion product. In Freed-Hopkins-Teleman's Loop groups and twisted K-theory III they define an $R(G)$-module structure on this additive group using induction of representations (hence a functorial description).

All references I'm reading just mention 'fusion rules', and cite Verlinde, whose 1988 paper is in the journal Nuclear Physics B. Surely we have a more recent discussion along the lines of the FHT construction mentioned above, and not in terms of linear combinations of coefficients of some irreps considered as generators...?

The fusion product arises from the braided monoidal structure on the modular tensor category of [adjectives] loop group representations at level $k$ itself. This is the value $Z(S^1)$ of Chern-Simons theory with gauge group $G$ and level $k$ on the circle, and accordingly its braided monoidal structure comes from thinking about what Chern-Simons assigns to a pair of pants / a sphere punctured at three points / a disk with two disks inside it, namely a "bilinear functor"

$$\boxtimes : Z(S^1) \times Z(S^1) \to Z(S^1)$$

which, in terms of the "basis" of $Z(S^1)$ given by the irreducible representations at level $k$, is completely specified by giving a vector space for each triple $(V_i, V_j, V_k)$ of irreducible representations, namely $\text{Hom}(V_i \boxtimes V_j, V_k)$. The dimensions of these vector spaces give the "fusion rules" in the Verlinde ring, and can be thought of as being obtained by inserting the representations $V_i, V_j, V_k^{\ast}$ at the three punctures of a sphere punctured at three points (or something like that).

There are as many ways to describe $\boxtimes$ as there are ways to describe what Chern-Simons assigns to surfaces with boundary: geometric quantization, quantum groups, conformal nets...

If you just want to pass directly to the Verlinde ring then dimensionally reduce everything I said on $S^1$, by which I mean consider $Z(S^1 \times S^1)$ and $S^1$ times a pair of pants, etc.

• "...along the lines of the FHT construction mentioned above" is perhaps the part of my question I should have emphasised much more strongly :-) – David Roberts Feb 17 '15 at 17:17
• @David: isn't this given in Loop groups and twisted K-theory I? – Qiaochu Yuan Feb 17 '15 at 17:19
• Not that I can see - they write "Using the Pontryagin product, we are able to define a fusion product on Rτ(LG) for any G at any primitive level τ. We do not, however, give a construction of this product in terms of representation theory." – David Roberts Feb 17 '15 at 20:15
• @David: if it's not in one of the FHT papers then I think it's an open question to give a description you'd be happy with. (Incidentally, I think the title of this question is misleading; your specifications certainly seem more specific than just "concise mathematical definition.") – Qiaochu Yuan Feb 17 '15 at 21:09
• @David: I'm not familiar with the literature on Chern-Simons, but I'm sure you can get a description of the fusion product in papers on loop group conformal nets, for example. – Qiaochu Yuan Feb 18 '15 at 9:05

For any triple $(M_0,M_1,M_\infty)$ of positive energy representations of $\widehat{LG}$ at a fixed level, restriction to small punctured discs produces a canonical (up to some scaling which doesn't matter here) action of the Lie algebra $\mathfrak{g}_X = \Gamma(X, \mathfrak{g} \otimes \mathscr{O}_X)$, where $X$ is the thrice punctured line $\mathbb{P}^1 \setminus \{0,1,\infty\}$. The multiplicity of $M_\infty^\vee$ in $M_0 \boxtimes M_1$ (i.e., the fusion rule) is given by the dimension of the space of coinvariants. Since you are only asking about dimension, you get the same answer from the dual space $\mathrm{Hom}_{\mathfrak{g}_X}(M_0 \otimes M_1 \otimes M_\infty, \mathbb{C})$ of conformal blocks. Looijenga has a brief treatment of the theory.

• "Since you are only asking about dimension" <- do you mean this in the abstract 'you', or me in particular? I'm not asking about dimension, if I inadvertently gave that impression. – David Roberts Feb 19 '15 at 7:04
• @DavidRoberts Fusion rules are defined as dimensions of certain vector spaces, so any procedure that produces that number gives you a definition of fusion rule. Could you say a bit more about what sort of answer you are seeking? – S. Carnahan Feb 19 '15 at 7:21
• I'm not interested in computations in the Verlinde ring, rather a conceptual--possibly even functorial--description of the fusion product on positive energy representations (in hindsight, this is a better way to say what I would like). – David Roberts Feb 19 '15 at 7:53