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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...?

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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.

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  • $\begingroup$ "...along the lines of the FHT construction mentioned above" is perhaps the part of my question I should have emphasised much more strongly :-) $\endgroup$ – David Roberts Feb 17 '15 at 17:17
  • $\begingroup$ @David: isn't this given in Loop groups and twisted K-theory I? $\endgroup$ – Qiaochu Yuan Feb 17 '15 at 17:19
  • $\begingroup$ 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." $\endgroup$ – David Roberts Feb 17 '15 at 20:15
  • $\begingroup$ @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.") $\endgroup$ – Qiaochu Yuan Feb 17 '15 at 21:09
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    $\begingroup$ @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. $\endgroup$ – Qiaochu Yuan Feb 18 '15 at 9:05
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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.

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    $\begingroup$ "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. $\endgroup$ – David Roberts Feb 19 '15 at 7:04
  • $\begingroup$ @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? $\endgroup$ – S. Carnahan Feb 19 '15 at 7:21
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    $\begingroup$ 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). $\endgroup$ – David Roberts Feb 19 '15 at 7:53

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