Fusion multiplicities as polytope volumes: -point and higher-genus su(2) fusion

Abstract

We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N > 3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide.

𝒩-point and higher-genus osp(1|2) fusion

“…Most results so far pertain to three-point fusion, 16,17 but also higher-genus and higher-point su͑2͒ fusions have been discussed. 9 Below we shall extend the latter results to osp͑1͉2͒.…”

“…That independence is novel compared to the similar situation for su͑2͒. 9 The basis loop-gluing super-diagram associated to ͑47͒ is 0 0 0 1. ͑51͒…”

“…4 In the case of su͑2͒ the BZ triangle is trivial but has led to characterizations of higherpoint and higher-genus couplings and fusions as discretized volumes of certain polytopes. 9 Here we shall discuss the generalization to osp͑1͉2͒.…”

Affine su(3) and su(4) fusion multiplicities as polytope volumes

“…Thus, those efforts are combining the work presented here with our recent results on higher-point and higher-genus su(2) fusion [5], and our general description of higher-point su(r + 1) tensor product multiplicities [4].…”

“…Its discretised volume (i.e., the number of integer points enclosed by it) is the tensor product multiplicity [3]. This idea was generalised to higher-point couplings in [4], and to affine su(2) fusions in [5] (that work also describes the extension to higher-point fusions and higher genus). The objective of the present work is the extension to affine su(3) and su(4) fusions.…”