Abstract
AbstractThe λ = 0 ’t Hooft limit of the 2d$ {\mathcal{W}_N} $minimal models is shown to be equivalent to the singlet sector of a free boson theory, thus paralleling exactly the structure of the free theory in the Klebanov-Polyakov proposal. In 2d, the singlet sector does not describe a consistent theory by itself since the corresponding partition function is not modular invariant. However, it can be interpreted as the untwisted sector of a continuous orbifold, and this point of view suggests that it can be made consistent by adding in the appropriate twisted sectors. We show that these twisted sectors account for the ‘light states’ that were not included in the original ’t Hooft limit. We also show that, for the Virasoro minimal models (N = 2), the twisted sector of our orbifold agrees precisely with the limit theory of Runkel & Watts. In particular, this implies that our construction satisfies crossing symmetry.
Highlights
On AdS4 is dual to the singlet sector of the 3d O(N ) vector model in the large N limit
The λ = 0 ’t Hooft limit of the 2d WN minimal models is shown to be equivalent to the singlet sector of a free boson theory, paralleling exactly the structure of the free theory in the Klebanov-Polyakov proposal
In 2d, the singlet sector does not describe a consistent theory by itself since the corresponding partition function is not modular invariant. It can be interpreted as the untwisted sector of a continuous orbifold, and this point of view suggests that it can be made consistent by adding in the appropriate twisted sectors
Summary
The minimal models we are interested in are the WN coset models su(N )k ⊕ su(N ) su(N )k+1. In order to understand the resulting representations in detail, it is convenient to describe the coset theory in terms of a Drinfeld-Sokolov (DS) reduction. From this perspective, the representations of the coset theory are labelled by (see for example [24] for an introduction to these matters). In the limit k → ∞, the coset representation (Λ+; Λ−) only depends on (Λ+ − Λ−); for example, for N = 2, this is just the familiar statement that, as k → ∞, h(r; s). Where the sum runs over all representations of su(N ), and Λ∗ is the conjugate representation to Λ
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