Boundary algebras and Kac modules for logarithmic minimal models

Abstract

Virasoro Kac modules were initially introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.

The representation theory of seam algebras

“…Indeed the use of the elements id = P k , e j = P k E j P k for 1 ≤ j < n and e n = [k] q P k E n P k of TL n+k (β ) to define b n,k (β ) adds a new relation that is not satisfied by the elements of the corresponding blob algebra. As noted in [1], the discrepancy is seen most easily in the case k = 1. Indeed the generators of b n,1 ≃ TL n+1 satisfy the additional relation e n e n−1 e n = e n , but those of the blob algebra do not.…”

“…With this notation, the last identity reads where id = P k ∈ TL n+k and e j = P k E j P k for 1 ≤ j < n and e n = [k] q P k E n P k 1 . (The content of the present section follows that of [1].) Clearly this subset is closed under addition and multiplication.…”

“…(The choice of word comes from the physical interest for boundary seam algebras and will not concern us.) Due to the fact that P k might not be defined, the range of the two integers n, k and of the complex number q (with β = q + q −1 ) will 1 To ease readability, we shall use capital letters for generators and elements of TL n+k and small ones for those of b n,k . be restricted as follows:…”

“…Isomorphisms between the two presentations (diagrammatic, and through generators and relations) are explicited in [15] for TL n and in [1] for b n,k . For the family of b n,k 's, the defining relations (2.7) and (2.8) allow one to enlarge the domain of the parameters (2.5).…”

“…The properties they share are listed without proof. (See [1,16] and references therein.) First, like P n , the Wenzl-Jones projector P k is an idempotent.…”