Relaxed Highest-Weight Modules I: Rank 1 Cases
04 Oct 2019-Communications in Mathematical Physics (Springer Berlin Heidelberg)-Vol. 368, Iss: 2, pp 627-663
TL;DR: In this paper, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to $${\mathfrak{s}\math frak{l}_2}
Abstract: Relaxed highest-weight modules play a central role in the study of many important vertex operator (super)algebras and their associated (logarithmic) conformal field theories, including the admissible-level affine models. Indeed, their structure and their (super)characters together form the crucial input data for the standard module formalism that describes the modular transformations and Grothendieck fusion rules of such theories. In this article, character formulae are proved for relaxed highest-weight modules over the simple admissible-level affine vertex operator superalgebras associated to $${\mathfrak{s}\mathfrak{l}_2}$$
and $${\mathfrak{osp} (1 \vert 2)}$$
. Moreover, the structures of these modules are specified completely. This proves several conjectural statements in the literature for $${\mathfrak{s}\mathfrak{l}_2}$$
, at arbitrary admissible levels, and for $${\mathfrak{osp} (1 \vert 2)}$$
at level $${-\frac{5}{4}}$$
. For other admissible levels, the $${\mathfrak{osp}(1 \vert 2)}$$
results are believed to be new.
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TL;DR: In this paper, the Verlinde formula for fusion rules in the Weyl vertex algebra was constructed, and a result that relates irreducible weight modules for the Wey vertex algebra to the affine Lie superalgebra gl(1|1)^ was given.
Abstract: In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way, we confirm the conjecture on fusion rules based on the Verlinde formula. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra gl(1|1)^.
43 citations
TL;DR: In this article, the embeddings of simple admissible affine vertex algebras were studied and a family of weight, logarithmic, and Whittaker modules were constructed.
Abstract: We study the embeddings of the simple admissible affine vertex algebras $${V_k(sl(2))}$$
and $${V_k({osp}(1,2))}$$
, $${k
otin {\mathbb Z}_{\ge 0}}$$
, into the tensor product of rational Virasoro and N = 1 Neveu–Schwarz vertex algebra with lattice vertex algebras. By using these realizations we construct a family of weight, logarithmic, and Whittaker $${\widehat{sl(2)}}$$
and $${\widehat{osp(1,2)}}$$
-modules. As an application, we construct all irreducible degenerate Whittaker modules for $${V_k(sl(2))}$$
.
29 citations
TL;DR: In this article, it was shown that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are C_1 -cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite-length generalized Vmodules has a braided tensor category structure.
Abstract: We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$
-cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$
at level k and the category $$KL_k(\mathfrak {g})$$
of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$
at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$
has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$
, including the category $$KL_{-1}(\mathfrak {sl}_n)$$
. Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$
consisting of objects with semisimple Cartan subalgebra actions.
28 citations
TL;DR: In this article, the fusion rules in the category of weight modules for the Weyl vertex algebra are described as the dimension of the vector space of intertwining operators between three irreducible modules.
Abstract: In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.
24 citations
Posted Content•
TL;DR: In this paper, the authors considered the problem of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank, and showed that this can be reduced to the classification of highest weight modules by generalising Olivier Mathieu's theory of coherent families.
Abstract: This is the second of a series of articles devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first studied the simple "rank-$1$" affine vertex superalgebras $L_k(\mathfrak{sl}_2)$ and $L_k(\mathfrak{osp}(1\vert2))$, with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalising Olivier Mathieu's theory of coherent families. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category $\mathscr{O}$ in one of these examples.
22 citations
Cites background from "Relaxed Highest-Weight Modules I: R..."
...More precisely, the simple relaxed highest-weight modules are the simple objects of a relaxed category R, see [1, 5] for the definition, that naturally generalises the well-known Bernšteı̆n– Gel’fand–Gel’fand category O ....
[...]
...The first [1] studied the simple “rank-1” affine vertex superalgebras Lk(sl2) and Lk(osp(1 |2)), with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones)....
[...]
...The next instalment of this series will address the important problem of computing the character of a general relaxed highest-weight module, thus generalising the rank-1 results of [1]....
[...]
References
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TL;DR: In this paper, the authors studied conformal field theories with a finite number of primary fields with respect to some chiral algebra and showed that the fusion rules are completely determined by the behavior of the characters under the modular group.
1,506 citations
TL;DR: In this article, it was shown that the characters of the integrable highest weight modules of affine Lie algebras and the minimal series of the Virasoro algebra give rise to conformal field theories.
Abstract: In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.
1,122 citations
TL;DR: In this article, the authors studied the spectrum of bosonic string theory on AdS3 and studied classical solutions of the SL(2,R) WZW model, including solutions for long strings with nonzero winding number.
Abstract: In this paper we study the spectrum of bosonic string theory on AdS3 We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with nonzero winding number We show that the model has a symmetry relating string configurations with different winding numbers We then study the Hilbert space of the WZW model, including all states related by the above symmetry This leads to a precise description of long strings We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string
596 citations
TL;DR: In this paper, the authors studied the spectrum of bosonic string theory on AdS_3 and showed that the model has a symmetry relating string configurations with different winding numbers, which leads to a precise description of long strings.
Abstract: In this paper we study the spectrum of bosonic string theory on AdS_3. We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with non-zero winding number. We show that the model has a symmetry relating string configurations with different winding numbers. We then study the Hilbert space of the WZW model, including all states related by the above symmetry. This leads to a precise description of long strings. We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string.
498 citations
TL;DR: It is shown that the modular invariant representations of the Virasoro algebra Vir are precisely the "minimal series" of Belavin et al.
Abstract: In this paper, we launch a program to describe and classify modular invariant representations of infinite-dimensional Lie algebras and superalgebras. We prove a character formula for a large class of highest weight representations L(λ) of a Kac-Moody algebra [unk] with a symmetrizable Cartan matrix, generalizing the Weyl-Kac character formula [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70]. In the case of an affine [unk], this class includes modular invariant representations of arbitrary rational level m = t/u, where t [unk] Z and u [unk] N are relatively prime and m + g ≥ g/u (g is the dual Coxeter number). We write the characters of these representations in terms of theta functions and calculate their asymptotics, generalizing the results of Kac and Peterson [Kac, V. G. & Peterson, D. H. (1984) Adv. Math. 53, 125-264] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1988) Adv. Math. 70, 156-234] for the u = 1 (integrable) case. We work out in detail the case [unk] = A1(1), in particular classifying all its modular invariant representations. Furthermore, we show that the modular invariant representations of the Virasoro algebra Vir are precisely the “minimal series” of Belavin et al. [Belavin, A. A., Polyakov, A. M. & Zamolodchikov, A. B. (1984) Nucl. Phys. B 241, 333-380] using the character formulas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1984) Lect. Notes Math. 1060, 230-245]. We show that tensoring the basic representation and modular invariant representations of A1(1) produces all modular invariant representations of Vir generalizing the results of Goddard et al. [Goddard P., Kent, A. & Olive, D. (1986) Commun. Math. Phys. 103, 105-119] and of Kac and Wakimoto [Kac, V. G. & Wakimoto, M. (1986) Lect. Notes Phys. 261, 345-371] in the unitary case. We study the general branching functions as well. All these results are generalized to the Kac-Moody superalgebras introduced by Kac [Kac, V. G. (1978) Adv. Math. 30, 85-136] and to N = 1 super Virasoro algebras. We work out in detail the case of the superalgebra B(0, 1)(1), showing, in particular, that restricting to its even part produces again all modular invariant representations of Vir. These results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations.
318 citations