Representations of the Nappi--Witten vertex operator algebra
TL;DR: The Nappi-Witten model is a Wess-Zumino Witten model in which the target space is the nonreductive Heisenberg group $H_4.
Abstract: The Nappi-Witten model is a Wess-Zumino-Witten model in which the target space is the nonreductive Heisenberg group $H_4$. We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $\mathsf{H}_4$. In particular, we classify the irreducible $\mathsf{H}_4$-modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic conformal field theory.
Citations
More filters
264 citations
Posted Content•
TL;DR: In this article, the authors used the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level minimal models.
Abstract: The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\widehat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.
3 citations
TL;DR: In this paper , the authors used the standard module formalism to classify the irreducible weight modules of the minimal model vertex operator algebra when the level of the weight module is admissible.
Abstract: The first part of this work uses the algorithm recently detailed in Kawasetsu and Ridout (Commun Contemp Math 24:2150037, 2022. arXiv:1906.02935 [math.RT]) to classify the irreducible weight modules of the minimal model vertex operator algebra $${\textsf {L} }_{{\textsf {k} }}(\mathfrak {sl}_{3})$$ , when the level $${\textsf {k} }$$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $$\widehat{\mathfrak {sl}}_{3}$$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family’s parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in Kawasetsu (Adv Math 393:108079, 2021. arXiv:2003.10148 [math.RT]), these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $$\mathfrak {sl}_{3}$$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $${\textsf {k} }=-\frac{3}{2}$$ . This gives the first nontrivial test of this formalism for a nonrational vertex operator algebra of rank greater than 1 and confirms the expectation that the methodology developed here will apply in much greater generality.
2 citations
TL;DR: In this paper , the authors used the standard module formalism to classify the irreducible weight modules of the minimal model vertex operator algebra Lk(sl3), when the level k is admissible.
Abstract: The first part of this work uses the algorithm recently detailed in Kawasetsu and Ridout (Commun Contemp Math 24:2150037, 2022. arXiv:1906.02935 [math.RT]) to classify the irreducible weight modules of the minimal model vertex operator algebra Lk(sl3), when the level k is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms ofsl3 on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family’s parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in Kawasetsu (Adv Math 393:108079, 2021. arXiv:2003.10148 [math.RT]), these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level sl3 minimal models. The second part of this work applies the standard module formalism to compute these explicitly when k = − 3 2 . This gives the first nontrivial test of this formalism for a nonrational vertex operator algebra of rank greater than 1 and confirms the expectation that the methodology developed here will apply in much greater generality.
1 citations
References
More filters
Book•
01 Jan 1983TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
Abstract: Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper
4,653 citations
"Representations of the Nappi--Witte..." refers background in this paper
...This identification not only algebraically formalises the conformal symmetry of these models, it also means that the representation theory of the corresponding affine Kac–Moody algebras [2] is available to organise the spectrum....
[...]
...C([2],2)+C([2,1],1) = 0 and C([1,1],2)+ 3C([1,1,1],1) = 0....
[...]
...12) C(∅,4)+ 3C([3],1) = 0, C([1],3)+C([2],2)+ 3C([2,1],1)= 0 and C([1,1],2)+ 3C([1,1,1],1) = 0....
[...]
...Finally, there are two ways to remove a part from [2,1], hence the second equation of (A....
[...]
...U =C(∅,4)E−4+C([1],3)I−1E−3+C([2],2)I−2E−2+C([1,1],2)I 2 −1E−2 (A....
[...]
TL;DR: In this article, a non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented, which is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
Abstract: A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.
2,231 citations
"Representations of the Nappi--Witte..." refers background in this paper
...Wess–Zumino–Witten models are examples of nonlinear sigma models that describe (noncritical) strings propagating on Lie groups or supergroups [1]....
[...]
...11b) C(∅,4)+ 2C([2],2) = 0 and C([1],3)+ 2C([2,1],1) = 0...
[...]
...+C([3],1)I−3E−1 +C([2,1],1)I−2I−1E−1 +C([1,1,1],1)I 3 −1E−1....
[...]
...U =C(∅,4)E−4 +C([1],3)I−1E−3 +C([2],2)I−2E−2 +C([1,1],2)I 2 −1E−2 (A....
[...]
Book•
31 Dec 1968
TL;DR: In this paper, a standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special function are related to (and derived from) simple well-known facts of representation theory.
Abstract: A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple well-known facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group $SU(2)$, and the hypergeometric function and representations of the group $SL(2,R)$, as well as many other classes of special functions.
2,064 citations
Book•
02 Oct 2012
TL;DR: In this paper, the notion of vertex operator algebra was introduced and a formal series and the formal delta function were derived from the axiomatic definition of a vertex operator and its application in the formal calculus.
Abstract: 1 Introduction.- 1.1 Motivation.- 1.2 Example of a vertex operator.- 1.3 The notion of vertex operator algebra.- 1.4 Simplification of the definition.- 1.5 Representations and modules.- 1.6 Construction of families of examples.- 1.7 Some further developments.- 2 Formal Calculus.- 2.1 Formal series and the formal delta function.- 2.2 Derivations and the formal Taylor Theorem.- 2.3 Expansions of zero and applications.- 3 Vertex Operator Algebras: The Axiomatic Basics.- 3.1 Definitions and some fundamental properties.- 3.2 Commutativity properties.- 3.3 Associativity properties.- 3.4 The Jacobi identity from commutativity and associativity.- 3.5 The Jacobi identity from commutativity.- 3.6 The Jacobi identity from skew symmetry and associativity.- 3.7 S3-symmetry of the Jacobi identity.- 3.8 The iterate formula and normal-ordered products.- 3.9 Further elementary notions.- 3.10 Weak nilpotence and nilpotence.- 3.11 Centralizers and the center.- 3.12 Direct product and tensor product vertex algebras.- 4 Modules.- 4.1 Definition and some consequences.- 4.2 Commutativity properties.- 4.3 Associativity properties.- 4.4 The Jacobi identity as a consequence of associativity and commutativity properties.- 4.5 Further elementary notions.- 4.6 Tensor product modules for tensor product vertex algebras.- 4.7 Vacuum-like vectors.- 4.8 Adjoining a module to a vertex algebra.- 5 Representations of Vertex Algebras and the Construction of Vertex Algebras and Modules.- 5.1 Weak vertex operators.- 5.2 The action of weak vertex operators on the space of weak vertex operators.- 5.3 The canonical weak vertex algebra ?(W) and the equivalence between modules and representations.- 5.4 Subalgebras of ?(W).- 5.5 Local subalgebras and vertex subalgebras of ?(W).- 5.6 Vertex subalgebras of ?(W) associated with the Virasoro algebra.- 5.7 General construction theorems for vertex algebras and modules.- 6 Construction of Families of Vertex Operator Algebras and Modules.- 6.1 Vertex operator algebras and modules associated to the Virasoro algebra.- 6.2 Vertex operator algebras and modules associated to affine Lie algebras.- 6.3 Vertex operator algebras and modules associated to Heisenberg algebras.- 6.4 Vertex operator algebras and modules associated to even lattices-the setting.- 6.5 Vertex operator algebras and modules associated to even lattices-the main results.- 6.6 Classification of the irreducible L?(?, O)-modules for g finite-dimensional simple and ? a positive integer.- References.
892 citations
"Representations of the Nappi--Witte..." refers background in this paper
...This parabolic Verma module carries the structure of a vertex algebra given by the standard affine state-field correspondence [35]....
[...]
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