Extended $W$ algebra of $sl_{2}$ type at positive rational level and its representations (String theory, integrable systems and representation theory)

"String theory, integrable systems and representation theory". July 30～August 2, 2013. edited by Koji Hasegawa and Yasuhiko Yamada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This paper is a Japanese expository version of my joint paper with S.Wood [TW]. In occasion to write a Japanese version paper of [TW], I explained the fundamental concept of conformal field theories and description of vertex operator algebras by using language of conformal field theories. I hope this will help researchers who are not familiar with conformal field theories. Now I explain the contents of this paper. Capter 2. Energy momentum tensors and field operators. In the first part of this paper, I explained relationship between the representation theory of Virasoro algebra and the energy momentum tensor. In the second part, I explained the concepts of field operators, the mutually locality of two field operators, and operator product expantions. Finally I explained the concepts of propagation of locality of operators. This is important to formulate vertex operator algebras and their modules. Chapter3. Vertex operator algebras and their representations. In this chapter, by using the concept of energy momentum tensor, mutually locality of two field operators and the propagation of the locality, I introduce the concept of vertex operator algebra (VOA). Then I introduce the universal enveloping algebra U(V) of a vertex operator algebra V. Then I define the zero mode algebra A0(V) as a subquotient of U(V). An finiteness condition, called Zhu's C2 cofiniteness condition, and Theorem 3-2 due to I. Frenkel and Zhu state that when Zhu's C2 condition is satisfied, the dimension of A0(V) is finite, and the set of irreducible V-modules and the set of irreducible A0-modules are in one to one correspondance. So the number of irreducible V-modules is finite. Chapter 4. Bosonic vertex operator algebras and lattice vertex operator algebras In this chapter I introduce the most important VOAs, the Bosonic VOA and the lattice VOA. Bosonic VOA has parameters k+, k- ∈ C* with k+k- = 1, and will be denoted by IIk+, k-. The central charge of energy momentum tensor of IIk+, k- is ck+, k- = 13 - 6(k+ + k-). So the Bosonic VOA IIk+, k- contains the Virasoro vertex operator algebra Virck+, k-. The abelian category of IIk+, k--mod consisting of IIk+, k--modules is semi-simple, and the simple objects of IIk+, k--modules are Fock modules Fβ ∋ uβ, β ∈ C. So we get Virck+, k--module Fβ. If k+ ∈ C＼Q, ⇔ k- ∈ C＼Q, then Fβ is not simple as Virck+, k--module if and only if β ∈ Z1/2α+ ⊕ Z1/2α-, where α+ = √k+/2, α- = √k-/2, α+・α- = -2. And for β ∈ Z1/2α+ ⊕ Z1/2α-, the structure of Virck+, k--module is very simple, which can be analyzed by Virasoro screening operators S+(z), S-(z). When k+ ∈ Q>0, ⇔ k- ∈ Q>0, the structure of Fβ, β ∈ Z1/2α+ + Z1/2α-, is very complicated. In this case the rank of the free abelian group Z1/2α+ + Z1/2α- is one. For k+ = p-/p+, k- = p+/p-; p+, p- ≥ 2 and p+ and p- are relatively prime integers, we can define lattice vertex operator algebra Vp+, p- which has sub-VOA IIk+, k- ⊂ Vp+, p-. When k+ = p-/p+, k- = p+/p-, it can be proved that the abelian category Vp+, p--mod consisting of Vp+, p--modules is semi-simple and has 2p+p- simple modules. Chapter 5. Free field realization of Virasoro vertex operator algebra. For k+, k- ∈ C*, k+k- = 1, the Virasoro vertex operator algebra Virck+, k- is obtained as sub-VOA of bosonic VOA IIk+, k- by using the screening operators S+(z), S-(z). When k+ = p-/p+, k- = p+/p-, p+, p- ≥ 2, p+, p- mutually prime, then the central charge is ck+, k- = 13 - 6(p+-p-)2/p+p-, which we write cp+;p- in this case. In this case the Virasoro vertex operator algebra Virck+, k- is not irreducible as VOA, and it has quotient VOA which is called minimal Virasoro algebra and we denote it MinVirck+, k-. The abelian category MinVirck+, k--mod consisting of MinVirck+, k--modules is semi-simple as an abelian category, and the number of simple objects is 1/2(p+ -1)(p- - 1). Using the minimal Virasoro VOA MinVirck+, k-, three Russian physicists Belavin, Polyakov and Zamolodchikov developed the conformal field theory, and get numbers of remarkable results about two dimensional critical phenomena [BPZ]. Since Vircp+, p- is a sub-VOA of k+, k-, for k+ = p-/p+, k- = p+/p-, and in this case the bosonic IIVOA k+, k- is a sub-VOA of the lattice VOA Vp+, p-. Using the screening operators S+(z), S-(z), we can define the sub-VOA Mp+, p- of Vp+, p- and Mp+, p- has Vircp+, p- as sub-VOA. The main purpose of our paper [TW] is analysis of the vertex operator algebra Mp+, p-, which we call the extended Virasoro algebra of type sl2 at positive rational levels. The results of our paper [TW] are the following. (1) The vertex operator algebra Mp+, p- is generated by T(z), W1, a(z), a = ±, 0 and the conformal dimension of W1, a(z) is Δ1 = (2p+ - 1)(2p- - 1). (2) The VOA Mp+, p- satisfies Zhu's C2 finiteness condition and we can determined the structure of finite dimensional algebra A0(Mp+, p-) explicitly. (3) We can determine all the simple modules of Mp+, p-, the number of them is 1 2 (p+ -1)(p- -1)+2p+p-, and we can determine the structure of these simple Mp+, p- -modules. (4) The VOA Mp+, p- has sub-VOA Vircp+, p- and has the quotient VOA Mp+, p-- module. (5) The simple MinVircp+, p-- modules can be considered as simple Mp+, p-- modules, and the number of simple modules is 1/2 (p+ - 1)(p- - 1). The method of the proof of these results are very complicated, and mainly use the lifting of base field C to the discrete valuation ring O = C[[ϵ]]. Chapter 6. On the lifting to discrete valuation ring. At first we lift to the valuation field K = C((ϵ)). And then by using the properties of Jack polynomial defined on K, we prove the C[[ϵ]] integrality of the lifting. Chapter 7. The representation theory of Vcp+, p- and of Wcp+, p-. We analyze the structure of Fβ, β ∈ Z1/2α+ + Z1/2α-, as Vcp+, p- - module carefully. The results are stated as Felder complex of Virasoro Fock modules Fβ, and we construct Vircp+, p- -modules X±[r, s] ⊂ K[r, s], 1 ≤ r ≤ p+, 1 ≤ s ≤ p-. Chapter 8. The extended W algebra Mp+, p- and their representations. First we construct operators E and F which we call Frobenius operators. The construction of E and F and analysis of their properties in [TW] are very complicated, and we need some simple method to analyze this part. In 2016 we succeeded this simplification. We are now prepareing a paper and it will appear in 2017. So in this resume we state the properties of E and F, and skip the proof of it. Using the properties of E and F, we prove that X±[r;s] ⊂ K±[r;s] are indeed Mp+, p- -modules and X±[r;s] are irreducible as Mp+, p- -modules. Then we prove the C2-cofiniteness of Mp+, p-, and we can determine the structure of the zero mode algebra A0(Mp+, p-) explicitly. Then using the structure of A0(Mp+, p-), we can prove that the set of irreducible modules is {L[r;s] = L[p+-r, p-s] 1 ≤ r ≤ p+ -1, 1 ≤ s ≤ p- -1} and {X±[r, s] 1 ≤ r ≤ p+, 1 ≤ s ≤ p-}, and L[r;s] = L[p+-r, p-s] are the lifting of the simple MinVircp+, p- -modules to simple Mp+, p- modules by quotient VOA map Mp+, p- → MinVircp+, p- → 0. In Chapter 8 we simply state the results and their proof are omited. These are the statements of our Japanese version of [TW]