Representations of certain irrational W-algebras

Abstract

U disertaciji proučavamo neke iracionalne $\mathcal{W}$-algebre i njihove reprezentacije. Glavni primjer koji istražujemo je prosta Bershadsky-Polyakov verteks algebra $\mathcal{W}_k(=\mathcal{W}_k(sl_3, f_\theta))$, tj. minimalna afina $\mathcal{W}$-algebra pridružena $sl_3$. T. Arakawa je dokazao u članku [9] da je $\mathcal{W}_k$ racionalna ako je $k$ polucijeli broj veći od $-3/2$. Mi proučavamo slučajeve kad je $\mathcal{W}_k$ iracionalna, te klasificiramo $\mathcal{W}_k$-module za neke $k$. Klasifikacija ireducibilnih modula koristi Zhuovu teoriju i formule za singularne vektore. Istaknimo da je Zhuova algebra realizirana kao kvocijent Smithove algebre iz [44]. Klasifikacija ireducibilnih jakih modula za algebru $\mathcal{W}_k$ (tj. modula s konačno-dimenzionalnim težinskim potprostorima za operator $L(0)$) je zato povezana s klasifikacijom konačno-dimenzionalnih reprezentacija Smithove algebre. U slučaju $k=-5/3$ pokazujemo da je $\mathcal{W}_k$ realizirana kao verteks podalgebra Weylove verteks algebre. Dokazujemo da tada $\mathcal{W}_k$ ima točno 6 ireducibilnih jakih modula. U slučaju $k=-9/4$, $\mathcal{W}_k$ je važan primjer logaritamske verteks algebre. Klasificiramo jake $\mathcal{W}_k$ module za $k=-9/4$ i dokazujemo da $\mathcal{W}_k$ ima točno 3 ireducibilna jaka modula. Da bi pokazali da je $\mathcal{W}_k$ iracionalna, konstruiramo neprebrojivu familiju težinskih modula za $\mathcal{W}_k$ izvan kategorije $\mathcal{O}$. Konstruiramo familiju singularnih vektora koja generalizira Arakawine formule za singularne vektore za $k$ polucijeli broj iz [9]. U slučajevima $k=-1$ i $k=0$ klasificiramo sve module u kategoriji $\mathcal{O}$. U slučaju $k=0$, dajemo eksplicitnu realizaciju verteks algebre $\mathcal{W}_k$ i njenih modula kao određene iracionalne podalgebre verteks algebri pridruženih rešetkama. Također proučavamo algebru $\mathcal{W}_k$ u slučaju kad je $k$ cijeli broj veći od -1.In this thesis we study certain irrational $\mathcal{W}$-algebras and their representations. Main example that we investigate is the simple Bershadsky-Polyakov vertex algebra $\mathcal{W}_k(=\mathcal{W}_k(sl_3, f_\theta))$, i.e. the minimal affine $\mathcal{W}$-algebra associated to $sl_3$. T. Arakawa proved in [9] that $\mathcal{W}_k$ is rational if $k$ is a half integer greater that $-3/2$. We study cases where $\mathcal{W}_k$ is irrational, and classify $\mathcal{W}_k$-modules for certain $k$. Main tools that we use to classify irreducible modules are Zhu's theory and formulas for singular vectors. Zhu algebra is realized as a quotient of the Smith algebra from [44]. Classification of irreducible strong modules for $\mathcal{W}_k$ (i.e. modules with finite dimensional weight subspaces for the the operator $L(0)$) is hence connected to the classification of finite dimensional representations of the Smith algebra. In the case $k=-5/3$, we show that $\mathcal{W}_k$ is realized as a vertex subalgebra of the Weyl vertex algebra. We prove that in this case $\mathcal{W}_k$ has exactly 6 irreducible strong modules. In the case $k=-9/4$ , $\mathcal{W}_k$ is an important example of a logarithmic vertex algebra. We classify strong $\mathcal{W}_k$-modules for $k=-9/4$ and prove that $\mathcal{W}_k$ has exactly 3 irreducible strong modules. In order to show that $\mathcal{W}_k$ is irrational, we construct an uncountable family of weight modules for $\mathcal{W}_k$ outside of category $\mathcal{O}$. We construct a family of singular vectors which generalizes Arakawa’s formulas for singular vectors for $k$ half integer from [9]. For $k=-1$ and $k=0$ we classify all modules in the category $\mathcal{O}$. In the case $k=0$, we give an explicit realization of the vertex algebra $\mathcal{W}_k$ and its modules as certain irrational subalgebras of lattice VOAs. We also study the algebra $\mathcal{W}_k$ when $k$ is an integer greater than -1