Afine verteks algebre tipa A na nivoima bliskim dopustivima

doctoral thesis

• Title (english): Affine vertex algebras of type A beyond admissible levels
• Author: Ivana Vukorepa
• Mentor: Dražen Adamović (mentor)
• Mentor: Ozren Perše (komentor)
• Committee member: Gordan Radobolja (predsjednik povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Committee member: Tomislav Šikić (član povjerenstva)
• Committee member: Marijana Butorac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-07-19, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U ovoj disertaciji proučavamo neke afine verteks algebre na nivoima bliskim dopustivima. Poseban naglasak je na prostoj afinoj verteks algebri \(L_{-5/2} (sl(4))\). Određujemo eksplicitnu formulu za singularni vektor konformne težine četiri u univerzalnoj afinoj verteks algebri \(V^{-5/2} (sl(4))\) i dokazujemo da generira maksimalni ideal u \(V^{-5/2} (sl(4))\). Klasificiramo ireducibilne \(L_{-5/2} (sl(4))\)– module u kategoriji \(\mathscr{O}\) i određujemo pravila fuzije za ireducibilne module u kategoriji jakih modula \(KL_{-5/2}\). Dokazujemo i da je \(KL_{-5/2}\) poluprosta tenzorska kategorija. U dokazima koristimo eksplicitnu formulu za singularni vektor i Zhuovu teoriju. Također koristimo svojstva konformnog ulaganja \(gl(4) \hookrightarrow sl(5)\) na nivou \(k = -5/2\) [6] i teoriju afinih \(\mathcal{W}\)-algebri. Pritom dokazujemo da je \(k = -5/2\) kolapsirajući nivo obzirom na subregularni nil-potentni element \(f_{subreg}\) i dokazujemo određene rezultate o iščezavanju i neiščezavanju funktora hamiltonske kvantne redukcije \(H_{f_{subreg}}\). U drugom dijelu disertacije dajemo novi pristup klasifikaciji ireducibilnih modula u kategoriji \(\mathscr{O}\) za proste afine verteks algebre na nivoima bliskim dopustivima. Uvodimo pojam skoro dopustivih i glavnih skoro dopustivih težina i pokazujemo kako se pomoću njih mogu opisati ireducibilni moduli u kategoriji \(\mathscr{O}\) za verteks algebru \(L_{-5/2}(sl(4))\). Pokazujemo da se takav pristup može primijeniti i za verteks algebre \(L_{-1}(sl(n))\), za \(n \geq 3\), čija je klasifikacija ireducibilnih modula u kategoriji \(\mathscr{O}\) određena u [12]. Za navedene verteks algebre konstruiramo široku familiju modula za koje vrijedi Kac-Wakimotova formula karaktera. Na kraju proučavamo verteks algebru \(L_{-7/2}(sl(6))\) i navodimo slutnju u slučaju višeg ranga.

Abstract (english)

In this thesis we study certain affine vertex algebras beyond admissible levels. Our primary focus is the simple affine vertex algebra \(L_{-5/2} (sl(4))\). We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra \(V^{-5/2} (sl(4))\)and show that it generates the maximal ideal in \(V^{-5/2} (sl(4))\). We classify irreducible \(L_{-5/2} (sl(4))\)–modules in the category \(\mathscr{O}\), and determine the fusion rules between irreducible modules in the category of ordinary modules \(KL_{-5/2}\). We also prove that \(KL_{-5/2}\) is a semi-simple tensor category. In our proofs we use an explicit formula for singular vectors and Zhu’s theory. We also use properties of conformal embedding \(gl(4) \hookrightarrow sl(5)\) at level \(k = -5/2\) [6] and the theory of affine \(\mathcal{W}\) –algebras. We show that \(k = -5/2\) is a collapsing level with respect to the subregular nilpotent element \(f_{subreg}\) and prove certain results on vanishing and non-vanishing of the quantum Hamiltonian reduction functor \(H_{f_{subreg}}\). In the second part of this thesis we give a new approach to the classification of irreducible modules in the category \(\mathscr{O}\) for simple affine vertex algebras beyond admissible levels. We introduce the notion of almost admissible and principal almost admissible weights and show how they can be used to describe irreducible modules in the category \(\mathscr{O}\) for vertex algebra \(L_{-5/2}(sl(4))\). We also show that such an approach can be applied to vertex algebras \(L_{-1}(sl(n))\), for \(n \geq 3\), whose classification of irreducible modules in the category \(\mathscr{O}\) was determined in [12]. For the above algebras, we construct a large family of modules to which the Kac-Wakimoto character formula applies. In the end, we study the vertex algebra \(L_{-7/2}(sl(6))\) and give the conjecture for the higher-rank cases.

• Keywords: afine verteks algebre; algebre verteks operatora; pravila fuzije; konformna ulaganja; teorija reprezentacija
• Keywords (english): affine vertex algebras; vertex operator algebras; fusion rules; conformal embeddings; representation theory
• Language: croatian
• URN:NBN: urn:nbn:hr:217:434185
• Promotion: 2022
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: vii, 101 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-10-05 10:37:24