Whittaker modules and fusion rules for the Weyl vertex algebra, affine vertex algebras and their orbifolds

doctoral thesis

• Title (croatian): Whittakerovi moduli i pravila fuzije za Weylovu verteks-algebru, afine verteks-algebre i njihove invarijantne podalgebre
• Author: Veronika Pedić Tomić
• Mentor: Dražen Adamović (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Tomislav Šikić (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

The topic of this thesis are two problems in the vertex operator algebra theory: determination of fusion rules and the orbifold problem. For the fusion rules problem we study the example of Weyl vertex algebra, also known as the \(\beta\gamma\) ghost system. This is a nonrational vertex algebra, hence we give a first proof of a Verlinde formula for non-rational VOAs and confirm the Verlinde type conjecture given by D. Ridout and S. Wood in [70]. For the orbifold problem, we extend a theorem given in the Dong-Mason quantum Galois theory paper [41], from the category of ordinary modules to the whole category of weak modules. The proof given by Dong and Mason necessarily involves Zhu’s theory, and therefore cannot be extended to the category of weak modules. In particular, we study the example of Whittaker modules for the Weyl vertex algebra and Heisenberg VOA. In the first part of this thesis, we calculate fusion rules in the category of weight modules for the Weyl vertex algebra. Our proof is entirely vertex-algebraic and it uses the theory of intertwining operators for vertex algebras and the fusion rules for the affine vertex superalgebra \(L_1(\mathfrak(gl)(1|1))\). Moreover, we explicitly construct the intertwining operators involved. We also prove a general irreducibility result which relates irreducible weight modules for the Weyl vertex algebra \(M\) to irreducible weight modules for \(L_1(\mathfrak(gl)(1|1))\). In the second part of this thesis we prove a theorem on irreducible weak \(V\)-module \(W\) and an automorphism \(g\) of finite order. Here either \(W \circ g^i \ncong W\) for all \(i\), in which case \(W\) is an irreducible \(V^{\langle g \rangle}\)–module, or \(W \cong = W \circ g\) in which case \(W\) is a direct sum of \(p\) irreducible \(V^{\langle g \rangle}\)–modules. The key idea of our proof was to consider a “big” module for the vertex algebra, constructed as a direct sum of modules \(\bigoplus_i W \circ g^i\). Moreover, we present a counterexample for the expansion of our theorem to the case of infinitedimensional group \(G\) for the irreducible Weyl algebra modules of Whittaker type.

Abstract (croatian)

U ovoj disertaciji proučavamo dvije teme teorije verteks-algebri: računanje pravila fuzije te problem podalgebre fiksnih točaka. Verteks-algebra za koju računamo pravila fuzije je Weylova verteks-algebra ili \(\beta\gamma\) sistem. To je iracionalna verteks-algebra i naš dokaz je prvi dokaz Verlindeove formule za slučaj iracionalnih verteks-algebri te potvrđujemo slutnju iznesenu u članku D. Ridouta i S. Wooda [70]. Za problem podalgebre fiksnih točaka proširujemo teorem C. Donga i G. Masona iz njihova članka o kvantnoj Galoisovoj teoriji [41], s kategorije jakih modula na cijelu kategoriju slabih modula. Dokaz iznesen u [41] ne može se proširiti na slabe module jer koristi Zhuovu teoriju. Svoj rezultat primjenjujemo na Weylovu verteks-algebru, ali i Heisenbergovu algebru verteks-operatora te za obje promatramo kategoriju Whittakerovih modula. U prvom dijelu disertacije računamo pravila fuzije u kategoriji težinskih modula Weylove verteks-algebre. Naš je dokaz potpuno uklopljen u teoriju verteks-algebri te koristi teoriju operatora ispreplitanja verteks-algebri i pravila fuzije za afinu verteks-superalgebru \(L_1(\mathfrak(gl)(1|1))\). Štoviše, eksplicitno smo konstruirali operatore ispreplitanja koji se javljaju u iskazu. Također, pokazali smo općeniti rezultat koji povezuje ireducibilne težinske module Weylove verteks-algebre \(M\) s ireducibilnim težinskim modulima za \(L_1(\mathfrak(gl)(1|1))\). U drugom dijelu disertacije dokazali smo teorem o ireducibilnim slabim \(V\)–modulima \(W\) i automorfizmu \(g\) konačnog reda. Naime, pokazali smo da je ili \(W \circ g^i \ncong W\), za sve \(i\), i u tom slučaju je \(W\) ireducibilan \(V^{\langle g \rangle}\)–modul, ili je \(W \cong = W \circ g\), i u tom slučaju je \(W\) direktna suma \(p\) ireducibilnih \(V^{\langle g \rangle}\)–modula. Glavna ideja našeg dokaza je bila konstruirati “veliki” ˇ modul za verteks-algebru, tako da uzmemo direktnu sumu modula \(\bigoplus_i W \circ g^i\). Nadalje, dajemo protuprimjer za proširenje našeg teorema na slučaj beskonačno-dimenzionalne grupe automorfizama \(G\) za ireducibilne Whittakerove module Weylove verteks-algebre.

• Keywords: vertex algebra; vertex operator algebra; ordinary modules; weak modules; Whittaker modules; Weyl vertex algebra; Heisenberg vertex operator algebra; quantum Galois theory; orbifold problem; fusion rules; intertwining operators; Verlinde formula; fusion algebra; simple current; vertex algebra automorphism; lattice vertex superalgebra; Verma module; Clifford vertex algebra
• Keywords (croatian): verteks-algebra; algebra verteks-operatora; jaki moduli; slabi moduli; Whittakerovi moduli; Weylova verteks-algebra; Heisenbergova algebra verteks-operatora; kvantna Galoisova teorija; problem podalgebre fiksnih točaka; pravila fuzije; operatori ˇ ispreplitanja; Verlindeova formula; fuzijska algebra; prosta struja; automorfizam verteksalgebri; verteks-superalgebra pridruzena rešetci; Vermaov modul; Cliffordova verteks-algebra
• Language: english
• URN:NBN: urn:nbn:hr:217:249471
• 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: viii, 91 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-01-24 12:12:23