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UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations Title Representations of Vertex Operator Algebras Permalink https://escholarship.org/uc/item/7gg7v1zd Author Yu, Nina Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ REPRESENTATIONS OF VERTEX OPERATOR ALGEBRAS Adissertationsubmittedinpartialsatisfactionofthe requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Nina Yu June 2013 The Dissertation of Nina Yu is approved: Professor Chongying Dong, Chair Professor Geoffrey Mason Professor Haisheng Li Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Nina Yu 2013 Table of Contents Abstract iv Acknowledgments v 1Introduction 1 2 Basics 5 3 Z-Graded Weak Modules and Regularity 12 3.1 Introduction.................................. 12 3.2 Preliminary .................................. 13 3.3 Universalenvelopingalgebra. 16 3.4 MainTheorem ................................ 18 V A4 4 Quantum Dimensions of Irreducible L2 -modules 22 4.1 Introduction.................................. 22 4.2 Preliminary.................................. 23 4.2.1 OnQuantumGaloisTheory . 23 V A4 4.2.2 The vertex operator algebra L2 .................. 24 4.2.3 Modular-invariance property and Quantum Dimensions . 27 4.3 MainResult.................................. 28 Bibliography 34 iii Abstract Representations of Vertex Operator Algebras by Nina Yu In this thesis we study the representation theory of vertex operator algebras. The thesis consists of two parts. The first part deals with the connection among rationality, regu- larity and C2-cofiniteness of vertex operator algebras. It is proved that if any Z-graded weak module for a vertex operator algebra V is completely reducible, then V is rational and C2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras. Motivated by classification of rational vertex operator algebras with central charge c =1. We compute the quantum dimensions of irreducible modules C V A4 of the rational and 2-cofinite vertex operator algebra L2 in the other part. This result will be used to determine the fusion rules for this algebra. iv Acknowledgments First I would like to thank my advisor Chongying Dong for sharing his knowledge and experience with me. Without his support and encouragement these years I would never have come this far. I have learned so much from him that I cannot list everything I need to thank him here. He has been a great advisor and I hope to work with him again in the future. Second, I would like to thank the mathematics department. The nice and friendly community provides me a great environment for work and fun. I would like to thank my mathematics fellows for keeping me company and supporting me all these years. I also would like to thank Professor Geoffrey Mason and Haisheng Li for being my PhD committee members and taking time in reading and providing me valuable advice on my thesis. Lastly I would like to thank my family for their constant support during these years and throughout my life. v Chapter 1 Introduction A vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics. Vertex operator algebras were first introduced by Richard Borcherds in 1986, motivated by the vertex operators arising from field insertions in two dimensional conformal field theory, a framework that is essential to define string theory. The axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists called chiral algebras. The theory of vertex operator algebras has developed rapidly in the last two decades due to its close relations with conformal field theory in physics and many branches in mathematics such as number theory, infinite dimensional group theory, and topological geometry. The main theme of this thesis is about the structure and representations of vertex operator algebras. In particular, the connection among rationality, regularity and C2-cofiniteness is investigated and quantum dimensions of irreducible modules of V A4 the vertex operator algebra L2 are computed. Rationality, regularity and C2-cofiniteness are probably the three most impor- tant notions in the representation theory of vertex operator algebras. Although there has been much research on rationality, regularity and C2-cofiniteness, the connection among them has not been understood fully. One of the most important conjectures (about 20 years old) in the theory of vertex operator algebras is that rationality implies C2-cofiniteness. We made progress in proving this conjecture. It is proved in [DY] that if any Z-graded weak module for a vertex operator algebra V is completely reducible, then the vertex operator algebra V is C2-cofinite. To give a positive answer to the conjecture, one only needs to establish the complete reducibility of any Z-graded weak module from 1 the complete reducibility of any Z+-graded weak module. Let V be a vertex operator algebra. A weak V -module is a pair (W, YW ) which satisfies the lower truncation condition, identity property and Jacobi identity. Notice that weak modules have no grading assumption. One important consequence of this definition is that weak modules admit a Virasoro representation. An admissible V - module M is a weak V -module which carries a Z+-graded weak V -module. An ordinary V -module M is a weak V -module M = λ CMλ which has a grading that matches the ⊕ 2 L(0)-action of the Virasoro representation as well as finite-dimensional graded pieces. The relations among these three types of modules are: ordinary modules admissible modules weak modules . { }✓{ }✓{ } Rationality, which is an analog of semisimplicity of a finite dimensional associative al- gebra or Lie algebra, asserts that the admissible module category, i.e., Z+-graded weak module category, is semisimple [DLM2, Z]. C2-cofiniteness tells us that a certain sub- space of a vertex operator algebra has finite codimension. Regularity, which is the strongest among the three concepts, means that any weak module is a direct sum of irreducible ordinary modules [DLM2]. Both rationality and C2-cofiniteness imply that there are finitely many irreducible admissible modules up to isomorphism and each ho- mogeneous subspace of any of these irreducile admissible modules is finite dimensional [DLM3, KL]. It follows from the definition that regularity implies rationality. It is shown in [L1] that regularity also implies C2-cofiniteness. The equivalence of regularity and rationality together with C2-cofiniteness is established in [ABD]. It is evident that the definition of rationality is natural, but the definition of regularity is not natural at all as the irreducible objects in the weak module category are assumed to be ordinary. It seems more natural to define regularity by semisimplicity of the weak module category. This is successfully achieved in the first part of this thesis. In fact, a stronger result is obtained. That is, if any Z-graded weak module is completely reducible, then the vertex operator algebra is regular. It is clear that if any Z-graded weak module is completely reducible, then the vertex operator algebra is rational. So it remains to prove that if any Z-graded weak module is completely reducible for a vertex operator algebra V , then V is C2-cofinite. It is well known that the graded dual V 0 of V is also a V -module [FHL]. By a result from [L1], if L (0) is semisimple on the unique maximal weak module inside the completion of V 0, then V is C2-cofinite. 2 The main idea is to use the universal enveloping algebra (V ) introduced in U [FZ] to prove that any weak module generated by a single vector is a quotient of a Z- graded weak module and that L(0) acts semisimply on any irreducible submodule of a Z-graded weak module. The second part is devoted to the study of quantum dimensions of irreducible V A4 c =1 modules of L2 . Motivated by studying rational vertex operator algebras with and better understanding of quantum dimensions, we compute quantum dimensions V A4 L A A of irreducible modules of L2 where 2 is the root lattice of type 1,and 4 is the alternating group which is a subgroup of the automorphism group of lattice vertex operator algebra VL2 . In the theory of vertex operator algebras there is a well known conjecture that + any rational vertex operator with central charge c =1is isomorphic to VL,orVL ,or G VZ↵ where L is a rank one positive definite even lattice, (↵,↵)=2and G = A4,S4,A5 is asubgroupofSO (3) in the E-series [K]. The vertex operator algebra VL for any positive definite even lattice L has been studied in [DM2]. The orbifold vertex operator algebras + VL of rank one lattices L also have been characterized in [DJ1, DJ2, DJ3, ZD]. But the G vertex operator algebra VZ↵ has not been understood fully as G is not a cyclic group. G = A V A4 C When 4, it proved in [DJ4] that L2 is rational and 2-cofinite. Irreducible V A4 V A4 modules of L2 are also classified there. But fusion rules for irreducible L2 -modules are still unknown. Let V = n ZVn be a vertex operator algebra and M = n 0Mλ+n a V - ⊕ 2 ⊕ ≥ module. The quantum dimension qdimV M of M over V is essentially the relative di- dim M dim M mension dim V .Onehastouselimitstoapproachdim V since both dim M and dim V are infinite. In [DJX], the properties of quantum dimension are discussed and the possi- ble values of the quantum dimension are obtained for rational vertex operator algebras. A criterion for simple currents of a rational vertex operator algebra is also given. These properties enable quantum dimensions of irreducible modules play an important role in determining fusion rules for these modules.
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