FORMAL CALCULUS, UMBRAL CALCULUS, AND BASIC AXIOMATICS OF VERTEX ALGEBRAS BY THOMAS J. ROBINSON A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of James Lepowsky and approved by New Brunswick, New Jersey October, 2009 ABSTRACT OF THE DISSERTATION Formal calculus, umbral calculus, and basic axiomatics of vertex algebras by Thomas J. Robinson Dissertation Director: James Lepowsky The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, “weak skew-associativity,” in order to exploit the S3-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result ii in the representation theory of vertex algebras. iii Acknowledgements I wish to thank James Lepowsky most particularly for his long standing support, pa- tience and guidance. I also wish to thank both Yi-Zhi Huang and Haisheng Li, who, among other things, each taught me large portions of the material relevant to this work. In addition, my thanks go to Stephen Greenfield, Amy Cohen, Ovidiu Costin, Richard Lyons and Richard Wheeden for their help when I first arrived at Rutgers. I would also like to thank Antun Milas, Robert Wilson, Doron Zeilberger and Louis Shapiro. Of course, I have also benefited greatly from conversations with other members of the faculty at Rutgers, but they are too numerous to name. I further wish to thank Mark Krusemeyer and Gail Nelson, my undergraduate professors, and also Mr. Suggs and Dr. Watson, my high school math teachers. And most certainly all the staff in the Rutgers Math Department who through the years have saved me from myself. Finally, I am grateful for partial support from NSF grant PHY0901237. iv Dedication To my Mom and Dad. v Table of Contents Abstract ........................................ ii Acknowledgements ................................. iv Dedication ....................................... v 1. Introduction ................................... 1 1.1. IntroductiontoChapter2 . 6 1.2. IntroductiontoChapter3 . 9 1.3. IntroductiontoChapter4 . 16 2. Exponentiated derivations, the formal Taylor theorem, and Fa`adi Bruno’s formula ................................... 22 2.1. The formal Taylor theorem: a traditional approach . ........ 22 2.2. The formal Taylor theorem from a different point of view . ....... 26 2.3. The formal Taylor theorem for iterated logarithms and exponentials . 29 2.4. Formal analytic expansions: warmup . 30 2.4.1. Case N = 0, the case “x” ...................... 30 Method1 ............................... 30 Method2 ............................... 31 Method3 ............................... 31 Method4 ............................... 31 2.4.2. Case N = −1, the case “exp x”................... 32 Method1 ............................... 32 Method2 ............................... 32 Method3 ............................... 33 Method4 ............................... 33 vi 2.4.3. Case N = 1, the case “log x” .................... 34 Method1 ............................... 34 Method2 ............................... 35 Method3 ............................... 35 Method4 ............................... 36 2.4.4. Case N = −2, the case “exp exp x”................. 37 Method1 ............................... 37 Method2 ............................... 38 Method3 ............................... 38 Method4 ............................... 39 2.5. Arecursion .................................. 39 2.6. Formal analytic expansions: warmup continued and completed . 44 2.7. Formal analytic expansions: Cases N ≥ 0, “iterated logarithms” . 45 2.7.1. Method1 ............................... 45 2.7.2. Method2 ............................... 46 2.7.3. Method3 ............................... 47 2.7.4. Method4 ............................... 50 2.8. Somecombinatorics ............................. 52 2.9. Substitution maps for iterated logs and exponentials . .......... 53 2.10. A glimpse of Fa`adi Bruno and umbral calculus . ...... 55 3. Formal calculus and umbral calculus .................... 58 3.1. Preliminaries ................................. 58 3.2. A restatement of the problem and further developments . ........ 62 3.3. Umbralconnection .............................. 68 3.4. Some historical, contextual results . ...... 73 3.5. Umbralshiftsrevisited . 80 3.6. Sheffersequences ............................... 81 3.7. TheexponentialRiordangroup . 86 vii 3.8. Matrix multiplication via generating functions . .......... 89 3.9. Someunderlyingcombinatorics . 93 3.10.TheVirasoroalgebra. 100 3.11. Umbral shifts revisited and generalized . ........ 103 3.12.Fa`adiBruno’sformula. 110 3.13. A standard quadratic representation of the Virasoro algebra of central charge1.................................... 112 4. Replacement axioms for the Jacobi identity for vertex algebras and their modules ..................................... 119 4.1. Formalcalculussummarized. 119 4.2. Formal calculus further developed . 122 4.3. Vacuum-free vertex algebras . 127 4.4. Modules.................................... 133 4.5. Vertex algebras with vacuum . 136 4.6. Modules for a vertex algebras with vacuum . 148 References ....................................... 151 Vita ........................................... 155 viii 1 Chapter 1 Introduction The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, “weak skew-associativity,” in order to exploit the S3-symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result in the representation theory of vertex algebras. In the next two paragraphs we very briefly review some highlights in the development of vertex operator algebras and umbral calculus; then we describe the evolution of the present work. Following that, we give a basic chapter-by-chapter outline, itself followed by a more extensive section-by-section outline of the main body of this work. 2 In mathematics, vertex operators first arose in the work of J. Lepowsky and R. Wil- son, who were seeking concrete representations of affine Lie algebras [LW1], and using this work they were able to give a new, natural interpretation of the Rogers-Ramanujan partition identities [LW2]. Vertex operators were used by I. Frenkel, J. Lepowsky and A. Meurman [FLM1] to construct a natural “moonshine module,” whose “character” is the modular function J(q), for the largest sporadic finite simple group, the Fischer- Griess Monster group [G]. This proved a conjecture of J. McKay and J. Thompson and part of the Conway-Norton conjectures [CN]. R. Borcherds [Bor1] introduced the first mathematical definition of vertex algebra which, in particular, extended the relations for the vertex operators used to construct the moonshine module. Using this and other ideas, Borcherds [Bor2] proved the remaining Conway-Norton conjectures [CN] for the moonshine module. A variant of the notion of vertex algebra, that of vertex operator algebra, was introduced in [FLM2] using a “Jacobi identity,” which was implicit