Generalized moonshine, II: Borcherds products

In this thesis, we introduce an SL(2,R)-action on the chiral de Rham complex on the upper half plane, and study the vertex subalgebra Ω<sup>ch</sup>(H,Γ) of Γ-invariant global sections which are holomorphic at all the cusps, for any congruence subgroup Γ. The thesis can be divided into three parts. The first part includes a recollection on the theory of vertex algebras and modular forms. The second part consists of a brief review of the construction of chiral de Rham complex by Malikov, Schechtman and Vaintrob, which will be applied to the upper half plane. We consider the vertex algebra Ω<sup>ch</sup>(H,Γ) associated to an arbitrary congruence subgroup Γ, and compute its character formula. We also give an explicit formula for the lifting of modular forms to Ω<sup>ch</sup>(H, Γ), and the lifting formula is essentially unique and universal. The last part discusses the relations between the modified Rankin-Cohen bracket and the elements in Ω<sup>ch</sup>(H,Γ), and some further properties about Ω<sup>ch</sup>(H,Γ).

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $\sigma$ of order $T$ with $M$ as a $\sigma$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a $\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.

Automorphic functions on the upper half plane, especially modular functions

A Holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra F4,6A2,2

Vertex algebras generated by Lie algebras

We study surjective homomorphisms f : ∏ I A i → B of not-necessarily-associative algebras over a commutative ring k, for I a generally infinite set; especially when k is a field and B is countable-dimensional over k. Our results have the following consequences when k is an infinite field, the algebras are Lie algebras, and B is finite-dimensional: If all the Lie algebras A i are solvable, then so is B. If all the Lie algebras A i are nilpotent, then so is B. If k is not of characteristic 2 or 3, and all the Lie algebras A i are finite-dimensional and are direct products of simple algebras, then (i) so is B, (ii) f splits, and (iii) under a weak cardinality bound on I, f is continuous in the pro-discrete topology. A key fact used in getting (i)–(iii) is that over any such field, every finite-dimensional simple Lie algebra L can be written L = [ x 1 , L ] + [ x 2 , L ] for some x 1 , x 2 ∈ L , which we prove from a recent result of J.M. Bois. The general technique of the paper involves studying conditions under which a homomorphism on ∏ I A i must factor through the direct product of finitely many ultraproducts of the A i . Several examples are given, and open questions noted.

In this paper, using a homological theory of graded Lie algebras and the representation theory of A(') , we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra HA(') up to level 4 and deduce some interesting combinatorial identities. INTRODUCTION The hyperbolic Kac-Moody Lie algebras have been considered as the next natural objects of study after the affine case. While the affine Kac-Moody Lie algebras have been extensively studied for their close connections to areas such as combinatorics, modular forms, and mathematical physics, many basic questions regarding the hyperbolic case are still unresolved. For example, the behavior of the root multiplicities is not well understood. Feingold and Frenkel [F-F] and Kac, Moody, and Wakimoto [K-M-W] made some progress in this area. They computed the level 2 root multiplicities for the hyperbolic Kac-Moody Lie algebras HA(1) and HE(1), respectively. In [Kangl] and [Kang2], we introduced an inductive program to study the higher level root multiplicities and the principally specialized affine characters of a certain class of Lorentzian Kac-Moody Lie algebras. More precisely, we realize these Lorentzian Kac-Moody Lie algebras. as the minimal graded Lie algebras L = efEz Ln with local part V + Lo + V*, where Lo is an affine Kac-Moody Lie algebra, V is the basic representation of Lo, and V* is the contragredient of V. Thus L = G/I, where G = ~nEZ Gn is the maximal graded Lie algebra with local part V+Lo+ V* and I = eDnEZ I' is the maximal graded ideal of G which intersects the local part trivially. By developing a homological theory, we determined the structure of homogeneous subspaces Ln = G/II, as modules over the affine Kac-Moody Lie algebra Lo for certain higher levels. The Hochschild-Serre spectral sequences played an important role in determining the structure of I,n. For the hyperbolic Kac-Moody Lie algebra HA(1), we were able to determine the structure of L4 for n = 1, 2, ... , 5, and computed the principally specialized affine characters up to level 5. But the root multiplicities were computed up to level 3 only. In this paper, using the representation theory of A(1) developed Received by the editors June 4, 1990 and, in revised form, September 12, 1991. 1991 Mathematics Subject Classification. Primary 1 7B67, 1 7B65; Secondary 1 7B10. This work was supported by NSF grant DMS 8806371. () 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

Vertex Operator Algebras and the Monster

7. Modular functions. In this section necessary facts concerning the modular group and modular functions will be briefly summarized. More complete expositions of this material may be found in [3], [5], [6], [9], [15], [16], and [19]. Let r be the modular group, the group of all 2 X 2 matrices with rational integer entries and determinant equal to one. The elements S = (0 1) and T = (I 1) generate r. To an element A = (b b) E r, one associates a fractional linear transformation A(z) == (az + b)/(cz + d). The matrices A and -A give the same transformation. Let 3C* = 3C U {ioo} U Q where 3C is the upper half plane and Q is the set of rational points on the real axis. A fundamental domain is the set

In this paper, we establish 5-dissections of two infinite products and thereby prove that their q-series coefficients vanish for two arithmetic progressions. Moreover, our results also imply that the signs of coefficients of these infinite products are periodic with period 5 from $$n>1$$.

Decomposition of the lattice vertex operator algebra [formula omitted

The varieties of semi-conformal vectors of affine vertex operator algebras

The nonperturbative analysis of background duality in orbifold conformal field theory

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).

Fourier series of modular graph functions