In this survey article we discuss the origin, theory and applications of left-symmetric algebras
(LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have …

4. Quantum Toda field theories 4.1. Vertex operator algebras 4.2. The VOA of the
Heisenberg algebra 4.3. Quantum integrals of motion 4.4. Liouville theory 4.5. Quantum …

A bstract We classify all non-invertible Kramers-Wannier duality defects in the E 8 lattice
Vertex Operator Algebra (ie the chiral (E 8) 1 WZW model) coming from ℤ m symmetries. We …

This is an improved and expanded edition of Kac's original introduction to algebraic aspects
of conformal field theory which was published by the AMS in 1996. This revised edition is …

In contrast with the finite dimensional case, one of the distinguished features in the theory of
infinite dimensional Lie algebras is the modular invariance of the characters of certain …

Vertex operator algebra theory is a new area of mathematics. It has been an exciting and
ever-growing subject from the beginning, starting even before R. Borcherds introduced the …

Vertex algebras are algebraic objects that encapsulate the concept of operator product
expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming …

Ideas from quantum field theory and string theory have had considerable impact on
mathematics over the past 20 years. Advances in many different areas have been inspired …

Now we return to the setting of Chapter 5. The axioms for a generalized vertex operator
algebra do not quite apply to the structure of Ω*, the vacuum space (3.29)(for the Heisenberg …

We formulate rational conformal field theory in terms of a symmetric special Frobenius
algebra A and its representations. A is an algebra in the modular tensor category of Moore …