Twisted modules for vertex operator algebras

In this contribution, I explain the general principles of twisted modules for vertex operator algebras in its powerful formulation using formal series, and derive new general relations satisfied by twisted and non-twisted vertex operators. I prove new “equivalence” and “construction” theorems, identifying a very restricted set of sufficient conditions in order to have a twisted module for a vertex operator algebra, and a simple way of constructing the twisted vertex operator map. I show how to apply these theorems in order to construct twisted modules for the Heisenber vertex operator algebra. I use the new nontwisted relations in the Heisenberg vertex operator algebra in order to understand properties of a certain central extension of a Lie algebra of differential operators on the circle: the connection between the structure of the central term in Lie brackets and the Riemann Zeta function at negative integers. I then use the twisted relations in order to construct in a simple way a family of representations for this algebra based on twisted modules for the Heisenberg vertex operator algebra. As a simple consequence of the twisted relations, the construction involves the Bernoulli polynomials at rational values in a fundamental way. This contribution is based on, and further extends, works by Lepowsky, by Milas and by the author, Lepowsky and Milas.