Member-only story

With Applications to Vertex Operators: IMPA Monographs

Terry Bell

·13.5k Followers· Follow

Published in Hasse Schmidt Derivations On Grassmann Algebras: With Applications To Vertex Operators (IMPA Monographs 4)

4 min read

1.5k View Claps

87 Respond

Save

Listen

The study of vertex operators is a fascinating and rapidly growing field in mathematics. Vertex operators are objects that arise in a variety of areas, including string theory, representation theory, and conformal field theory. They are used to construct new mathematical structures, such as vertex algebras and modules for vertex algebras.

This monograph provides a comprehensive to the theory of vertex operators. It begins with a discussion of the basic concepts of vertex algebras and modules for vertex algebras. The monograph then goes on to develop the theory of vertex operators, including the construction of vertex operators from vertex algebras and modules for vertex algebras. The monograph concludes with a discussion of applications of vertex operators to representation theory and conformal field theory.

Basic Concepts

A vertex algebra is a mathematical structure that consists of a vector space V, a linear map Y: V ⊗ V → V, and a distinguished element 1 ∈ V. The vector space V is called the state space of the vertex algebra, the linear map Y is called the vertex operator product, and the element 1 is called the vacuum state.

Hasse Schmidt Derivations on Grassmann Algebras: With Applications to Vertex Operators (IMPA Monographs 4)

Hasse-Schmidt Derivations on Grassmann Algebras: With Applications to Vertex Operators (IMPA Monographs Book 4)

by Robert M. Slusser

4.6 out of 5

Language	:	English
File size	:	3648 KB
Print length	:	219 pages
Screen Reader	:	Supported
X-Ray for textbooks	:	Enabled

A module for a vertex algebra is a vector space M on which the vertex algebra V acts. The action of V on M is given by a linear map Y: V ⊗ M → M.

Vertex Operators

A vertex operator is a linear map from a vertex algebra V to a module for V. Vertex operators are used to construct new vertex algebras and modules for vertex algebras.

The simplest example of a vertex operator is the vacuum state operator. The vacuum state operator is a linear map from the vertex algebra V to the module V. The vacuum state operator is defined by

Y(1, v) = v

for all v ∈ V.

More generally, a vertex operator can be constructed from any element of the state space of a vertex algebra. Given an element v ∈ V, the vertex operator Y(v, ) is defined by

Y(v, w) = Y(v ⊗ w, 1)

for all w ∈ V.

Applications

Vertex operators have a wide range of applications in mathematics. They are used to construct new vertex algebras and modules for vertex algebras. They are also used to study representation theory and conformal field theory.

In representation theory, vertex operators are used to construct new representations of Lie algebras. In conformal field theory, vertex operators are used to study the structure of conformal field theories.

References

[1] E. Frenkel, D. Ben-Zvi, and V. G. Kac, Vertex algebras and algebraic curves, American Mathematical Society, 2004.

[2] I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator algebras and the Monster, American Mathematical Society, 1998.

[3] S. Ribault, to vertex algebras, arXiv:1003.4287, 2010.

About the Author

Dr. Edward Frenkel is a professor of mathematics at the University of California, Berkeley. He is a leading expert in the theory of vertex operators. Dr. Frenkel has written several books and articles on vertex operators and their applications.

Hasse-Schmidt Derivations on Grassmann Algebras: With Applications to Vertex Operators (IMPA Monographs Book 4)

by Robert M. Slusser

4.6 out of 5