Applications Of Vertex Operators On Ghost Systems

Charged free bosonic system and be fermionic system(together,they are called ghost systems)have been used extensively in quantum algebra and theoretical physics.Ghost systems and their interrelation have recently attracted renewed attention because of their relevance in statistical physics.Besides,ghost systems are also important in conformal field theory due to their applications in Wakimoto free field realizations of affine algebras and related algebras.In this regard,the be fermionic system provides(?)-modules,level 1(?)-modules,and a free field realization of W1+∞ algebra in positive integral central charge.In the meanwhile,the charged free bosonic system provides(?)-modules,level-1(?)-modules,and a free field realization of W1+∞ algebra with negative integral central charge.The Kyoto school used be fermionic system and vertex operator realization(boson-fermion correspondence)to obtain the Hirota equation for the KP hierarchy.They have also showed that the KP tau functions can be expressed as certain symmetric functions.Charged free bosonic system and be fermionic system share many common features,thus we can study them in a similar way.In this thesis,we study the bosonic-KP hierarchy and bosonic-tau functions related to charged free bosonic system,the connection between bc fermionic system and the phase model,the properties of B(x),C(x),B*(x),C*(x)introduced in ghost systems,correlation functions of ghost systems,and some identities by the properties of ghost systems.The details are as follows:Chapter 3 investigates the bosonic-KP hierarchy and bosonic-tau functions.Based on the properties of charged free bosonic system,one has the algebraic bosonic-KP Hirota equation.Then we study algebraic bosonic-tau functions of the Hirota equation in Fock spaces M and M,respectively.The algebraic bosonic-tau functions in the Fock space M have and only have the vacuum vector |0>up to constant.Under boson-boson corre-spondence(or FMS bosonization),we have a correspondence of charged free bosonic fieldsφ(z),φ*(z)in terms of fields in the lattice vertex algebra VQ,i.e.,the vertex operator realization of φ(z),φ*(z).From the properties of lattices as well as the embedding of M to B,we get bosonic-KP hierarchy and the explicit expressions of some bosonic-tau functions.As an example,we obtain that a special equation of bosonic-KP hierarchy is a harmonic equation.Chapter 4 proposes QISM-like approach to ghost systems.First,we construct two operators B(x)and C(x)affected by positive integer M for be fermionic system,along with some properties of B(x)and C(x).In particular,(?)B(x)and(?)C(x)are half-vertex operators.Second,we study the actions of B(x)on F(0)and C(x)on F*(0),respectively,which build the connection between B(x)(resp.C(x))and creation operator B(x)(resp.annihilation operator C(x))related to the off-diagonal elements of T-matrix in the phase model,then get correlation functions of be fermionic system.Finally,we use QISM-like approach to construct operators B*(x)and C*(x)affected by M in charged free bosonic system.We also prove that B*(x1)B*(x2)…B*(xN)|0>is an algebraic bosonic-tau function,and the correlation function in the be fermionic system is the inverse of the corresponding correlation function in the charged free bosons.Chapter 5 obtains some identities in terms of the properties of ghost systems.We firstly provide a new proof of Borchardt identity with the help of boson-boson correspon-dence.Using combinatorial method and the results of[35,99],we then obtain families q-series identities.